The coincidence of the homologies of integral currents and of integral singular chains, via cosheaves

Abstract

We consider the notion of metric spaces being locally Lipschitz contractible introduced by Yamaguchi, and a category of metric spaces satisfying this condition. Many objects in metric geometry including CAT-spaces and Alexandrov spaces, belong to this category. We consider the homology of integral currents with compact support in a metric space, introduced by Ambrosio and Kirchheim, and prove that it and the usual integral singular homology are isomorphic on the category. The proof of it is based on the theory of cosheaves. A method to compare the homologies associated to cosheaves is also proved in this paper.

1 Introduction

There are homology theories of metric spaces depending on metric structures. For instance, we denote by $S_k^\mathrm {Lip}(X)$ the free abelian group based on the set of all Lipschitz maps from a k-simplex $\triangle ^k$ with a standard Euclidean metric to a metric space X. Then, it is a subgroup of the group of usual integral singular k-chains $S_k(X)$, and further, $S_\bullet ^\mathrm {Lip}(X) = \bigoplus _{k=0}^\infty S_k^\mathrm {Lip}(X)$ becomes a subcomplex of $S_\bullet (X) = \bigoplus _{k=0}^\infty S_k(X)$. By the definition, the homology $H_*^\mathrm {Lip}(X)$ of $S_\bullet ^\mathrm {Lip}(X)$ depends on the metric structure of X. This group $H_*^\mathrm {Lip}(X)$ is called the singular Lipschitz homology of X. Here and hereafter, the symbol $\bullet$ means degrees of a chain complex and $*$ denotes a fixed degree (of a homology). Yamaguchi introduced the notion of metric spaces being locally Lipschitz contractible (abbreviated to LLC) and proved

Theorem 1.1

[19] For every LLC metric space X,  the inclusion $S_\bullet ^\mathrm {Lip}(X) \hookrightarrow S_\bullet (X)$ induces an isomorphism $H_*^\mathrm {Lip}(X) \rightarrow H_*^\mathrm {sing}(X).$

Here, $H_*^\mathrm {sing}$ denotes the usual integral singular homology, that is the homology of $S_\bullet$. We can regard $S_\bullet ^\mathrm {Lip}$ as a functor from the category of metric spaces and locally Lipschitz maps to the category of chain complices and chain maps, and the correspondence $S_\bullet ^\mathrm {Lip}\rightarrow S_\bullet$ as a natural transformation between the functors.

Ambrosio and Kirchheim introduced currents in general metric spaces [1]. Currents are generalizations of operations of integrating smooth forms on submanifolds. We will recall the precise definition and fundamental properties of metric currents in Sect. 3. A restricted class ${\mathbf {I}}_\bullet ^\mathrm {c}(X)$ consisting of all integral currents with compact support in X becomes a chain complex, due to [1]. Its homology is denoted by $H_*^\mathrm {IC}(X)$ in this paper. On the other hands, Riedweg and Schäppi introduced the notion of metric spaces admitting locally strong Lipschitz contractions (see Sect. 2). They defined a natural transformation $[\,\cdot \,]$ from $S_\bullet ^\mathrm {Lip}$ to ${\mathbf {I}}_\bullet ^\mathrm {c}$ and claimed

Theorem 1.2

[14] On the category of metric spaces admitting locally strong Lipschitz contractions and locally Lipschitz maps,  the natural transformations $S_\bullet \hookleftarrow S_\bullet ^\mathrm {Lip}\xrightarrow {[\,\cdot \,]} {\mathbf {I}}_\bullet ^\mathrm {c}$ induce isomorphisms between their homologies.

The assumptions in Theorems 1.1 and 1.2 are related. Indeed, we will prove that a metric space admitting locally strong Lipschitz contractions is LLC (Lemma 2.7). A main result of the present paper is

Theorem 1.3

On the category of LLC metric spaces and locally Lipschitz maps,  the natural transformations $H_*^\mathrm {sing}\leftarrow H_*^\mathrm {Lip}\xrightarrow {[\,\cdot \,]_*} H_*^\mathrm {IC}$ are isomorphisms.

As a direct corollary to Theorem 1.3, we have

Corollary 1.4

The functors $H_*^\mathrm {Lip}$ and $H_*^\mathrm {IC}$ can be extended to functors on the category of all locally Lipschitz contractible metric spaces and all continuous maps,  such that they are naturally isomorphic to $H_*^\mathrm {sing}$. In particular,  $H_*^\mathrm {Lip}$ and $H_*^\mathrm {IC}$ are homotopy invariants for LLC metric spaces.

The proofs of Theorems 1.1 and 1.2 were done directly. We will give a versatile proof of Theorem 1.3 using the theory of cosheaves. Here, cosheaves are categorically dual notion of sheaves, which were introduced by Bredon [3]. Indeed, in the course of the proof of Theorem 1.3, we give a technique to compare homologies associated to cosheaves (Theorem 4.9). Notice that Mongodi [9] considered that a chain complex consisting of metric currents in a metric space X, and proved that its homology coincides with the usual singular homology if X is locally Lipschitz contractible CW-complex. His proof was done by verifying that the homology of currents is actually a homology theory, that is, it satisfies the axioms of Eilenberg and Steenrod. Furthermore, he used the uniqueness of homology theory to prove the result. We should remark that there exists an LLC metric space which does not have the homotopy type of CW-complices (Sect. 5). So, our result can not be proved via the uniqueness of homology.

A way to compare the homologies using cosheaves was also discussed by De Pauw [6]. There, he consider the chain complices of usual currents in a subset of Euclidean spaces. Our proof of Theorem 1.3 is similar to an argument in that paper. However, our formulation as in Theorem 4.9 did not appear. We consider that such a formulation is important and is very useful.

1.1 Organization

The present paper mainly consists of three parts dealing with: Lipschitz contractions, metric currents, and cosheaves. The first two parts are subjects in geometry (and analysis) and the third part is purely an algebraic-topological subject. Our main theorem in the paper is Theorem 1.3. However, we consider that the proof of it is very important. It is based on Theorem 4.9. If the reader purely is interested in algebraic topology, the author recommend to firstly read Sect. 4.

In Sect. 2, we review and define the notion of local Lipschitz contractibility and its variants. We prove that the local Lipschitz contractibility is weaker than other conditions. In Sect. 3, we review the notion of metric currents and its fundamental theory introduced and investigated by Ambrosio and Kirchheim [1]. Furthermore, we give proofs of several remarkable properties which are needed to prove our main results. In Sect. 4, we recall the notion of cosheaves and its fundamental properties. We prove an important Theorem 4.9 which is a general method to compare homologies associated to cosheaves. We generalize the local Lipschitz contractibility in terms of local triviality of homology theories (Lemma 4.12). At the end of the section, we prove Theorem 1.3 using Theorem 4.9 and Lemma 4.12. Finally, in Sect. 5, we give several remarks about our results. In particular, we provide an example of an locally Lipschitz contractible metric space which does not admit the homotopy types of CW-complices.

2 Variants of local Lipschitz contractibility

Let us fix terminologies. In this section, X and Y always denote metric spaces. For $L \ge 0$, a map $f:X \rightarrow Y$ is said to be L-Lipschitz if it satisfies

$$\begin{aligned} d(f(x),f(y)) \le L d(x,y) \end{aligned}$$

for all $x,y \in X$. We say that f is Lipschitz if f is L-Lipschitz for some $L \ge 0$. The Lipschitz constant of f is the minimum of all L such that f is L-Lipschitz, and is denoted by $\mathrm {Lip}(f)$.

A map $f:X \rightarrow Y$ is said to be locally Lipschitz if for any $x \in X$, there exists an open set U in X containing x such that the restriction $f |_U$ is Lipschitz.

A map $f:X \rightarrow Y$ is called a bi-Lipschitz embedding if it satisfies

$$\begin{aligned} L^{-1} d(x,x') \le d(f(x), f(x')) \le L d(x,x') \end{aligned}$$

for all $x,x' \in X$ where $L \ge 1$ is some number. If a bi-Lipschitz embedding is bijective, then it is called a bi-Lipschitz homeomorphism. A locally bi-Lipschitz homeomorphism is a homeomorphism such that it and its inverse are locally Lipschitz.

A homotopy $h:X \times [0,1] \rightarrow Y$ is called a Lipschitz homotopy if it is a Lipschitz map, i.e., there exists a constant $C \ge 0$ such that

$$\begin{aligned} d(h(x,t), h(x',t')) \le C (d(x,x') + |t-t'|) \end{aligned}$$

for every $x,x' \in X$ and $t,t' \in [0,1]$. For a homotopy h, we write $h_t = h(\cdot ,t)$ for each $t \in [0,1]$. We will also consider locally Lipschitz homotopies, which are homotopies being locally Lipschitz.

Let U and V be subsets of X with $U \subset V$. A Lipschitz homotopy $h:U \times [0,1] \rightarrow V$ is called a Lipschitz contraction if there exists a point $x_0 \in V$ such that $h_0$ is the inclusion $U \rightarrow V$ and $h_1 \equiv x_0$ is a constant map. In this case, we say that U is Lipschitz contractible to$x_0$inV and that h is a Lipschitz contraction from U to $x_0$ in V.

Definition 2.1

([19], cf. [8]) Let X be a metric space. We say that X is locally Lipschitz contractible, for short LLC, if for any $x \in X$ and any $r > 0$, there exists $r' \in (0,r]$ such that $U(x,r')$ is Lipschitz contractible to x in U(x, r). This property is also called the LLC-condition.

Here, U(z, s) always denotes the open metric ball centered at z with radius s in a metric space. Notice that we use a version of the definition of LLC-condition reformulated in [8]. Obviously, if a metric space X is covered by open sets $X_i$ such that each open set $X_i$ is LLC, then X is LLC.

We introduce a notion which seems to be weaker than the LLC-condition.

Definition 2.2

We say that a metric space X is weakly locally Lipschitz contractible, for short WLLC, if for any $x \in X$ and any open set $U \subset X$ with $x \in U$, there exists an open set $V \subset X$ with $x \in V \subset U$ such that V is Lipschitz contractible in U to some point of U.

Obviously, every LLC space is WLLC. Furthermore, we have

Lemma 2.3

Let U be a subset of a metric space V. Suppose that U is Lipschitz contractible in V. Then,  for any $x \in U,$U is Lipschitz contractible to x in V.

In particular,  every WLLC metric space is LLC.

Proof

By the assumption, there is a Lipschitz homotopy $h:U \times [0,1] \rightarrow V$ such that $h_0 = \mathrm {id}_U$ and $h_1$ is a constant map. The image of $h_1$ is denoted by $y \in V$. Furthermore, let us take an arbitrary point $x \in U$. Then, we consider a map $k:U \times [0,1] \rightarrow V$ defined by

$$\begin{aligned} k(z,t) = \left\{ \begin{array}{ll} h(z, 2 t) &{}\quad \text {if } t \le 1/2, \\ h(x, 2- 2 t) &{}\quad \text {if } t \ge 1/2 \end{array} \right. \end{aligned}$$

for $z \in U$ and $t \in [0,1]$. This is well-defined. Indeed, $h(z,1) =y= h(x,1)$. Furthermore, k is Lipschitz. Actually, for any $z \in U$ and $s,t \in [0,1]$ with $s \le 1/2 \le t$, we have

$$\begin{aligned} d(k(z,s), k(z,t))= & {} d(h(z,2 s), h(x,2-2t)) \\\le & {} d(h(z,2s),h(z,1)) + d(h(x,1), h(x,2-2t)) \\\le & {} \mathrm {Lip}(h) (1-2s) + \mathrm {Lip}(h) (2 t -1) \\= & {} 2 \mathrm {Lip}(h) (t-s). \end{aligned}$$

For $s, t \in [0,1]$ with $s, t \le 1/2$ or with $s, t \ge 1/2$, we also have

$$\begin{aligned} d(k(z,s), k(z,t)) \le 2 \mathrm {Lip}(h) |s-t|. \end{aligned}$$

For any $z,w \in U$ and $s \in [0,1]$, we obtain

$$\begin{aligned} d(k(z,s),k(w,s)) \le \mathrm {Lip}(h) d(z,w). \end{aligned}$$

By the definition, $k_1$ is a constant map of the value x. Therefore, U is Lipschitz contractible to x in V.

Let us take a WLLC metric space X. Then, for any $x \in X$ and $r > 0$, there exist an $r'>0$ such that $U(x,r')$ is Lipschitz contractible in U(x, r). By the former statement, $U(x,r')$ is also Lipschitz contractible to x in U(x, r). Hence, we know that X is LLC. This completes the proof. $\square$

Proposition 2.4

The LLC-condition is inherited to open subsets and is preserving under locally bi-Lipschitz homeomorphisms. Namely,  if X is LLC and U is an open subset of X,  and if $f:X \rightarrow Y$ is a locally bi-Lipschitz homeomorphism,  then U and Y are LLC.

Proof

Let X be an LLC metric space and U its open set. For any $x \in U$ and $r > 0$, we can take $r' > 0$ such that $U(x,r') = U \cap U(x,r') \subset U \cap U(x,r)$. Here, U(z, s) denotes the open ball in the whole space X. Since X is LLC, there is an $r'' > 0$ such that $U(x,r'')$ is Lipschitz contractible in $U(x,r')$. Hence, U is LLC.

Let $f:X \rightarrow Y$ be a locally bi-Lipschitz homeomorphism. Hence, for any $x \in X$, there is an open neighborhood U of x such that $f|_{U}:U \rightarrow f(U)$ is a bi-Lipschitz homeomorphism. By the former statement, U is LLC. Therefore, we may assume that f itself is bi-Lipschitz. Let $C \ge 1$ be a constant which bounds $\mathrm {Lip}(f)$ and $\mathrm {Lip}(f^{-1})$. We prove that Y is LLC. For any $x \in X$ and $r > 0$, we have

$$\begin{aligned} f(U(x, C^{-1}r)) \subset U(f(x), r) \subset f(U(x,Cr)). \end{aligned}$$

Since X is LLC, there exist an $r' > 0$ and a Lipschitz contraction $h:U(x, Cr') \times [0,1] \rightarrow U(x, C^{-1}r)$ to x. We consider a map $k:f(U(x,Cr')) \times [0,1] \rightarrow U(f(x), r)$ defined by

$$\begin{aligned} k(f(y),t) = f(h(y,t)) \end{aligned}$$

for $y \in U(x,Cr')$ and $t \in [0,1]$. By the construction, k gives a Lipschitz contraction from $U(f(x),r')$ to f(x) in U(f(x), r). This completes the proof. $\square$

In [8], the author and Yamaguchi defined a notion stronger than the LLC-condition as follows.

Definition 2.5

[8] A metric space X is said to be strongly locally Lipschitz contractible, for short SLLC, if for any $x \in X$, there exist $r> 0$ and a Lipschitz contraction $h:U(x,r) \times [0,1] \rightarrow U(x,r)$ to x such that d(x, h(y, t)) is monotone nonincreasing in t for every $y \in U(x,r)$. Such an h is called a strong Lipschitz contraction.

It is clear that any SLLC space is LLC. Obviously, every Banach manifold is SLLC.

2.1 Strong Lipschitz contractions in the sense of [14]

The terms of strong Lipschitz contractions were used in two papers. One of them was in [8] as in Definition 2.5 and another one was introduced by Riedweg and Schäppi in [14] as in the following definition.

Definition 2.6

[14] A metric space Xadmits locally strong Lipschitz contractions if for any $x \in X$, there exist $r > 0$ and $\gamma > 0$ such that every subset $S \subset U(x,r)$ admits a Lipschitz contraction $\varphi :S \times [0,1] \rightarrow X$ to some point in X whose Lipschitz constant is controlled as

$$\begin{aligned} d(\varphi (z,t),\varphi (z',t')) \le \gamma \mathrm {diam} (S) |t-t'| + \gamma d(z,z') \end{aligned}$$

(2.1)

for all $z,z' \in S$ and $t,t' \in [0,1]$.

In [14], a map $\varphi$ satisfying (2.1) was called a strong Lipschitz contraction of S.

Lemma 2.7

Any metric space admitting locally strong Lipschitz contractions is (weakly) locally Lipschitz contractible.

Proof

Let a metric space X admit locally strong Lipschitz contractions. For any $x \in X$, there are $r > 0$ and $\gamma > 0$ such that any subset S of U(x, r) admits $\varphi _S$ satisfying (2.1). Let us take $r'' < r' \le r$. Then, the Lipschitz contraction $\varphi = \varphi _{U(x,r'')}:U(x,r'') \times [0,1] \rightarrow X$ satisfies

$$\begin{aligned} d(\varphi (z,t), \varphi (z',t')) \le 2 \gamma r'' |t-t'| + \gamma d(z,z') \end{aligned}$$

for all $z,z' \in U(x,r'')$ and $t,t' \in [0,1]$. In particular, $\varphi (z,t)$ is contained in $U(x, 3 \gamma r'')$ for any $z \in U(x,r'')$ and $t \in [0,1]$. Hence, if $3 \gamma r'' \le r'$, then $\varphi$ is a Lipschitz contraction of $U(x,r'')$ in $U(x,r')$. Therefore, we know that X is WLLC. By Lemma 2.3, X is LLC. This completes the proof. $\square$

2.2 Examples

Many objects in metric geometry, related to restrictions of sectional curvature, are known to be SLLC. Such spaces are called CAT-spaces which are metric spaces of curvature locally bounded from above and Alexandrov spaces which have a local curvature bound from below. Furthermore, Ohta introduced generalizations of CAT-spaces in the view point of convexities of distance functions. Let us briefly recall the definitions of them.

A metric space X is geodesic if any two points p, q in X admit a curve $c:[0,1] \rightarrow X$ such that $c(0)=p$ and $c(1) = q$ and that $d(c(t),c(t'))=|t-t'| d(p,q)$. Such a curve c is called a geodesic segment.

Definition 2.8

[10] Let $k \in (0,2]$. An open set U of a geodesic space X is called a $C_k$-domain for k if for any three points $x,y,z \in U$, and any geodesic segment $c:[0,1] \rightarrow X$ between $c(0)=y$ and $c(1)=z$, we have

$$\begin{aligned} d(x,c(t))^2 \le (1-t) d(x,y)^2 +t d(x,z)^2 - \frac{k}{2} t (1-t) d(y,z)^2. \end{aligned}$$

(2.2)

Let $L_1, L_2 \ge 0$. An open set U in a geodesic space X is called a $C_L$-domain for $(L_1,L_2)$ if for any three points $x,y,z \in U$, any geodesic segments $c, {\bar{c}}:[0,1] \rightarrow X$ with $c(0)={\bar{c}}(0)=x$, $c(1)=y$, and ${\bar{c}}(1)=z$, and for all $t \in [0,1]$, we have

$$\begin{aligned} d(c(t),{\bar{c}}(t)) \le \left( 1+L_1 \frac{\min \{d(x,y)+d(x,z), 2 L_2\}}{2}\right) t d(y,z) \end{aligned}$$

(2.3)

A $C_2$-domain is usually called a CAT(0)-domain. If U satisfies the opposite inequality of (2.2) for $k=2$, we say that U has nonnegative curvature (in the sense of Alexandrov). In a complete Riemannian manifold, a CAT(0)-domain (or a nonnegatively curved domain) actually has the nonpositive (or nonnegative, respectively) sectional curvature. In the same way, there are definitions of synthetic sectional curvature bound from above (and from below) by a real number $\kappa$, for metric spaces, in terms of geodesic triangle comparison. See details [4]. Ohta’s results (Corollaries 2.4 and 3.2, Lemma 2.9 and Proposition 3.1 in [10]) and their proofs imply

Proposition 2.9

Both a $C_k$-domain and a $C_L$-domain in a geodesic metric space are SLLC,  where $k \in (0,2]$ and $L_1, L_2 \ge 0$ are arbitrary. In particular,  a CAT-space is SLLC.

Let us give a proof only for CAT(0)-domains, for convenience.

Proof

Let U be a CAT(0)-domain in a geodesic space and $x \in U$. For $R > 0$, any geodesic joining two points in the ball U(x, R) is contained in U(x, 2R). Taking R with $U(x,2 R) \subset U$, we define a map

$$\begin{aligned} h:U(x, R) \times [0,1] \rightarrow U(x,R) \end{aligned}$$

by $h(y,t) = c_y(t)$. Here, $c_y:[0,1] \rightarrow X$ is a geodesic with $c_y(0)=y$ and $c_y(1) = x$. Using the condition (2.2) for $k=2$ twice, we have

$$\begin{aligned} d(h(y,t), h(z,t))^2 \le (1-t)^2 d(y,z)^2. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} d(h(y,t), h(z,s)) \le d(y,z) + R |t-s|. \end{aligned}$$

Hence, h is a Lipschitz contraction to x. Furthermore, by the definition, it is a strong Lipschitz contraction in the sense of Definition 2.5. This completes the proof. $\square$

Furthermore, in [14], CAT-spaces are proved to admit locally strong Lipschitz contractions.

A complete geodesic space with curvature locally bounded from below, in a synthetic sense, is called an Alexandrov space. Any complete Riemannian manifold and the Gromov–Hausdorff limit of manifolds having a uniform lower sectional curvature bound are Alexandrov spaces. See details [4, 5]. A main result in [8] states that every finite dimensional Alexandrov space is SLLC. The proof of this fact is not trivial. For instance, we can observe that the same proof of Lemma 2.9 does not work for a geodesic space of nonnegative curvature. To prove the result in [8], we used the theory of gradient flow of distance functions founded by Perelman and Petrunin [11, 13].

The author does not know whether arbitrary (finite dimensional) Alexandrov space admits locally strong Lipschitz contractions.

2.3 Lipschitz extensions

We recall McShane–Whitney’s Lipschitz extension theorem.

Theorem 2.10

[7, 18] Let X be a metric space and A a subset of X. Let $f:A \rightarrow {\mathbb {R}}$ be an L-Lipschitz function. Then,  the following functions

$$\begin{aligned}&X \ni x \mapsto \inf _{a \in A}\, (f(a) + L d(x,a))\quad \text {and} \\&X \ni x \mapsto \sup _{a \in A}\, (f(a) - L d(x,a)) \end{aligned}$$

are L-Lipschitz on X and extensions of f.

We will often use this theorem in the present paper. The set of all real-valued Lipschitz functions on a metric space Y is denoted by $\mathrm {Lip}(Y)$. Using the above Lipschitz extension theorem, we have

Lemma 2.11

Let A be a compact set in a metric space X. If a sequence $f_j \in \mathrm {Lip}(A)$ converges to $f \in \mathrm {Lip}(A)$ as $j \rightarrow \infty$ pointwise on A with $\sup _j \mathrm {Lip}(f_j) < \infty ,$ then there are Lipschitz extensions ${\tilde{f}}_j$ of $f_j$ to X which converges to some Lipschitz extension ${\tilde{f}}$ of f pointwise on X as $j \rightarrow \infty$.

Proof

Let $f_j$ converge to f in $\mathrm {Lip}(A)$ as $j \rightarrow \infty$ pointwise on A, with $L := \sup _j \mathrm {Lip}(f_j) < \infty$. Theorem 2.10 ensures that the functions ${\tilde{f}}_j$ and ${\tilde{f}}$ defined as

$$\begin{aligned} {\tilde{f}}_j(x) = \min _{a \in A} f_j(a) + L d(a,x) \quad \text {and}\quad {\tilde{f}}(x) = \min _{a \in A} f(a) + L d(a,x) \end{aligned}$$

are Lipschitz extensions of $f_j$ and f to X. Let us fix $x \in X {\setminus } A$. For each j, we take $y_j, y \in A$ such that

$$\begin{aligned} {\tilde{f}}_j(x) = f_j(y_j) + L d(y_j,x) \quad \text {and}\quad {\tilde{f}}(x) = f(y) + L d(y,x). \end{aligned}$$

Then, we have

$$\begin{aligned} {\tilde{f}}_j(x) \le f_j(y) + L d(y,x) \end{aligned}$$

for every j. Hence, we obtain $\varlimsup _{j \rightarrow \infty } {\tilde{f}}_j(x) \le {\tilde{f}}(x)$. Furthermore, a subsequence $\{y_{k(j)}\}_j$ of $\{y_j\}_j$ may converge to some $y_\infty \in A$ as $j \rightarrow \infty$. Then, we have

$$\begin{aligned} {\tilde{f}}(x) \le f(y_\infty ) + L d(y_\infty ,x) = \lim _{j \rightarrow \infty } f_j(y_{k(j)}) + L d(y_{k(j)},x). \end{aligned}$$

Therefore, $\varliminf _{j \rightarrow \infty } {\tilde{f}}_j(x) \ge {\tilde{f}}(x)$. This completes the proof. $\square$

Let $\mathrm {Lip}_{\mathrm b}(X)$ denotes the set of all real-valued bounded Lipschitz functions on X.

Lemma 2.12

Let A be a subset of a metric space X. Then,  the maps $\mathrm {Lip}(X) \rightarrow \mathrm {Lip}(A)$ and $\mathrm {Lip}_{\mathrm b}(X) \rightarrow \mathrm {Lip}_{\mathrm b}(A)$ assigning the function restricted to A are surjective.

Proof

By Theorem 2.10, the restriction map $\mathrm {Lip}(X) \rightarrow \mathrm {Lip}(A)$ is surjective. Let $f \in \mathrm {Lip}_{\mathrm b}(A)$, we have a Lipschitz extension ${\tilde{f}}$ of f to X by using Theorem 2.10. Let $C > 0$ satisfy that $-C \le f(a) \le C$ for all $a \in A$. Then, the function $\min \{C, \max \{-C, {\tilde{f}}\}\}$ is bounded Lipschitz on X and an extension of f. Hence, the map $\mathrm {Lip}_{\mathrm b}(X) \rightarrow \mathrm {Lip}_{\mathrm b}(A)$ is surjective. $\square$

2.4 Locally-Lipschitz homotopy invariance of the singular Lipschitz homology

For a metric space X, its singular Lipschitz chain complex $S_\bullet ^\mathrm {Lip}(X)$ was defined in the introduction. As for the usual singular chain complex $S_\bullet (X)$, the complex $S_\bullet ^\mathrm {Lip}(X)$ has a canonical augmentation defined as

$$\begin{aligned} \varepsilon :S_0^\mathrm {Lip}(X) \ni \sum _{i=1}^N a_i x_i \mapsto \sum _{i=1}^{\mathbf {N}}a_i \in {\mathbb {Z}}, \end{aligned}$$

(2.4)

where $x_i \in X$ and $a_i \in {\mathbb {Z}}$. For the extended chain complex

$$\begin{aligned} \cdots \rightarrow S_k^\mathrm {Lip}(X) \rightarrow S_{k-1}^\mathrm {Lip}(X) \rightarrow \cdots \rightarrow S_0^\mathrm {Lip}(X) \xrightarrow {\varepsilon } {\mathbb {Z}}, \end{aligned}$$

its homology is denoted by ${\tilde{H}}_*^\mathrm {Lip}(X)$, and is called the reduced singular Lipschitz homology of X.

Lemma 2.13

Let $h:X \times [0,1] \rightarrow Y$ be a locally Lipschitz homotopy. Then the induced homomorphisms between the (reduced) singular Lipschitz homologies coincide as follows.

$$\begin{aligned} h_0{}_*= & {} h_1{}_*:H_k^\mathrm {Lip}(X) \rightarrow H_k^\mathrm {Lip}(Y), \\ h_0{}_*= & {} h_1{}_*:{\tilde{H}}_0^\mathrm {Lip}(X) \rightarrow {\tilde{H}}_0^\mathrm {Lip}(Y). \end{aligned}$$

Proof

Recall that the continuous homotopy h satisfies $h_0{}_*= h_1{}_*$ as a map between the usual (reduced) singular homologies. The proof of this fact is done by giving a chain homotopy equivalence between chain maps $h_0{}_\#$ and $h_1{}_\#$ from $S_\bullet (X)$ to $S_\bullet (Y)$. This chain homotopy is constructed by a decomposition of the prism $\triangle ^k \times [0,1]$ into simplices. Such a prism decomposition is given by a combinatorial or a piecewisely linear way. Therefore, the standard chain homotopy gives a chain homotopy equivalence between $h_0{}_\#$ and $h_1{}_\#$ as chain maps from $S_\bullet ^\mathrm {Lip}(X)$ to $S_\bullet ^\mathrm {Lip}(Y)$. Hence, the maps $h_0{}_*$ and $h_1{}_*$ from $H_k^\mathrm {Lip}(X)$ to $H_k^\mathrm {Lip}(Y)$ are the same. The reduced version is proved by a similar way. $\square$

3 Chain complices consisting of metric currents

In this section, we denote by X and Y metric spaces. We recall the notion of currents in metric spaces introduced by Ambrosio and Kirchheim [1]. Here, we note that we will use a slightly modified definition from the original one. For the reason why we use the modified definition, see Remark 5.3.

3.1 Basics of measure theory

Let us denote by $\mu$ a Borel measure on a metric space X. The support$\mathrm {spt}(\mu )$ of $\mu$ is defined by

$$\begin{aligned} \mathrm {spt}(\mu ) = \{x \in X \mid \mu (U(x,r))> 0 \text { for any } r > 0\} \end{aligned}$$

which is a closed subset of X. The measure $\mu$ is said to be finite if $\mu (X) < \infty$. The outer measure obtained from $\mu$ is denoted by the same symbol as $\mu$.

We say that $\mu$ is concentrated on a subset A of X if $\mu (X {\setminus } A) = 0$. It is known that if $\mu$ is concentrated on a separable set, then $\mu$ is concentrated on its support. If $\mu$ is finite and is concentrated on a separable set, then its support is separable.

We say that $\mu$ is tight if for any $\varepsilon > 0$, there exists a compact subset $K \subset X$ such that $\mu (X {\setminus } K) < \varepsilon$. If $\mu$ is finite, then $\mu$ is tight if and only if $\mu$ is concentrated on a $\sigma$-compact set. In this case, $\mu$ is concentrated on its support. Furthermore, $\mathrm {Lip}_{\mathrm b}(X)$ is dense in $L^1(X,\mu )$ if $\mu$ is a finite tight Borel measure on X.

Let ${\mathcal {M}}$ be a family of finite Borel measures on X. The infimum $\bigwedge _{\nu \in {\mathcal {M}}} \nu$ of ${\mathcal {M}}$ is given by

$$\begin{aligned} \bigwedge _{\nu \in {\mathcal {M}}} \nu (B) := \inf \left\{ \sum _{j=1}^\infty \mu _j (B_j) \right\} \end{aligned}$$

for all Borel sets $B \subset X$, where the infimum runs over among all countable family $\{\mu _j\}_{j=1}^\infty \subset {\mathcal {M}}$ and all Borel partitions $\{B_j\}$ of B. Here, a Borel partition $\{B_j\}$ of B is a disjoint countable family consisting of Borel sets satisfying $\bigcup _j B_j = B$. By the definition, $\bigwedge _{\nu \in {\mathcal {M}}} \nu (B) \le \nu '(B)$ for any $\nu ' \in {\mathcal {M}}$ and Borel set $B \subset X$. In particular, ${\mathcal {M}}$ is a finite Borel measure. Furthermore, if some $\nu ' \in {\mathcal {M}}$ is tight, then $\bigwedge _{\nu \in {\mathcal {M}}} \nu$ is tight.

For another metric space Y with a measurable map $f:X \rightarrow Y$, we denote by $f_\# \mu$ the push-forward measure of $\mu$ by f which is defined by $f_\# \mu (B) = \mu (f^{-1}(B))$ for all Borel sets $B \subset Y$.

3.2 Metric currents

From now on, k denotes a nonnegative integer.

We set ${\mathcal {D}}^0(X) = \mathrm {Lip}_{\mathrm b}(X)$ and ${\mathcal {D}}^k(X) = \mathrm {Lip}_{\mathrm b}(X) \times [\mathrm {Lip}(X)]^k$ for $k \ge 1$. We will often abbreviate an element $(f,\pi _1,\ldots ,\pi _k) \in {\mathcal {D}}^k(X)$ by $(f,\pi )$.

Definition 3.1

[1] A multilinear functional $T:{\mathcal {D}}^k(X) \rightarrow {\mathbb {R}}$ is called a k-current in X if it satisfies the following three conditions.

• (1) T is continuous in the following sense. Let $f \in \mathrm {Lip}_{\mathrm b}(X)$ and $\pi _i^j \in \mathrm {Lip}(X)$ where $i \in \{1,\ldots , k\}$ and $j \in {\mathbb {N}}$ with $\sup _{i,j} \mathrm {Lip}(\pi _i^j) < \infty$. If $\pi _i^j$ converges to some function $\pi _i$ as $j \rightarrow \infty$ pointwise on X for each i, then we have $\lim _{j \rightarrow \infty } T(f,\pi ^j) \rightarrow T(f,\pi )$, where $\pi ^j = (\pi _1^j,\ldots , \pi _k^j)$ and $\pi = (\pi _1,\ldots , \pi _k)$.

• (2) T satisfies the locality as follows. For $(f,\pi ) \in {\mathcal {D}}^k(X)$, if $\pi _i$ is constant on $\{f \ne 0\}$ for some i, then $T(f,\pi ) = 0$.

• (3) T has finite mass in the following sense. There is a finite tight Borel measure $\mu$ on X such that

  $$\begin{aligned} |T(f,\pi )| \le \prod _{i=1}^k \mathrm {Lip}(\pi _i) \int _X |f| \, d \mu \end{aligned}$$

  holds, for all $(f,\pi ) \in {\mathcal {D}}^k(X)$. Here, when $k=0$, the value $\prod _{i=1}^k \mathrm {Lip}(\pi _i)$ is regarded as 1.

The minimal measure of $\mu$’s satisfying the finite mass axiom (3) as above for T is called the mass measure of T and is denoted by $\Vert T\Vert$. We say that a current T has compact support if $\Vert T\Vert$ has compact support, or equivalently, $\Vert T\Vert$ is concentrated on a compact set. The set of all k-currents in X is denoted by ${\mathbf {M}}_k(X)$ and its subset consisting of currents having compact support is denoted by ${\mathbf {M}}_k^\mathrm {c}(X)$.

Typical and essential examples of currents are as follows.

Example 3.2

For an $L^1$-function $\theta$ on ${\mathbb {R}}^k$ in the Lebesgue measure ${\mathcal {L}}^k$, a k-current is defined as

for $(f,\pi ) \in {\mathcal {D}}^k({\mathbb {R}}^k)$. Here, $\partial \pi _i / \partial x_j$ are defined for almost everywhere ${\mathbb {R}}^k$ and are bounded integrable functions, due to Rademacher’s theorem.

There is a useful characterization of the mass measures of currents as follows.

Proposition 3.3

(Proposition 2.7 in [1]) Let T be a metric k-current in X. For every Borel set B in X,  we have

$$\begin{aligned} \Vert T\Vert (B) = \sup \sum _{j=1}^\infty T(\chi _{B_j}, \pi ^j), \end{aligned}$$

where the supremum runs among all Borel partitions $\{B_j\}$ of B and all Lipschitz maps $\pi ^j = (\pi ^j_i)_{1 \le i \le k}: X \rightarrow {\mathbb {R}}^k$ with $\mathrm {Lip}(\pi _i^j) \le 1$ for all i.

We remark that, in this proposition, $\chi _{B_j}$ is not Lipschitz, but $T(\chi _{B_j}, -)$ is well-defined (see the above of Lemma 3.4 later).

For k-currents T, S in X and a Borel set B in X, we have

$$\begin{aligned}&\Vert T+S\Vert (B) \le \Vert T\Vert (B)+\Vert S\Vert (B), \\&\Vert -T\Vert (B) = \Vert T\Vert (B). \end{aligned}$$

These properties follow from the definition. The second property induces that $\mathrm {spt}(-T)=\mathrm {spt}(T)$.

We recall fundamental operations to obtain currents. From now on, let $T \in {\mathbf {M}}_k(X)$. For another metric space with a Lipschitz map $\phi :X \rightarrow Y$, we have the push-forward$\phi _\# T$ of T by $\phi$ defined by

$$\begin{aligned} \phi _\# T (f, \pi ) = T(f \circ \phi , \pi \circ \phi ) \end{aligned}$$

for $(f,\pi ) \in {\mathcal {D}}^k(Y)$, which is a k-current in Y. Furthermore, we have $\Vert \phi _\# T\Vert \le \mathrm {Lip}(\phi )^k \phi _\# \Vert T\Vert$ as measures on Y. If $\phi$ is a bi-Lipschitz embedding, then by Lemma 2.12, $\phi _\#:{\mathbf {M}}_k(X) \rightarrow {\mathbf {M}}_k(Y)$ is injective. Since $\mathrm {Lip}_{\mathrm b}(X)$ is dense in $L^1(X,\Vert T\Vert )$, the k-current T can be extended to a multilinear functional on $L^1(X,\Vert T\Vert ) \times [\mathrm {Lip}(X)]^k$ in a unique way. Therefore, for any Borel set $A \subset X$, the functional $T \lfloor A$ is defined by

$$\begin{aligned} T \lfloor A(f,\pi ) = T(\chi _A f, \pi ) \end{aligned}$$

for $(f,\pi ) \in {\mathcal {D}}^k(X)$, where $\chi _A$ denotes the characteristic function of A, which is called the restriction of T to A and is also a k-current in X.

Lemma 3.4

Let T be a current in X and A a Borel set in X. The mass measure of $T \lfloor A$ coincides with the restricted measure $\Vert T\Vert \lfloor A$ to A defined as

$$\begin{aligned} \Vert T\Vert \lfloor A (B) := \Vert T\Vert (A \cap B) \end{aligned}$$

for every Borel set $B \subset X$. In particular,  the support of $T \lfloor A$ is contained in $\mathrm {spt}(T) \cap {\bar{A}},$ where ${\bar{A}}$ is the closure of A in X.

Proof

We take a metric current T in X of degree k and a Borel set A. By Proposition 3.3,

$$\begin{aligned} \Vert T \lfloor A\Vert (B) = \sup \sum _{j=1}^\infty T (\chi _A \chi _{B_j}, \pi ^j) \end{aligned}$$

holds, where the supremum runs among all Borel partitions $\{B_j\}$ of B and all k-tuples $\pi ^j = (\pi _i^j)_{1 \le i \le k}$ of 1-Lipschitz functions $\pi _i^j:X \rightarrow {\mathbb {R}}$. Since $\chi _{A \cap B_j} = \chi _A \chi _{B_j}$ and a Borel partition $\{B_j\}$ of B gives a Borel partition $\{B_j \cap A\}$ of $A \cap B$, the value $\Vert T\Vert (A \cap B)$ actually coincides with $\Vert T \lfloor A\Vert (B)$, by Proposition 3.3 again. Therefore, we have $\Vert T \lfloor A\Vert = \Vert T\Vert \lfloor A$. Hence, we obtain $\mathrm {spt}(T \lfloor A) = \mathrm {spt}(\Vert T\Vert \lfloor A) \subset \mathrm {spt}\Vert T\Vert \cap {\bar{A}} = \mathrm {spt}\, T \cap {\bar{A}}$. This completes the proof. $\square$

It is clear that $T \lfloor \mathrm {spt}(T) = T$ as currents in X. The restriction is a current in the whole space X in general. However, if T has compact support, T itself can be regard as a current in its support as follows.

Lemma 3.5

Let $T \in {\mathbf {M}}_k^\mathrm {c}(X)$. Then,  there is a unique k-current $T'$ in $\mathrm {spt}(T)$ such that $T = \iota _\# T'$, where $\iota :\mathrm {spt}(T) \rightarrow X$ is the inclusion.

Proof

Let us take $T \in {\mathbf {M}}_k^\mathrm {c}(X)$ and set $A = \mathrm {spt}(T)$. A functional $T':{\mathcal {D}}^k(A) \rightarrow {\mathbb {R}}$ is defined as follows. For $(f,\pi ) \in {\mathcal {D}}^k(A)$, by Lemma 2.12, we have an extension $({\tilde{f}}, {{\tilde{\pi }}}) \in {\mathcal {D}}^k(X)$ of $(f,\pi )$. Then, we set

$$\begin{aligned} T'(f,\pi ) = T({\tilde{f}}, {\tilde{\pi }}). \end{aligned}$$

First, we check that this value is independent on the choice of $({\tilde{f}}, {\tilde{\pi }})$ and show that $T'$ is multilinear and satisfies the locality. Let ${\hat{f}} \in \mathrm {Lip}_{\mathrm b}(X)$ be another extension of f. By the finite mass axiom, we have $T({\tilde{f}}, {\tilde{\pi }}) = T({\hat{f}}, {\tilde{\pi }})$. Furthermore, if ${\hat{f}}$ is a bounded Borel function on X with ${\hat{f}} |_A = f$, then $T({\hat{f}}, {\tilde{\pi }}) = T({\tilde{f}}, {\tilde{\pi }})$, because T can be regarded as a functional on $L^1(X,\Vert T\Vert ) \times [\mathrm {Lip}(X)]^k$. This implies the linearity of $T'(f, \pi )$ in f. Here, recall that T as the functional on $L^1(X,\Vert T\Vert ) \times [\mathrm {Lip}(X)]^k$ satisfies the strengthened locality, continuity and finite mass axiom, as stated in [1]. For another Lipschitz extension ${\hat{\pi }} \in [\mathrm {Lip}(X)]^k$ of $\pi$, the strengthened locality of T implies $T({\tilde{f}}, {\tilde{\pi }}) = T({\tilde{f}}, {\hat{\pi }})$. Furthermore, the strengthened locality of T implies the multilinearity of $T'(f,\pi )$ in $\pi$ and the locality of $T'$.

Let us define a finite Borel measure $\mu$ on A by $\mu (B) = \Vert T\Vert (B)$ for any Borel set $B \subset A$. The tightness of $\Vert T\Vert$ ensures that $\mu$ is tight. For any $(f, \pi ) \in {\mathcal {D}}^k(A)$, we have

$$\begin{aligned} |T'(f,\pi )| = |T({\tilde{f}}, {\tilde{\pi }})| \le \prod _{i=1}^k \mathrm {Lip}({\tilde{\pi }}_i) \int _X |{\tilde{f}}| \, d \Vert T\Vert = \prod _{i=1}^k \mathrm {Lip}(\pi _i) \int _X |f| \, d \mu . \end{aligned}$$

Here, we note that a Lipschitz extension ${\tilde{\pi }}_i$ of $\pi _i$ can be chosen as $\mathrm {Lip}({\tilde{\pi }}_i) = \mathrm {Lip}(\pi _i)$. Hence, $T'$ has finite mass.

Finally, we prove that $T'$ is continuous. Let $\pi _i^j \in \mathrm {Lip}(A)$, $i =1,\ldots ,k$, $j \in {\mathbb {N}}$ with $\sup _{i,j} \mathrm {Lip}(\pi _i^j) \le L < \infty$ such that $\pi _i^j$ converges to $\pi _i$ as $j \rightarrow \infty$ pointwise on A, for every i. By Lemma 2.11, we have extensions ${\tilde{\pi }}_i^j$ of $\pi _i^j$ to X with $\sup _{i,j} \mathrm {Lip}({\tilde{\pi }}_i^j) \le L$, such that ${\tilde{\pi }}_i^j$ converges to some Lipschitz extension ${\tilde{\pi }}_i$ of $\pi _i$ pointwise on X as $j \rightarrow \infty$, for each i. By the continuity of T, we have

$$\begin{aligned} \lim _{j \rightarrow \infty } T'(f, \pi ^j) = \lim _{j \rightarrow \infty } T({\tilde{f}}, {\tilde{\pi }}^j) = T({\tilde{f}}, {\tilde{\pi }}) = T'(f,\pi ). \end{aligned}$$

Hence T is continuous. Therefore, $T' \in {\mathbf {M}}_k^\mathrm {c}(A)$. By the definition, $\iota _\# T' = T$. Since $\iota _\#$ is injective, such a $T'$ is unique. This completes the proof. $\square$

As a corollary to Lemma 3.5, the push-forward operator $\phi _\#:{\mathbf {M}}_k^\mathrm {c}(X) \rightarrow {\mathbf {M}}_k^\mathrm {c}(Y)$ is also defined, for a locally Lipschitz map $\phi :X \rightarrow Y$, as follows. For $T \in {\mathbf {M}}_k^\mathrm {c}(X)$, taking $T'$ as in Lemma 3.5 for T, and set $\phi _\# T := \iota _\# (\phi |_{\mathrm {spt}(T)})_\# T' \in {\mathbf {M}}_k^\mathrm {c}(Y)$, where $\iota :\phi (\mathrm {spt}(T)) \rightarrow Y$ is the inclusion. By the construction, the pushforwards have the following functorial property:

$$\begin{aligned} (\psi \circ \phi )_\# = \psi _\# \phi _\#:{\mathbf {M}}_k^\mathrm {c}(X) \rightarrow {\mathbf {M}}_k^\mathrm {c}(Z) \end{aligned}$$

for locally Lipschitz mappings $\phi :X \rightarrow Y$ and $\psi :Y \rightarrow Z$.

The boundary$\partial T$ of T is a functional on ${\mathcal {D}}^{k-1}(X)$ defined by

$$\begin{aligned} \partial T(f,\pi ) = T(1, f, \pi ). \end{aligned}$$

It satisfies the continuity and locality, but does not have finite mass in general. It is trivial that

$$\begin{aligned} \partial \phi _\# = \phi _\# \partial \end{aligned}$$

(3.1)

holds on ${\mathbf {M}}_k^\mathrm {c}(X)$, for a locally Lipschitz map $\phi :X \rightarrow Y$. By the locality, we have

$$\begin{aligned} \partial \partial T = 0. \end{aligned}$$

(3.2)

Definition 3.6

[1] A k-current T in X is normal if its boundary has finite mass, or equivalently, $\partial T \in {\mathbf {M}}_{k-1}(X)$. Here, when $k = 0$, we always regard T as normal and $\partial T = 0$. We denote the set of all normal k-currents in X by ${\mathbf {N}}_k(X)$ and set ${\mathbf {N}}_k^\mathrm {c}(X) = {\mathbf {N}}_k(X) \cap {\mathbf {M}}_k^\mathrm {c}(X)$.

By the definition, (3.2) and (3.1), the group ${\mathbf {N}}_\bullet (X) = \bigoplus _{k=0}^\infty {\mathbf {N}}_k(X)$ becomes a chain complex with the boundary map $\partial$ and $\phi _\#:{\mathbf {N}}_\bullet (X) \rightarrow {\mathbf {N}}_\bullet (Y)$ is a chain map for any Lipschitz map $\phi :X \rightarrow Y$. Furthermore, by Slicing Theorem 5.6 in [1], if $T \in {\mathbf {N}}_k(X)$, then we have

$$\begin{aligned} \mathrm {spt}(\partial T) \subset \mathrm {spt}(T). \end{aligned}$$

(3.3)

Hence, ${\mathbf {N}}_\bullet ^\mathrm {c}(X) = \bigoplus _{k=0}^\infty {\mathbf {N}}_k^\mathrm {c}(X)$ is a subcomplex of ${\mathbf {N}}_\bullet (X)$. Furthermore, by (3.1), $\phi _\#:{\mathbf {N}}_\bullet ^\mathrm {c}(X) \rightarrow {\mathbf {N}}_\bullet ^\mathrm {c}(Y)$ is well-defined as a chain map, for any locally Lipschitz map $\phi :X \rightarrow Y$.

Lemma 3.7

Let $T \in {\mathbf {N}}_k^\mathrm {c}(X)$. Then,  there is a unique $T' \in {\mathbf {N}}_k^\mathrm {c}(\mathrm {spt}\, T)$ such that $\iota _\# T' = T,$ where $\iota :\mathrm {spt}\, T \rightarrow X$ is the inclusion.

Proof

Let us take $T \in {\mathbf {N}}_k^\mathrm {c}(X)$. By Lemma 3.5, there is a unique $T' \in {\mathbf {M}}_k^\mathrm {c}(K)$ such that $\iota _\# T' = T$, where $K = \mathrm {spt}\, T$. On the other hands, since $\partial T \in {\mathbf {M}}_{k-1}^\mathrm {c}(X)$, by using Lemma 3.5 again, we have $S \in {\mathbf {M}}_{k-1}^\mathrm {c}(K)$ such that $\iota _\# S = \partial T$. Then, we obtain

$$\begin{aligned} \iota _\# S = \partial T = \partial \iota _\# T' = \iota _\# \partial T'. \end{aligned}$$

Since $\iota _\#$ is injective from Lemma 2.12, we conclude that $\partial T' = S$ and hence $\Vert \partial T'\Vert =\Vert S\Vert$. Therefore, $T' \in {\mathbf {N}}_k^\mathrm {c}(K)$. This completes the proof. $\square$

3.3 Integral currents

For a metric space X, we denote by ${\mathcal {H}}^k = {\mathcal {H}}_X^k$ the k-dimensional Hausdorff measure on X. A subset S of X is called a countably${\mathcal {H}}^k$-rectifiable set if there are countably many Borel subsets $B_j$ of ${\mathbb {R}}^k$ and Lipschitz maps $\phi _j:B_j \rightarrow X$ such that

$$\begin{aligned} {\mathcal {H}}^k \left( S {\setminus } \bigcup _{j=1}^\infty \phi _j(B_j) \right) = 0. \end{aligned}$$

Definition 3.8

[1] A k-current T in X is said to be rectifiable if $\Vert T\Vert$ is concentrated on a countably ${\mathcal {H}}^k$-rectifiable set and is absolutely continuous in ${\mathcal {H}}^k$. When $k=0$, a rectifiable 0-current T is represented as

$$\begin{aligned} T(f) = \sum _{j=1}^\infty \theta _j f(x_j) \end{aligned}$$

for $f \in \mathrm {Lip}_{\mathrm b}(X)$, where $\theta _j \in {\mathbb {R}}$ and $x_j \in X$. Such a T is written as

$$\begin{aligned} T = \sum _{j=1}^\infty \theta _j [x_j]. \end{aligned}$$

(3.4)

When $k \ge 1$, a k-current T in X is called an integer rectifiable current if it is rectifiable and for any open set $O \subset X$ and a Lipschitz function $\phi$, there is an integral valued $L^1$-function $\theta$ on ${\mathbb {R}}^k$ such that

as currents in ${\mathbb {R}}^k$. An integer rectifiable 0-currentT in X is defined as (3.4) such that $\theta _j \in {\mathbb {Z}}$ for all j.

For integral rectifiable 0-current T represented as (3.4), the families $\{x_j\}$ and $\{\theta _j\}$ can be finite sets.

Definition 3.9

[1] A k-current in X is said to be integral if it is integer rectifiable and normal. We denote by ${\mathbf {I}}_k(X)$ the set of all integral k-currents in X and set ${\mathbf {I}}_k^\mathrm {c}(X) = {\mathbf {I}}_k(X) \cap {\mathbf {M}}_k^\mathrm {c}(X)$.

The Boundary Rectifiability Theorem 8.6 in [1] says that ${\mathbf {I}}_\bullet (X) = \bigoplus _{k=0}^\infty {\mathbf {I}}_k(X)$ becomes a subcomplex of ${\mathbf {N}}_\bullet (X)$. The group ${\mathbf {I}}_\bullet ^\mathrm {c}(X) = \bigoplus _{k=0}^\infty {\mathbf {I}}_k^\mathrm {c}(X)$ is also a subcomplex of ${\mathbf {I}}_\bullet (X)$. As mentioned in the introduction, we will compare the homology of ${\mathbf {I}}_\bullet ^\mathrm {c}(X)$, denote by $H_*^\mathrm {IC}(X)$, with the singular (Lipschitz) homology of X. Furthermore, we can check that the chain map $\phi _\#:{\mathbf {I}}_\bullet ^\mathrm {c}(X) \rightarrow {\mathbf {I}}_\bullet ^\mathrm {c}(Y)$ is well-defined, for a locally Lipschitz map $\phi :X \rightarrow Y$. Therefore, ${\mathbf {I}}_\bullet ^\mathrm {c}$ can be regarded as a covariant functor from the category of all metric spaces and all locally Lipschitz maps to the category of all chain groups and all chain maps.

Lemma 3.10

For any $T \in {\mathbf {I}}_k^\mathrm {c}(X),$ there is a unique $T' \in {\mathbf {I}}_k^\mathrm {c}(\mathrm {spt}\, T)$ such that $\iota _\# T' = T,$ where $\iota :\mathrm {spt}\, T \rightarrow X$ is the inclusion.

Proof

Let $T \in {\mathbf {I}}_k^\mathrm {c}(X)$. By Lemma 3.7, there is a unique $T' \in {\mathbf {N}}_k^\mathrm {c}(K)$ such that $\iota _\# T' = T$, where $K = \mathrm {spt}\, T$. From the definition, there is a countably ${\mathcal {H}}^k$-rectifiable set $S \subset X$ such that $\Vert T\Vert (X {\setminus } S) = 0$. Then, we note that $S \cap K$ is also a countably ${\mathcal {H}}^k$-rectifiable set in K. Hence, we have $\Vert T'\Vert (K {\setminus } S)=\Vert T\Vert (X {\setminus } S)=0$. For a Borel set $B \subset K$ with ${\mathcal {H}}^k(B)=0$, we have $\Vert T'\Vert (B) = \Vert T\Vert (B)=0$. Therefore, $T'$ is a rectifiable current. We prove that $T'$ is integer rectifiable. For any open set O in K, there is an open set U in X such that $O = K \cap U$. Let us take a Lipschitz map $\phi :K \rightarrow {\mathbb {R}}^k$. By Lemma 2.11, there is an extension $\psi :X \rightarrow {\mathbb {R}}^k$ such that $\psi$ is Lipschitz. Since T is integer rectifiable, there is an integrable function $\theta :{\mathbb {R}}^k \rightarrow {\mathbb {Z}}$ such that

Now, we note that $T \lfloor U = (\iota _\# T') \lfloor U = \iota _\# (T' \lfloor O)$. Therefore, we obtain

This implies that $T'$ is integer rectifiable. This completes the proof. $\square$

3.4 Locally Lipschitz homotopy invariance of the homology of currents

We check that the homology $H_*^\mathrm {IC}(X)$ is independent on locally Lipschitz homotopy. Furthermore, we have

Lemma 3.11

Let $h:X \times [0,1] \rightarrow Y$ be a locally Lipschitz homotopy. Then,  we have $H_*({\mathbf {N}}_\bullet ^\mathrm {c}(h_0)) = H_*({\mathbf {N}}_\bullet ^\mathrm {c}(h_1))$ as morphisms from $H_*({\mathbf {N}}_\bullet ^\mathrm {c}(X))$ to $H_*({\mathbf {N}}_\bullet ^\mathrm {c}(Y))$. Furthermore,  we have $H_*^\mathrm {IC}(h_0) = H_*^\mathrm {IC}(h_1)$ as morphisms from $H_*^\mathrm {IC}(X)$ to $H_*^\mathrm {IC}(Y)$.

To prove Lemma 3.11, let us recall a product of currents considered in [1, 17]. Let T be a k-current in a metric space X. Then, we define a functional $T \times [0,1]:{\mathcal {D}}^{k+1}(X \times [0,1]) \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} (T \times [0,1]) (f, \pi ) = \sum _{i=1}^{k+1} (-1)^i \int _0^1 T \left( f_t \frac{\partial \pi _i}{\partial t}, {\hat{\pi }}_i{}_t \right) dt \end{aligned}$$

for $(f,\pi ) \in {\mathcal {D}}^{k+1}(X \times [0,1])$, where ${\hat{\pi }}_i{}_t = (\pi _j{}_t)_{j \ne i}$. Here, the partial derivative $\partial \pi _i(x,t) / \partial t$ is defined for ${\mathcal {L}}^1$-a.e. t and $\Vert T\Vert$-a.e. x, by Rademacher’s Theorem and Fubini’s Theorem. The functional $T \times [0,1]$ does not satisfy the continuity in general. Ambrosio and Kirchheim, and Wenger proved

Proposition 3.12

([17, Theorem 3.2], [1, Proposition 10.2]) Let $T \in {\mathbf {N}}_k(X)$ with bounded support. Then,  $T \times [0,1] \in {\mathbf {N}}_{k+1}(X \times [0,1])$ with boundary

$$\begin{aligned} \partial (T \times [0,1]) = T \times [0] - T \times [1] - (\partial T) \times [0,1]. \end{aligned}$$

Here,  $(T \times [t])(f, \pi ) = T(f_t, \pi _t)$ for $(f,\pi ) \in {\mathcal {D}}^k(X \times [0,1])$.

In addition,  if $T \in {\mathbf {I}}_k(X),$ then $T \times [0,1] \in {\mathbf {I}}_{k+1}(X \times [0,1])$.

By the construction, if T has compact support, then $T \times [0,1]$ has compact support. Indeed, we have, for $(f,\pi ) \in {\mathcal {D}}^{k+1}(X \times [0,1])$ with $\mathrm {Lip}(\pi _i) \le 1$,

$$\begin{aligned} |(T \times [0,1])(f,\pi )| \le (k+1) \int _0^1 \int _X |f_t| \, d \Vert T\Vert (x) \, d t. \end{aligned}$$

Hence, the support of $T \times [0,1]$ is contained in $\mathrm {spt}(T) \times [0,1]$.

Proof of Lemma 3.11

Let us denote by ${\mathbf {C}}_k$ one of ${\mathbf {N}}_k^\mathrm {c}$ or ${\mathbf {I}}_k^\mathrm {c}$. Let $h:X \times [0,1] \rightarrow Y$ be a locally Lipschitz homotopy. Let us define a map

$$\begin{aligned} P:{\mathbf {C}}_k(X) \rightarrow {\mathbf {C}}_{k+1}(Y) \end{aligned}$$

by

$$\begin{aligned} P(T) = h_\# (T \times [0,1]). \end{aligned}$$

Then, by Proposition 3.12, we have

$$\begin{aligned} \partial P(T) + P(\partial T) = h_\# (T \times [0] - T \times [1]) = h_0{}_\# T - h_1{}_\# T. \end{aligned}$$

Hence, the map P is a chain homotopy between $h_0{}_\#$ and $h_1{}_\#$. Therefore, the induced maps between homologies are the same. This completes the proof. $\square$

3.5 Mayer–Vietoris type property

Lemma 3.13

(Localization Lemma 5.3 in [1]) Let T be a normal k-current in a metric space X and $f:X \rightarrow {\mathbb {R}}$ a Lipschitz function. For almost all $s \in {\mathbb {R}},$ the restriction $T \lfloor \{f \le s \}$ of T to the sublevel set of f is a normal current in X. Furthermore,  if T is integral,  then so is $T \lfloor \{f \le s\}$ for a.e. $s \in {\mathbb {R}}$.

As a corollary to Lemma 3.13, we know that $T \lfloor \{f > s\} = T - T \lfloor \{f \le s\}$ is also a normal current, for a.e. $s \in {\mathbb {R}}$. Furthermore, if T has a compact support, then $T \lfloor \{f \le s\}$ and $T \lfloor \{f > s\}$ have compact support.

Lemma 3.14

Let U and V be open subsets in a metric space X. Then,  there is an exact sequence as follows.

$$\begin{aligned} 0 \rightarrow {\mathbf {N}}_\bullet ^\mathrm {c}(U \cap V) \xrightarrow {\psi } {\mathbf {N}}_\bullet ^\mathrm {c}(U) \oplus {\mathbf {N}}_\bullet ^\mathrm {c}(V) \xrightarrow {\varepsilon } {\mathbf {N}}_\bullet ^\mathrm {c}(U \cup V) \rightarrow 0, \end{aligned}$$

where $\psi$ and $\varepsilon$ are defined as

$$\begin{aligned} \psi (T) = (i_\# T, -i'_\#T), \quad \varepsilon (S,S') = j_\# S+j_\#' S'. \end{aligned}$$

Here,  $i:U \cap V \rightarrow U,$$i':U \cap V \rightarrow V,$$j:U \rightarrow U \cup V,$$j':V \rightarrow U \cup V$ are the inclusions.

Furthermore,  if we replace ${\mathbf {N}}_\bullet ^\mathrm {c}$ with ${\mathbf {I}}_\bullet ^\mathrm {c},$ then we obtain an exact sequence

$$\begin{aligned} 0 \rightarrow {\mathbf {I}}_\bullet ^\mathrm {c}(U \cap V) \xrightarrow {\psi } {\mathbf {I}}_\bullet ^\mathrm {c}(U) \oplus {\mathbf {I}}_\bullet ^\mathrm {c}(V) \xrightarrow {\varepsilon } {\mathbf {I}}_\bullet ^\mathrm {c}(U \cup V) \rightarrow 0. \end{aligned}$$

Proof

Let us denote by ${\mathbf {C}}_k$ one of ${\mathbf {N}}_k^\mathrm {c}$ and ${\mathbf {I}}_k^\mathrm {c}$. Let us take open sets U and V in X and consider the following sequence

$$\begin{aligned} 0 \rightarrow {\mathbf {C}}_k(U \cap V) \xrightarrow {\psi } {\mathbf {C}}_k(U) \oplus {\mathbf {C}}_k(V) \xrightarrow {\varepsilon } {\mathbf {C}}_k(U \cup V) \rightarrow 0. \end{aligned}$$

Since $i_\#$ and $i'_\#$ are injective, $\psi$ is injective. By the definition, we have $\varepsilon \psi = 0$. We prove that $\varepsilon$ is surjective. Let us take $T \in {\mathbf {C}}_k(U \cup V)$. We denote by d the distance function from $X {\setminus } V$. Since $\mathrm {spt}(T)$ is compact and is contained in $U \cup V$, there is an $r > 0$ such that

$$\begin{aligned} \mathrm {spt}(T) \cap \{d \le r\} \subset U. \end{aligned}$$

By Lemma 3.13, we can take r such that $T \lfloor \{d \le r\}$ is in ${\mathbf {C}}_k$. Then, $S := T \lfloor \{d \le r\}$ is regarded as a current in ${\mathbf {C}}_k(U)$ and $S' := T \lfloor \{d > r\}$ can be regarded as a current in ${\mathbf {C}}_k(V)$, by Lemmas 3.4 and 3.5. By the construction, we have $T = j_\# S+j_\#' S'$. Hence, the map $\varepsilon$ is surjective. Next, we take an element $(S,S') \in {\mathbf {C}}_k(U) \oplus {\mathbf {C}}_k(V)$ with $j_\# S+ j_\#' S' = 0$. Since $\mathrm {spt}(-S) = \mathrm {spt}(-j_\# S) = \mathrm {spt}(j_\# S) = \mathrm {spt}(S)$, we know that $\mathrm {spt}(S) = \mathrm {spt}(S')$ and it is contained in $U \cap V$. By Lemma 3.5, the current S can be regarded as a current in $U \cap V$, say T. Then, we have $\psi (T) = (i_\#T,-i_\#' T) = (S,S')$. This completes the proof. $\square$

3.6 A natural transformation $[\,\cdot \,]$ from $S_\bullet ^\mathrm {Lip}$ to ${\mathbf {I}}_\bullet ^\mathrm {c}$

In the introduction, we already define the complex $S_\bullet ^\mathrm {Lip}(X)$ of singular Lipschitz chains in a metric space X, which is a subcomplex of the usual integral singular chain complex $S_\bullet (X)$. Following [14], we define a chain map $[\,\cdot \,]:S_\bullet ^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_\bullet ^\mathrm {c}(X)$. For each singular Lipschitz simplex $\sigma :\triangle ^k \rightarrow X$, which is just a Lipschitz map, a k-current $[\sigma ]$ in X is defined by

By the definition, we have

$$\begin{aligned} \mathrm {spt}([\sigma ]) \subset \mathrm {im}(\sigma ). \end{aligned}$$

Its ${\mathbb {Z}}$-linear extension gives a group homomorphism $[\,\cdot \,]:S_k^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_k^\mathrm {c}(X)$. Then, we have

$$\begin{aligned} \mathrm {spt}([c]) \subset \mathrm {im}(c) \end{aligned}$$

for every Lipschitz chain c. Here, for a singular chain $c = \sum _{\sigma } a_\sigma \sigma$, its image is defined by

$$\begin{aligned} \mathrm {im}(c) = \bigcup _{a_\sigma \ne 0} \mathrm {im}(\sigma ). \end{aligned}$$

We note that, Stokes’s theorem for Lipschitz functions on $\triangle ^k$ holds as the following form.

$$\begin{aligned} \int _{\triangle ^k} \det \frac{\partial (f_1,\dots ,f_k)}{\partial (s_1,\dots ,s_k)} \, d {\mathcal {L}}^k(s) = \int _{\partial \triangle ^k} f_1 \det \frac{\partial (f_2,\ldots , f_k)}{\partial (t_1,\ldots , t_{k-1})}\, d {\mathcal {L}}^{k-1}(t) \end{aligned}$$

for $f_1,\ldots , f_k \in \mathrm {Lip}(\triangle ^k)$, where $\int _{\partial \triangle ^k}$ is the sum of integrations over $(k-1)$-faces of $\triangle ^k$ with orientations, and t is an intrinsic coordinate of each face. This formula is actually proved by a standard smoothing argument, and is represented as

where $\iota _i:\triangle ^{k-1} \rightarrow \triangle ^k$ is an orientation preserving isometric embedding into a face of $\triangle ^k$. From this formulation, the map $[\,\cdot \,]:S_\bullet ^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_\bullet ^\mathrm {c}(X)$ is known to be a chain map. Furthermore, it is natural in the sense that $[\phi _\# c] = \phi _\# [c]$ for a locally Lipschitz map $\phi :X \rightarrow Y$ to another metric space Y and $c \in S_\bullet ^\mathrm {Lip}(X)$.

3.7 The groups of 0-chains

Let X be a metric space. Obviously, the groups $S_0^\mathrm {Lip}(X)$ and $S_0(X)$ are the same. For the group ${\mathbf {I}}_0^\mathrm {c}(X)$ of integral 0-currents, the following two lemmas hold.

Lemma 3.15

The map $[\,\cdot \,]:S_0^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_0^\mathrm {c}(X)$ is an isomorphism.

Proof

Let us take $c = \sum _{i=1}^N a_i x_i \in S_0^\mathrm {Lip}(X) = S_0^\mathrm {Lip}(X)$. Then, $[c] = \sum _{i=1}^N a_i [x_i] \in {\mathbf {I}}_0^\mathrm {c}(X)$ from the definition. Hence, the map $[\,\cdot \,]:S_0^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_0^\mathrm {c}(X)$ is surjective. We prove that it is injective. We assume $[c]=0$, that is, $\sum _{i=1}^N a_i f(x_i) = 0$ for every bounded Lipschitz map $f:X \rightarrow {\mathbb {R}}$. Fix an index i, we can take a function $f \in \mathrm {Lip}_{\mathrm b}(X)$ such that $f(x_i)=1$ and $f(x_j) = 0$ for all $j \ne i$. Hence, $a_i = 0$ for all i. It implies $c=0$. This completes the proof.

$\square$

Let us consider the map $\varepsilon :{\mathbf {N}}_0^\mathrm {c}(X) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \varepsilon T = T(1) \end{aligned}$$

for all $T \in {\mathbf {N}}_0^\mathrm {c}(X)$. The restriction of $\varepsilon$ to the group ${\mathbf {I}}_0^\mathrm {c}(X)$ is also represented as the same symbol $\varepsilon :{\mathbf {I}}_0^\mathrm {c}(X) \rightarrow {\mathbb {Z}}$, where we note that the target group can be ${\mathbb {Z}}$. We have $\partial \varepsilon S = 0$ for all $S \in {\mathbf {N}}_1^\mathrm {c}(X)$, by the locality axiom of currents. Hence, the both $\varepsilon$’s are augmentations of the complices ${\mathbf {N}}_\bullet ^\mathrm {c}(X)$ and ${\mathbf {I}}_\bullet ^\mathrm {c}(X)$.

Lemma 3.16

Let $\varepsilon :S_0^\mathrm {Lip}(X) \rightarrow {\mathbb {Z}}$ be the standard augmentation defined in (2.4) which is denoted by the same symbol as the map $\varepsilon :{\mathbf {I}}_0^\mathrm {c}(X) \rightarrow {\mathbb {Z}}.$ Then,  we have $\varepsilon \circ [\,\cdot \,] = \varepsilon .$

Proof

The lemma follows directly from the definitions. $\square$

Lemma 3.17

Let $X_0$ be a metric space consisting of a single point. Then,  we have $H_0^\mathrm {IC}(X_0) \cong {\mathbb {Z}}$ and $H_k^\mathrm {IC}(X_0) = 0$ for $k \ge 1$. In addition,  $H_0({\mathbf {N}}_\bullet ^\mathrm {c}(X_0)) \cong {\mathbb {R}}$ and $H_k({\mathbf {N}}_\bullet ^\mathrm {c}(X_0)) = 0$ for $k \ge 1$.

Proof

By the definition, we have ${\mathbf {I}}_0^\mathrm {c}(X_0) \cong {\mathbb {Z}}$ and ${\mathbf {N}}_0^\mathrm {c}(X_0) \cong {\mathbb {R}}$. By Theorem 3.9 in [1], normal currents have the following property: if T is a normal k-current in a metric space Y, then $\Vert T\Vert$ is absolutely continuous in the k-dimensional Hausdorff measure ${\mathcal {H}}^k$. Therefore, we have ${\mathbf {I}}_k^\mathrm {c}(X_0) = 0 = {\mathbf {N}}_k^\mathrm {c}(X_0)$ for all $k \ge 1$. This implies the conclusion of the lemma. $\square$

4 A way comparing homologies by using cosheaves

In this section, we recall the notion of cosheaves and give a technique to compare two homologies associated to cosheaves. Let $\mathsf {Ab}$ and ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$ denote the categories of all abelian groups with group homomorphisms and of all chain complices of abelian groups with chain maps, respectively. Throughout the present paper, any chain complex was and will be indexed by nonnegative integers. A complex $C \in {\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$ is represented as $C = C_\bullet = (C_\bullet , \partial ) = (\cdots \rightarrow C_k \xrightarrow {\partial _k} C_{k-1} \rightarrow \cdots \xrightarrow {\partial _1} C_0) = \bigoplus _{k \ge 0} C_k$. We recall that the m-th homology $H_m(C_\bullet )$ of $C_\bullet$ is defined as $H_m(C_\bullet ) = \mathrm {Ker}\, \partial _m / \mathrm {Im}\, \partial _{m-1}$ if $m \ge 1$ and as $H_0(C_\bullet ) = C_0 / \mathrm {Im}\, \partial _1$. For a complex $C_\bullet = (C_\bullet , \partial ) \in {\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$ and an abelian group $A \in \mathsf {Ab}$, a homomorphism $\varepsilon :C_0 \rightarrow A$ is called an augmentation of $C_\bullet$ if $\varepsilon \circ \partial _1 = 0$ holds, and the augmentation $\varepsilon$ is also denoted by $\varepsilon :C_\bullet \rightarrow A$. Let ${\mathsf {C}}$ be one of $\mathsf {Ab}$ and ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$. We also define the reduced homology ${\tilde{H}}_*(C_\bullet )$ of the complex $C_\bullet$ augmented by $\varepsilon$ as ${\tilde{H}}_m(C_\bullet ) = H_m(C_\bullet )$ if $m \ge 1$ and ${\tilde{H}}_0(C_\bullet ) = H_0(C_\bullet \xrightarrow {\varepsilon } A) = \mathrm {Ker}\, \varepsilon / \mathrm {Im}\, \partial _1$.

4.1 Cosheaves

Cosheaves were introduced by Bredon [3]. Proofs of statements below about (pre)cosheaves, we refer to the book [3].

In this section, X always denotes a topological space. The set ${\mathsf {O}}(X)$ of all open subsets of X is regarded as a category in a usual way, that is, open sets $U \in {\mathsf {O}}(X)$ are objects and a morphism $U \rightarrow V$ uniquely exists if and only if $U \subset V$, for $U, V \in {\mathsf {O}}(X)$.

Let ${\mathsf {C}}$ denote one of the categories $\mathsf {Ab}$ or ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$. A precosheaf on X (of ${\mathsf {C}}$-valued) is a covariant functor ${\mathfrak {A}}$ from ${\mathsf {O}}(X)$ to ${\mathsf {C}}$. We will use only a precosheaf ${\mathfrak {A}}$ such that ${\mathfrak {A}}(\emptyset ) = 0$. For a precosheaf ${\mathfrak {A}}$ on X and $U, V \in {\mathsf {O}}(X)$, the morphism induced by $U \subset V$ is denoted by $i_{V,U} = i_{V,U}^{\,{\mathfrak {A}}}:{\mathfrak {A}}(U) \rightarrow {\mathfrak {A}}(V)$. A precosheaf ${\mathfrak {A}}$ is said to be flabby if $i_{X,U}$ is injective for every $U \in {\mathsf {O}}(X)$, or equivalently, $i_{V,U}$ is injective for every U and V in ${\mathsf {O}}(X)$ with $U \subset V$.

Definition 4.1

[3] A precosheaf ${\mathfrak {A}}$ on X is called a cosheaf if it satisfies the following: For any family of open sets ${\mathcal {U}} = \{U_\alpha \}_{\alpha \in A}$ of X, setting $U = \bigcup _{\alpha \in A} U_\alpha$, the short complex

$$\begin{aligned} \bigoplus _{\alpha ,\beta \in A} {\mathfrak {A}}(U_\alpha \cap U_\beta ) \xrightarrow {\Phi } \bigoplus _{\alpha \in A} {\mathfrak {A}}(U_\alpha ) \xrightarrow {\varepsilon } {\mathfrak {A}}(U) \rightarrow 0 \end{aligned}$$

(4.1)

is exact, where $\Phi = \sum _{\alpha , \beta } i_{U_\alpha , U_\alpha \cap U_\beta } - i_{U_\beta , U_\alpha \cap U_\beta }$ and $\varepsilon = \sum _\alpha i_{U,U_\alpha }$.

There is a useful characterization of precosheaves to be cosheaves.

Proposition 4.2

[3, Chapter VI, Proposition 1.4] A precosheaf ${\mathfrak {A}}$ on X is cosheaf if and only if it satisfies the following two conditions.

• (a) For any open sets $U,V \in {\mathsf {O}}(X),$ the short complex

  $$\begin{aligned} {\mathfrak {A}}(U \cap V) \xrightarrow {\Psi } {\mathfrak {A}}(U) \oplus {\mathfrak {A}}(V) \xrightarrow {\varepsilon } {\mathfrak {A}}(U \cup V) \rightarrow 0 \end{aligned}$$

  is exact,  where $\varepsilon = i_{U \cup V, U} + i_{U \cup V, V}$ and $\Psi = i_{U, U \cap V} - i_{V, U \cap V}$.

• (b) If a family $\{U_\alpha \}_{\alpha \in A}$ of open sets in X is directed,  that is,  for any $\alpha , \alpha ' \in A,$ there is $\alpha '' \in A$ such that $U_\alpha \cup U_{\alpha '} \subset U_{\alpha ''},$ then the map

  $$\begin{aligned} \varinjlim _{\alpha \in A} {\mathfrak {A}}(U_\alpha ) \rightarrow {\mathfrak {A}}\left( \bigcup _{\alpha \in A} U_\alpha \right) \end{aligned}$$

  is an isomorphism.

The short sequence (4.1) can be extended on the left side as a chain complex as follows. For $k \ge 1$, let us define a map

$$\begin{aligned} \Phi _k = \Phi _k^{{\mathcal {U}}, {\mathfrak {A}}}:\bigoplus _{\alpha _0,\ldots , \alpha _k \in A} {\mathfrak {A}}(U_{\alpha _0 \dots \alpha _k}) \rightarrow \bigoplus _{\alpha _0,\ldots , \alpha _{k-1} \in A} {\mathfrak {A}}(U_{\alpha _0 \dots \alpha _{k-1}}) \end{aligned}$$

associated to a family ${\mathcal {U}} = \{U_\alpha \}_{\alpha \in A}$ of open sets, by,

$$\begin{aligned} \Phi _k = \sum _{\alpha _0,\ldots , \alpha _k} \sum _{p=0}^k (-1)^p\, i_{{\hat{U}}_{\alpha _p}, U_{\alpha _0 \dots \alpha _k}} \end{aligned}$$

where $U_{\alpha _0 \dots \alpha _k}$ denotes the intersection $\bigcap _{p=0}^k U_{\alpha _p}$ and ${\hat{U}}_{\alpha _p}$ is $\bigcap _{j \ne p} U_{\alpha _j}$. Here, $\Phi _1^{{\mathcal {U}}, {\mathfrak {A}}}$ is nothing but $\Phi$ in (4.1). We set

$$\begin{aligned} {\check{C}}_k({\mathcal {U}}, {\mathfrak {A}}) := \bigoplus _{\alpha _0,\ldots , \alpha _k \in A} {\mathfrak {A}}(U_{\alpha _0 \dots \alpha _k}). \end{aligned}$$

The maps $(\Phi _k^{{\mathcal {U}}, {\mathfrak {A}}})_{k \ge 1}$ satisfies $\Phi _{k-1}^{{\mathcal {U}}, {\mathfrak {A}}} \circ \Phi _k^{{\mathcal {U}}, {\mathfrak {A}}} = 0$ for $k \ge 2$. So, the group

$$\begin{aligned} {\check{C}}_\bullet ({\mathcal {U}}, {\mathfrak {A}}) := \bigoplus _{k \ge 0} {\check{C}}_k({\mathcal {U}},{\mathfrak {A}}) \end{aligned}$$

becomes a chain complex with boundary map $\Phi = \Phi ^{{\mathcal {U}}, {\mathfrak {A}}} = (\Phi _k^{{\mathcal {U}},{\mathfrak {A}}})_{k \ge 1}$ and $\varepsilon :{\check{C}}_0({\mathcal {U}}, {\mathfrak {A}}) \rightarrow {\mathfrak {A}}(\bigcup _\alpha U_\alpha )$ is an augmentation of this complex.

Proposition 4.3

[3, Chapter VI, Corollary 4.5] Let ${\mathfrak {A}}$ be a flabby cosheaf on X. Then the sequence

$$\begin{aligned} \cdots \rightarrow {\check{C}}_k({\mathcal {U}}, {\mathfrak {A}}) \xrightarrow {\Phi _k} {\check{C}}_{k-1}({\mathcal {U}}, {\mathfrak {A}}) \rightarrow \cdots \xrightarrow {\Phi _1} {\check{C}}_0({\mathcal {U}}, {\mathfrak {A}}) \xrightarrow {\varepsilon } {\mathfrak {A}}(X) \rightarrow 0 \end{aligned}$$

is exact,  for any open covering ${\mathcal {U}}$ of X.

For two open coverings ${\mathcal {U}} = \{U_i\}_{i \in I}$ and ${\mathcal {V}} = \{V_j\}_{j \in J}$ of X, we say that ${\mathcal {V}}$ is a refinement of ${\mathcal {U}}$ if there is a map $\lambda :J \rightarrow I$ between index sets such that $V_j \subset U_{\lambda (j)}$ holds for every $j \in J$. Such a map $\lambda$ is called a refinement projection from ${\mathcal {V}}$ to ${\mathcal {U}}$, and is denoted by $\lambda :{\mathcal {V}} \prec {\mathcal {U}}$. The refinement projection $\lambda$ induces a map $\lambda _\#:{\check{C}}_k({\mathcal {V}}, {\mathfrak {A}}) \rightarrow {\check{C}}_k({\mathcal {U}},{\mathfrak {A}})$ defined by

$$\begin{aligned} \lambda _\#:{\mathfrak {A}}(V_{j_0 \dots j_k}) \rightarrow {\mathfrak {A}}(U_{\lambda (j_0)\dots \lambda (j_k)}) \end{aligned}$$

for each component of ${\check{C}}_k({\mathcal {V}}, {\mathfrak {A}})$. Actually, we have

Lemma 4.4

$\lambda _\#:{\check{C}}_\bullet ({\mathcal {V}},{\mathfrak {A}}) \rightarrow {\check{C}}_\bullet ({\mathcal {U}},{\mathfrak {A}})$ is a chain map.

Proof

Let us consider the following diagram

This diagram commutes from the definition. It implies the conclusion of the lemma. $\square$

4.2 Double complices

If ${\mathfrak {A}}$ is a precosheaf of ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$-valued, then we denote it by ${\mathfrak {A}}= {\mathfrak {A}}_\bullet$. In this case, ${\check{C}}_\bullet ({\mathcal {U}}, {\mathfrak {A}}_\bullet )$ becomes a double complex by the boundary maps $\Phi$ and $\partial$. Indeed, the following holds.

Lemma 4.5

We have $\partial \Phi = \Phi \partial ,$ where $\partial$ acts on each component of ${\check{C}}_p({\mathcal {U}}, {\mathfrak {A}}_\bullet ) = \bigoplus _{i_0,\ldots , i_p} {\mathfrak {A}}_\bullet (U_{i_0 \dots i_p})$ for all p.

Furthermore,  if $\lambda :{\mathcal {V}} \prec {\mathcal {U}}$ is a refinement projection,  then the map $\lambda _\#:{\check{C}}_\bullet ({\mathcal {V}}, {\mathfrak {A}}_\bullet ) \rightarrow {\check{C}}_\bullet ({\mathcal {U}}, {\mathfrak {A}}_\bullet )$ is a chain map of double complices.

Proof

By the definition, the following diagram

commutes. Here, ${\hat{U}}_{i_j} = \bigcap _{\ell \ne j} U_{i_\ell }$. The first statement follows from this and the definition of $\Phi$.

The second statement follows from Lemma 4.4 and a similar argument done there. $\square$

Now, let us consider an abstract double complex $A = A_{\bullet , \bullet } = \bigoplus _{i,j \ge 0} A_{i,j}$ of nonnegative degrees. We denote its boundary maps by

$$\begin{aligned} \Phi = \Phi _i = \Phi _{i,j}:A_{i,j} \rightarrow A_{i-1,j}\quad \text {and}\quad \partial = \partial _j = \partial _{i,j}:A_{i,j} \rightarrow A_{i,j-1}. \end{aligned}$$

Here, we set $\Phi _0 = 0$ and $\partial _0 = 0$. The total complex ${\mathcal {A}}_\bullet$ of A is defined by ${\mathcal {A}}_m = \bigoplus _{i+j=m} A_{i,j}$ together with the boundary map $\sum _{i+j=m} \Phi _i + (-1)^j \partial _j:{\mathcal {A}}_m \rightarrow {\mathcal {A}}_{m-1}$. Let $B = B_\bullet = \bigoplus _{j \ge 0} B_j$ be a chain complex with the boundary map $\partial ^B = \partial ^B_j:B_j \rightarrow B_{j-1}$. Suppose that there is a map $\varepsilon :A_{0,\bullet } \rightarrow B_\bullet$ consisting of morphisms $\varepsilon _j:A_{0,j} \rightarrow B_j$ such that $\varepsilon _j \partial _{j+1} = \partial _{j+1}^B \varepsilon _{j+1}$ and $\varepsilon _j \Phi _{1,j} = 0$ for every $j \ge 0$. Here, we set $\partial ^B_0 = 0$. This situation is presented as in the following diagram

In this case, a morphism $\varepsilon _*:H_m({\mathcal {A}}_\bullet ) \rightarrow H_m(B_\bullet )$ is defined in a canonical way. Then, the following is well-known. Actually, by a simple diagram chasing, one can easily get

Lemma 4.6

Let $A = (A_{i,j},\Phi ,\partial ),B = (B_j,\partial ^B)$ and $\varepsilon$ be as above. Let $m \ge 0$. If the sequence

$$\begin{aligned} A_{m-j,j} \xrightarrow {\Phi } A_{m-j-1,j} \xrightarrow {\Phi } \cdots \xrightarrow {\Phi } A_{0,j} \xrightarrow {\varepsilon } B_j \rightarrow 0 \end{aligned}$$

is exact,  for each $0 \le j \le m,$ then the map $\varepsilon _*:H_m({\mathcal {A}}_\bullet ) \rightarrow H_m(B_\bullet )$ is surjective. Namely, for any $c \in B_m$ with $\partial ^B c=0,$ there exist elements $c_{m-k,k} \in A_{m-k,k}$ for $0 \le k \le m$ satisfying

$$\begin{aligned} \varepsilon c_{0,m} = c \quad \text {and}\quad \Phi c_{m-k, k} = \partial c_{m-k-1, k+1} \end{aligned}$$

for every k with $0 \le k \le m-1$.

4.3 Local triviality of precosheaves

Let X be a space. A precosheaf ${\mathfrak {H}}$ on X is said to be locally trivial (or be locally zero) if for any $x \in X$ and $U \in {\mathsf {O}}(X)$ with $x \in U$, there is $V \in {\mathsf {O}}(X)$ with $x \in V \subset U$ such that the map $i_{V,U}^{\,{\mathfrak {H}}}:{\mathfrak {H}}(V) \rightarrow {\mathfrak {H}}(U)$ is trivial.

A topological space is said to be paracompact if any open covering of it admits a locally finite refinement. Recall that any metric space is paracompact.

Theorem 4.7

[3, Chapter VI, Theorem 4.6] Let X be a paracompact topological space. If ${\mathfrak {H}}$ is a locally trivial precosheaf on X,  then for any open covering ${\mathcal {U}} = \{U_i\}$ of X,  there is an open refinement ${\mathcal {V}} = \{V_j\}$ of ${\mathcal {U}}$ with a refinement projection $\lambda :{\mathcal {V}} \prec {\mathcal {U}}$ such that the map

$$\begin{aligned} {\mathfrak {H}}(V_{j_0 \dots j_m}) \rightarrow {\mathfrak {H}}(U_{\lambda (j_0) \dots \lambda (j_m)}) \end{aligned}$$

is trivial for every indices $j_0,\ldots , j_m$ of ${\mathcal {V}}$.

4.4 Local triviality of spaces

Let us denote by $\mathsf {Met}$ the category of metric spaces and locally Lipschitz maps and by $\mathsf {Top}$ the category of topological spaces and continuous maps. Let ${\mathsf {D}}$ be one of $\mathsf {Met}$ and $\mathsf {Top}$, and ${\mathsf {C}}$ denote one of $\mathsf {Ab}$ and ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$. Let us consider a covariant functor $H:{\mathsf {D}} \rightarrow {\mathsf {C}}$. Then, for each $X \in {\mathsf {D}}$, we obtain a precosheaf $H:{\mathsf {O}}(X) \rightarrow {\mathsf {C}}$ on X. We say that a space X is H-locally trivial if the precosheaf H on X is locally trivial. In this terminologies, the following holds.

Proposition 4.8

Let $H:{\mathsf {D}} \rightarrow {\mathsf {C}}$ be a covariant functor. The H-local triviality is inherited to open subsets.

Furthermore,  when ${\mathsf {D}} = \mathsf {Met},$ the H-local triviality is stable under locally bi-Lipschitz homeomorphisms.

Proof

Let $X \in {\mathsf {D}}$ and $X'$ an open subset of X. We assume that X is H-locally trivial. Let us take $x \in X'$ and an open neighborhood U of x in $X'$. Since U is open in X and X is H-locally trivial, there is $V \in {\mathsf {O}}(X)$ with $x \in V \subset U$ such that $H(\iota )=0$, where $\iota :V \rightarrow U$ is the inclusion. Since $V \in {\mathsf {O}}(X')$, we conclude that $X'$ is H-locally trivial.

Furthermore, we assume that X is a metric space and take another metric space Y. Let $f:X \rightarrow Y$ be a locally bi-Lipschitz homeomorphism. To prove the statement, we may assume that f is a bi-Lipschitz homeomorphism. Let us take $y \in Y$ and an open neighborhood V of y in Y. Set $x = f^{-1}(y) \in X$, $L = \max \{\mathrm {Lip}(f), \mathrm {Lip}(f^{-1})\}$. By the H-local triviality of X, we obtain $r > 0$ such that $U(x,r) \subset f^{-1}(V)$ and $H(\iota ) = 0$, where $\iota :U(x,r) \rightarrow f^{-1}(V)$ is the inclusion. Then, $U(y, L^{-1}r) \subset V$ and the inclusion $\iota ':U(y, L^{-1}r) \rightarrow V$ is decomposed as $\iota ' = f^{-1} \circ \iota \circ f$. Hence, we obtain $H(\iota ') = 0$. This completes the proof. $\square$

4.5 A way to compare homologies by using cosheaves

In this subsection, we prove the following important

Theorem 4.9

Let ${\mathfrak {A}}_\bullet$ and ${\mathfrak {A}}_\bullet '$ be flabby cosheaves on a paracompact topological space X of ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$-valued together with a natural transformation $\eta :{\mathfrak {A}}_\bullet \rightarrow {\mathfrak {A}}_\bullet '$. Let A and $A'$ be precosheaves on X of $\mathsf {Ab}$-valued together with natural transformations $\zeta :A \rightarrow A'$, $\xi :{\mathfrak {A}}_0 \rightarrow A$ and $\xi ':{\mathfrak {A}}_0' \rightarrow A'$ such that $\xi ' \eta = \zeta \xi$. Suppose that $\xi :{\mathfrak {A}}_\bullet \rightarrow A$ and $\xi ':{\mathfrak {A}}_\bullet ' \rightarrow A'$ are augmentations. Furthermore, we assume that $\eta :{\mathfrak {A}}_0(U) \rightarrow {\mathfrak {A}}_0'(U)$ is surjective and $\zeta :A(U) \rightarrow A'(U)$ is injective for each $U \in {\mathsf {O}}(X)$. Then,  the following holds.

• (1) If the precosheaves ${\tilde{H}}_p({\mathfrak {A}}_\bullet )$ are locally trivial on X for $0 \le p \le m$, then $\eta _*:H_m({\mathfrak {A}}_\bullet (U)) \rightarrow H_m({\mathfrak {A}}_\bullet '(U))$ is injective for every $U \in {\mathsf {O}}(X)$. Here,  ${\tilde{H}}_p({\mathfrak {A}}_\bullet )$ is the p-th homology of the augmented complex $\xi :{\mathfrak {A}}_\bullet \rightarrow A$.

• (2) If the precosheaves ${\tilde{H}}_p({\mathfrak {A}}_\bullet )$ and $H_q({\mathfrak {A}}_\bullet ')$ are locally trivial on X for $0 \le p \le m-1$ and $1 \le q \le m,$ then $\eta _*:H_m({\mathfrak {A}}_\bullet (U)) \rightarrow H_m({\mathfrak {A}}_\bullet '(U))$ is surjective for every $U \in {\mathsf {O}}(X)$.

Note that the surjectivity of $\eta$ implies that $H_0({\mathfrak {A}}_\bullet (U)) \rightarrow H_0({\mathfrak {A}}_\bullet '(U))$ is always surjective, for every open set $U \subset X$.

Proof

Let $X,{\mathfrak {A}}_\bullet ,{\mathfrak {A}}_\bullet ',A,A',\eta ,\zeta ,\xi ,\xi '$ be in the assumption. First, we assume that the precosheaves ${\tilde{H}}_p({\mathfrak {A}}_\bullet )$ are locally trivial on X for $0 \le p \le m$. We prove that $\eta _*:H_m({\mathfrak {A}}_\bullet (X)) \rightarrow H_m({\mathfrak {A}}_\bullet '(X))$ is injective. By Theorem 4.7, we obtain a sequence of open coverings $\{{\mathcal {U}}_p\}_{p=0}^m$ of X such that ${\mathcal {U}}_p$ is a refinement of ${\mathcal {U}}_{p+1}$ together with a refinement projection $\lambda _p:{\mathcal {U}}_p \prec {\mathcal {U}}_{p+1}$ and that

$$\begin{aligned} (\lambda _p)_*:{\tilde{H}}_p({\mathfrak {A}}_\bullet (U_{i_0\dots i_\ell })) \rightarrow {\tilde{H}}_p({\mathfrak {A}}_\bullet (\lambda _p(U_{i_0 \dots i_\ell }))) \end{aligned}$$

are trivial maps for all finite elements $U_{i_0},\ldots , U_{i_\ell } \in {\mathcal {U}}_p$, for each $p = 0,\ldots , m-1$. Here, we write $\lambda _p(U_{i_0 \dots i_\ell }) = \cap _{a=0}^\ell \lambda _p (U_{i_a})$. For ${\mathcal {U}}= {\mathcal {U}}_p$, we consider the following diagram.

Here, $\partial$ denotes the boundary map of the complex ${\mathfrak {A}}_\bullet$. Let us take $c \in {\mathfrak {A}}_m(X)$ with $\partial c = 0$. Since ${\mathfrak {A}}_\bullet$ is a flabby cosheaf, by Lemma 4.6, there are elements $c_{p,m-p} \in {\check{C}}_p({\mathcal {U}}_0,{\mathfrak {A}}_{m-p})$ such that

$$\begin{aligned} \varepsilon c_{0,m} = c \quad \text {and}\quad \partial c_{p,m-p} = \Phi c_{p+1,m-p-1} \end{aligned}$$

for every p with $1 \le p \le m-1$. Furthermore, we suppose that c satisfies $\partial ' c' = \eta c$ for some $c' \in {\mathfrak {A}}_{m+1}'(X)$, where $\partial '$ is the boundary map of ${\mathfrak {A}}_\bullet '$. Then, to prove that $\eta _*$ is injective, it suffices to show that there is an element ${\bar{c}} \in {\mathfrak {A}}_{m+1}(X)$ such that $\partial {\bar{c}} = c$. Since $\varepsilon :{\check{C}}_0({\mathcal {U}}_0, {\mathfrak {A}}_{m+1}') \rightarrow {\mathfrak {A}}_{m+1}'(X)$ is surjective, there exists $c_{0,m+1}' \in {\check{C}}_0({\mathcal {U}}_0, {\mathfrak {A}}_{m+1}')$ such that $\varepsilon c_{0,m+1}' = c'$. Then, we have

$$\begin{aligned} \varepsilon (\partial ' c_{0,m+1}' - \eta c_{0,m}) = 0. \end{aligned}$$

Therefore, there is an element $c_{1,m}' \in {\check{C}}_1({\mathcal {U}}_0, {\mathfrak {A}}_m')$ such that

$$\begin{aligned} \Phi ' c_{1,m}' = \partial ' c_{0,m+1}' - \eta c_{0,m}. \end{aligned}$$

So, we have

$$\begin{aligned} \Phi ' \partial ' c_{1,m}' = - \eta \partial c_{0,m} = -\eta \Phi c_{1,m-1} = - \Phi ' \eta c_{1,m-1}. \end{aligned}$$

Hence, there is an element $c_{2,m-1}' \in {\check{C}}_2({\mathcal {U}}_0, {\mathfrak {A}}_{m-1}')$ such that $\Phi ' c_{2,m-1}' = \partial ' c_{1,m}' + \eta c_{1,m-1}$. Repeating such a diagram chasing, we obtain elements $c_{p+1,m-p}' \in {\check{C}}_{p+1}({\mathcal {U}}_0, {\mathfrak {A}}_{m-p}')$ such that

$$\begin{aligned}&\Phi ' c_{p+1.m-p}' = \partial ' c_{p,m-p+1}' + (-1)^{p+1} \eta c_{p, m-p} \end{aligned}$$

for $1 \le p \le m$. Since $\eta :{\check{C}}_{m+1}({\mathcal {U}}_0,{\mathfrak {A}}_0) \rightarrow {\check{C}}_{m+1}({\mathcal {U}}_0,{\mathfrak {A}}_0')$ is surjective, there is a $c_{m+1,0} \in {\check{C}}_{m+1}({\mathcal {U}}_0,{\mathfrak {A}}_0)$ such that $\eta c_{m+1,0} = c_{m+1,0}'$. Then, we have

$$\begin{aligned} \zeta \xi \Phi c_{m+1,0} = (-1)^{m+1} \zeta \xi c_{m,0}. \end{aligned}$$

Since $\zeta$ is injective, we obtain

$$\begin{aligned} \xi \Phi c_{m+1,0} = (-1)^{m+1} \xi c_{m,0}. \end{aligned}$$

By the property of $\lambda _0:{\mathcal {U}}_0 \prec {\mathcal {U}}_1$, there is a $c_{m,1} \in {\check{C}}_m({\mathcal {U}}_1, {\mathfrak {A}}_1)$ such that

$$\begin{aligned} \partial c_{m,1} = (\lambda _0)_\# (\Phi c_{m+1,0} + (-1)^m c_{m,0}). \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \partial \Phi c_{m,1} = (\lambda _0)_\# (-1)^m \partial c_{m-1,1}. \end{aligned}$$

Hence, by the property of $\lambda _1$, we obtain $c_{m-1,2} \in {\check{C}}_{m-1}({\mathcal {U}}_2, {\mathfrak {A}}_2)$ such that

$$\begin{aligned} \partial c_{m-1,2} = (\lambda _1)_\# (\Phi c_{m,1} + (-1)^{m+1} (\lambda _0)_\# c_{m-1,1}) \end{aligned}$$

Repeating this argument, we have elements $c_{m-p,p+1} \in {\check{C}}_{m-p}({\mathcal {U}}_{p+1}, {\mathfrak {A}}_{p+1})$ satisfying

$$\begin{aligned} \partial c_{m-p,p+1} = (\lambda _p)_\# \Phi c_{m-p+1,p} + (-1)^{m+p} ({\tilde{\lambda }}_p)_\# c_{m-p,p} \end{aligned}$$

for $1 \le p \le m$, where ${\tilde{\lambda }}_p {}_\# = \lambda _p {}_\# \circ \cdots \circ \lambda _0 {}_\#$. Then, setting ${\bar{c}} = \varepsilon c_{0,m+1} \in {\mathfrak {A}}_{m+1}(X)$, we have

$$\begin{aligned} \partial {\bar{c}} = \varepsilon c_{0,m} = c. \end{aligned}$$

This implies that $\eta _*:{\tilde{H}}_m({\mathfrak {A}}_\bullet (X)) \rightarrow {\tilde{H}}_m({\mathfrak {A}}_\bullet '(X))$ is injective. That is, this completes the proof of (1).

Next, we prove that $\eta _*:H_m({\mathfrak {A}}_\bullet (X)) \rightarrow H_m({\mathfrak {A}}_\bullet '(X))$ is surjective, assuming the assumption of (2). By a thing mentioned at before starting the proof, we may assume that $m \ge 1$. Let us take $c' \in {\mathfrak {A}}_m'(X)$ with $\partial ' c' = 0$. Let ${\mathcal {W}}$ be an arbitrary open covering of X. Then, there are $c_{p,m-p}' \in {\check{C}}_p({\mathcal {W}},{\mathfrak {A}}_{m-p}')$ such that

$$\begin{aligned} \left\{ \begin{aligned} \varepsilon c_{0,m}'&= c' \\ \partial ' c_{p,m-p}'&= \Phi ' c_{p+1,m-p-1}' \end{aligned} \right. \end{aligned}$$

(4.2)

for all $1 \le p \le m-1$. Since $\eta :{\mathfrak {A}}_0 \rightarrow {\mathfrak {A}}_0'$ is surjective, there is a $c_{m,0} \in \check{C}_m({\mathcal {W}}, {\mathfrak {A}}_0)$ such that

$$\begin{aligned} \eta c_{m,0} = c_{m,0}'. \end{aligned}$$

(4.3)

By the assumption, we have

$$\begin{aligned} \zeta \xi \Phi c_{m,0} = \xi ' \eta \Phi c_{m,0} = \xi ' \Phi ' c_{m,0}' = \xi ' \partial ' c_{m-1,1}' = 0. \end{aligned}$$

Since $\zeta :A \rightarrow A'$ is injective, we conclude

$$\begin{aligned} \xi \Phi c_{m,0}= 0. \end{aligned}$$

(4.4)

Now, let us consider the following sequence of open coverings of X such that

$$\begin{aligned} {\mathcal {V}}_0 \prec {\mathcal {U}}_1 \prec {\mathcal {V}}_1 \prec \cdots \prec {\mathcal {V}}_{k-1} \prec {\mathcal {U}}_{k} \prec {\mathcal {V}}_{k} \prec \cdots \prec {\mathcal {V}}_{m-1} \prec {\mathcal {U}}_m. \end{aligned}$$

Here, the refinement projections are denoted by

$$\begin{aligned} \lambda _{k-1}:{\mathcal {V}}_{k-1} \prec {\mathcal {U}}_k \quad \text {and}\quad \mu _\ell :{\mathcal {U}}_\ell \prec {\mathcal {V}}_\ell \end{aligned}$$

for $1 \le k \le m$ and $1 \le \ell \le m-1$. By Theorem 4.7, we may assume that the induced maps

$$\begin{aligned} \begin{aligned}&(\lambda _{k-1})_*:{\tilde{H}}_{k-1}({\mathfrak {A}}_\bullet (V_{j_0 \dots j_p})) \rightarrow {\tilde{H}}_{k-1}({\mathfrak {A}}_\bullet (\lambda _{k-1}(V_{j_0 \dots j_p}))) \\&(\mu _\ell )_*:H_\ell ({\mathfrak {A}}_\bullet '(U_{i_0 \dots i_p})) \rightarrow H_\ell ({\mathfrak {A}}_\bullet '(\mu _k(U_{i_0 \dots i_p})) \end{aligned} \end{aligned}$$

(4.5)

are trivial, for all $p \ge 0$, $V_{j_0},\ldots , V_{j_p} \in {\mathcal {V}}_{k-1}$, $U_{i_0},\ldots , U_{i_p} \in {\mathcal {U}}_k$, $1 \le k \le m$ and $1 \le \ell \le m-1$. As seen above, we have a sequence $\{ c_{p,m-p}' \in {\check{C}}_p({\mathcal {V}}_0, {\mathfrak {A}}_{m-p}') \}_{0 \le p \le m}$ and an element $c_{m,0} \in {\check{C}}_m({\mathcal {V}}_0,{\mathfrak {A}}_m)$ satisfying (4.2)–(4.4). By (4.4) and the triviality of (4.5), there is an element $c_{m-1,1} \in {\check{C}}_{m-1}({\mathcal {U}}_1, {\mathfrak {A}}_1)$ such that

$$\begin{aligned} \partial c_{m-1,1} = \lambda _0{}_\# \Phi c_{m,0}. \end{aligned}$$

Then, we have

$$\begin{aligned} \partial \Phi c_{m-1,1}= & {} \lambda _0{}_\# \Phi ^2 c_{m,0} = 0, \\ \partial ' \eta c_{m-1,1}= & {} \lambda _0{}_\# \Phi ' c_{m,0}' = \partial ' \lambda _0{}_\# c_{m-1,1}'. \end{aligned}$$

By the second equality and the triviality of $(\mu _1)_*$, there is an element $c_{m-1,2}' \in {\check{C}}_{m-1}({\mathcal {V}}_1, {\mathfrak {A}}_2')$ such that

$$\begin{aligned} \partial ' c_{m-1,2}' = \mu _1{}_\# (\eta c_{m-1,1} - \lambda _0{}_\# c_{m-1,1}'). \end{aligned}$$

The rest equality and the triviality of $(\lambda _1)_*$ guarantee the existence of an element $c_{m-2,2} \in {\check{C}}_{m-2}({\mathcal {U}}_2, {\mathfrak {A}}_2)$ satisfying

$$\begin{aligned} \partial c_{m-2,2} = \lambda _1{}_\# \mu _1{}_\# \Phi c_{m-1,1}. \end{aligned}$$

Repeating such an argument, we obtain sequences of elements $c_{m-p,p+1}' \in {\check{C}}_{m-p}({\mathcal {V}}_p, {\mathfrak {A}}_{p+1})$ and $c_{m-p,p} \in {\check{C}}_{m-p}({\mathcal {U}}_p, {\mathfrak {A}}_p)$ such that

$$\begin{aligned} \partial ' c_{m-p,p+1}'= & {} \mu _p{}_\# (\eta c_{m-p,p}-\nu _{p-1} c_{m-p,p}'), \\ \partial c_{m-p,p}= & {} \lambda _{p-1}{}_\# \mu _{p-1} {}_\# \Phi c_{m-p+1,p-1} \end{aligned}$$

for all $2 \le p \le m$, where $\nu _{p-1} = \lambda _{p-1}{}_\# \mu _{p-1}{}_\# \cdots \lambda _1{}_\# \mu _1 {}_\# \lambda _0 {}_\#$. Let us consider elements $c := \varepsilon c_{0,m} \in {\mathfrak {A}}_m(X)$ and ${\bar{c}}' := \varepsilon c_{0,m+1}' \in {\mathfrak {A}}_{m+1}'(X)$. By the construction, they satisfy

$$\begin{aligned} \eta c = c' + \partial ' {\bar{c}}'. \end{aligned}$$

Therefore, we know that $\eta _*:H_m({\mathfrak {A}}_\bullet (X)) \rightarrow H_m({\mathfrak {A}}_\bullet '(X))$ is surjective. This completes the proof of Theorem 4.9. $\square$

4.6 Spaces of currents as cosheaves

Let X be a metric space. For each open set $U \in {\mathsf {O}}(X)$, we obtain chain complicies ${\mathbf {N}}_\bullet ^\mathrm {c}(U)$ and ${\mathbf {I}}_\bullet ^\mathrm {c}(U)$. Assignments $U \mapsto {\mathbf {N}}_\bullet ^\mathrm {c}(U)$ and $U \mapsto {\mathbf {I}}_\bullet ^\mathrm {c}(U)$ are precosheaves on X of ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$-valued, by the definition. We have

Lemma 4.10

The precosheaves ${\mathbf {N}}_\bullet ^\mathrm {c}$ and ${\mathbf {I}}_\bullet ^\mathrm {c}$ on X are actually flabby cosheaves.

Proof

Let us denote by ${\mathbf {C}}_k$ one of ${\mathbf {N}}_k^\mathrm {c}$ and ${\mathbf {I}}_k^\mathrm {c}$. By Lemma 2.12, the precosheaf ${\mathbf {C}}_k$ is flabby. By Lemma 3.14, we already know that ${\mathbf {C}}_\bullet$ satisfies the condition (a) in Proposition 4.2. We prove the condition (b) in Proposition 4.2. Let $\{U_\alpha \}$ be a directed family of open sets. Since ${\mathbf {C}}_k$ is flabby, the map ${\mathbf {C}}_k(U_\alpha ) \rightarrow {\mathbf {C}}_k(U)$ is injective for every $\alpha$, where $U = \bigcup _\alpha U_\alpha$. Because the functor of taking the direct limit is exact, the canonical map

$$\begin{aligned} \varinjlim {\mathbf {C}}_k(U_\alpha ) \rightarrow {\mathbf {C}}_k(U) \end{aligned}$$

is injective. Let us prove that this map is surjective. Let $T \in {\mathbf {C}}_k(U)$. Since T has a compact support and $\{U_\alpha \}$ is directed, there is an $\alpha$ such that $\mathrm {spt}(T) \subset U_\alpha$. By Lemma 3.5, T can be regarded as a current in $U_\alpha$. This implies that the considered map is surjective. This completes the proof. $\square$

4.7 Singular (Lipschitz) cosheaves

For a topological space X, the singular chain complex $S_\bullet$ of each open set gives a flabby precosheaf on X. In general, $S_\bullet$ is not a cosheaf. Nevertheless, taking subdivisions infinitely many times, we obtain a cosheaf on X as follows.

Example 4.11

[3] For each topological space X, let us consider a sequence of barycentric subdivisions

$$\begin{aligned} S_\bullet (X) \xrightarrow {\mathrm {Sd}} S_\bullet (X) \xrightarrow {\mathrm {Sd}} \cdots \xrightarrow {\mathrm {Sd}} S_\bullet (X) \xrightarrow {\mathrm {Sd}} \cdots \end{aligned}$$

and its direct limit, denoted by ${\mathfrak {S}}_\bullet (X)$. In this case, for degree k, the direct limit ${\mathfrak {S}}_k(X)$ is represented as the quotient group of $S_k(X)$ identifying c and $c'$ whenever $\mathrm {Sd}^m c = \mathrm {Sd}^{m'} c'$ for some $m, m' \ge 0$.

Then, a correspondence ${\mathsf {O}}(X) \ni U \mapsto {\mathfrak {S}}_\bullet (U) \in {\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$ is a flabby cosheaf on X. Indeed, because $S_\bullet$ is flabby and the direct limit $\varinjlim$ is an exact functor, ${\mathfrak {S}}_\bullet$ is flabby. To prove that ${\mathfrak {S}}_\bullet$ is a cosheaf, it suffices to check the properties (a) and (b) in Proposition 4.2. However, it is trivial, by the definition. Finally, noticing that the identity map on $S_\bullet$ and the subdivision are chain homotopy equivalent, we have a natural isomorphism

$$\begin{aligned} \eta _X:H_*(X) \cong H_*({\mathfrak {S}}_\bullet (X)). \end{aligned}$$

Here, the naturality means the functorial sense, that is, for a continuous map $f:X \rightarrow Y$ between topological spaces, the induced maps $f_*:H_*(X) \rightarrow H_*(Y)$ and $f_*:H_*({\mathfrak {S}}_\bullet (X)) \rightarrow H_*({\mathfrak {S}}_\bullet (Y))$ satisfy $\eta _Y f_*= f_*\eta _X$.

In Example 4.11, instead of a topological space and singular chains with a metric space and singular Lipschitz chains, respectively, we obtain a flabby cosheaf ${\mathfrak {S}}_\bullet ^\mathrm {Lip}$ on a metric space X of ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$-valued. That is, we set

$$\begin{aligned} {\mathfrak {S}}_\bullet ^\mathrm {Lip}(U) = \varinjlim \left( S_\bullet ^\mathrm {Lip}(U) \xrightarrow {\mathrm {Sd}} S_\bullet ^\mathrm {Lip}(U) \xrightarrow {\mathrm {Sd}} \cdots \right) \end{aligned}$$

for each $U \in {\mathsf {O}}(X)$. Here, we note that the subdivision preserves the Lipschitz-ness of singular chains. So, the map $\mathrm {Sd}:S_\bullet ^\mathrm {Lip}\rightarrow S_\bullet ^\mathrm {Lip}$ is well-defined. Furthermore, since the canonical chain homotopy equivalence maps between the identity and the subdivision on $S_\bullet$ also preserve the Lipschitz-ness of singular chains, they give chain homotopy equivalence between $S_\bullet ^\mathrm {Lip}$ and ${\mathfrak {S}}_\bullet ^\mathrm {Lip}$. Therefore, we have a natural isomorphism

$$\begin{aligned} H_*^\mathrm {Lip}(X) \cong H_*({\mathfrak {S}}_\bullet ^\mathrm {Lip}(X)) \end{aligned}$$

between their homologies.

Let us recall that the natural map $[\,\cdot \,] = [\,\cdot \,]_X:S_\bullet ^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_\bullet ^\mathrm {c}(X)$ for each metric space X was defined in Sect. 3.6. Obviously, we have $[\mathrm {Sd}\, c] = [c]$ for singular Lipschitz chain c. Hence, we can define a natural map

$$\begin{aligned}{}[\,\cdot \,]:{\mathfrak {S}}_\bullet ^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_\bullet ^\mathrm {c}(X). \end{aligned}$$

4.8 Local Lipschitz contractibility implying local triviality

We prove

Lemma 4.12

If a metric space X is locally Lipschitz contractible,  then it is H-locally trivial. Here,  H is one of the precosheaves $H_k^\mathrm {sing},$${\tilde{H}}_k^\mathrm {Lip},$$H_k^\mathrm {Lip}$ and $H_k^\mathrm {IC}$ for $k \ge 0$.

Recall that $H_k^\mathrm {Lip}= {\tilde{H}}_k^\mathrm {Lip}$ for $k \ge 1$, which are non-reduced homologies.

Proof

Since an LLC metric space X is locally contractible in the usual sense, the statement for $H = H_*^\mathrm {sing}$ holds. For another H, the statement follows from Lemmas 3.11, 3.17 and 2.13. $\square$

4.9 Cone inequalities implying H-locally triviality

Riedweg and Schäppi introduced the notion of metric spaces satisfying the cone inequalities. We translate this notion in terms of precosheaves.

Let X be a metric space and ${\mathfrak {C}}_\bullet = ({\mathfrak {C}}_\bullet , \partial )$ a flabby precosheaf on X of ${\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$-valued. Furthermore, A is an $\mathsf {Ab}$-valued precosheaf on X together with a natural transformation $\varepsilon :{\mathfrak {C}}_0 \rightarrow A$ such that $\varepsilon \partial = 0$. That is, $\varepsilon :{\mathfrak {C}}_\bullet \rightarrow A$ is an augmentation. The map $\varepsilon$ may be trivial. Moreover, we suppose the following.

• (1) For any $j \ge 0$, $c \in {\mathfrak {C}}_j(X)$, there is a unique compact set K(c) such that for every $V \in {\mathsf {O}}(X)$ with $K(c) \subset V$, there is an element $c' \in {\mathfrak {C}}_j(V)$ such that $i_\# c' = c$. Here, $i:V \rightarrow U$ is the inclusion and $i_\# = {\mathfrak {C}}_\bullet (i)$ denotes the induced map;

• (2) For any $j \ge 1$, $U \in {\mathsf {O}}(X)$, $c \in {\mathfrak {C}}_j(U)$, we have $K(\partial c) \subset K(c)$.

For instance, $S_\bullet$ and $S_\bullet ^\mathrm {Lip}$ satisfy these conditions for the canonical augmentations. In these cases, K(c) is the image of a singular chain c. Furthermore, ${\mathbf {N}}_\bullet ^\mathrm {c}$ and ${\mathbf {I}}_\bullet ^\mathrm {c}$ also satisfy the conditions, for augmentations $\varepsilon :{\mathbf {N}}_0^\mathrm {c}(X) \rightarrow {\mathbb {R}}$ and $\varepsilon :{\mathbf {I}}_0^\mathrm {c}(X) \rightarrow {\mathbb {Z}}$ defined by $\varepsilon (T) = T(1)$. In these cases, K(S) is the support of a current S.

Definition 4.13

[14] Let $X, {\mathfrak {C}}_\bullet , A, \varepsilon$ be as above. We say that X admits the cone inequality for${\mathfrak {C}}_j$ if for any $x \in X$, there exist $r > 0$ and a continuous non-decreasing function $F:[0,\infty ) \rightarrow [0,\infty )$ with $F(0) = 0$ such that

• when $j \ge 1$, for every $c \in {\mathfrak {C}}_j(X)$ with $K(c) \subset U(x,r)$ and $\partial c = 0$, there is $c' \in {\mathfrak {C}}_{j+1}(X)$ such that $\partial c' = c$ with $\mathrm {diam}\, K(c') \le F(\mathrm {diam}\, K(c))$;

• when $j = 0$, for every $c \in {\mathfrak {C}}_0(X)$ with $K(c) \subset U(x,r)$ and $\varepsilon c = 0$, there is $c' \in {\mathfrak {C}}_1(X)$ such that $\partial c' = c$ with $\mathrm {diam}\, K(c') \le F(\mathrm {diam}\, K(c))$.

Lemma 4.14

If X admits the cone inequality for ${\mathfrak {C}}_j,$ then the precosheaf ${\tilde{H}}_j({\mathfrak {C}}_\bullet )$ is locally trivial. Here,  ${\tilde{H}}_j({\mathfrak {C}}_\bullet )$ denotes the j-th homology of the augmented complex $\cdots \xrightarrow {\partial } {\mathfrak {C}}_j \xrightarrow {\partial } {\mathfrak {C}}_{j-1} \xrightarrow {\partial } \cdots \xrightarrow {\partial } {\mathfrak {C}}_0 \xrightarrow {\varepsilon } A$.

Proof

Let X admit a local cone inequality for ${\mathfrak {C}}_j$. Then, for $x \in X$, there exists $r > 0$ and $F:[0,\infty ) \rightarrow [0,\infty )$ satisfying the condition written above Definition 4.13. For any $s \in (0,r)$, we choose $s' \in (0,s)$ with

$$\begin{aligned} s' + F(2 s') < s. \end{aligned}$$

Let us take any $c \in {\mathfrak {C}}_j(X)$ with $K(c) \subset U(x,s')$. Such a c is considered as an element in ${\mathfrak {C}}_j(U(x,s'))$, because ${\mathfrak {C}}_j$ is flabby and it satisfies the condition (1). We suppose that $\partial c = 0$ when $j \ge 1$ and that $\varepsilon c = 0$ when $j = 0$. Since $s'< s < r$, there exists $c' \in {\mathfrak {C}}_{j+1}(X)$ such that $\partial c' = c$ and

$$\begin{aligned} \mathrm {diam}\, \,K(c') \le F(\mathrm {diam}\, K(c)). \end{aligned}$$

For any $y \in K(c) \subset K(c')$ and $z \in K(c')$, we have

$$\begin{aligned} d(x,z) \le d(x,y) + d(y,z)< s' + F(2 s') < s. \end{aligned}$$

Hence, $K(c') \subset U(x,s)$. So, the $c'$ can be regarded as an element in ${\mathfrak {C}}_{j+1}(U(x,s))$. Therefore, the morphism

$$\begin{aligned} {\tilde{H}}_j({\mathfrak {C}}_\bullet (U(x,s'))) \rightarrow {\tilde{H}}_j({\mathfrak {C}}_\bullet (U(x,s))) \end{aligned}$$

is trivial. This completes the proof. $\square$

Riedweg and Schäppi claimed

Theorem 4.15

[14, Theorems 3.2 and 3.4] If a metric space X admits the cone inequalities for $H_k,$${\tilde{H}}_k^\mathrm {Lip}$ and $H_k^\mathrm {IC},$ for all k with $0 \le k \le m,$ then the canonical maps $H_m(X) \leftarrow H_m^\mathrm {Lip}(X) \rightarrow H_m^\mathrm {IC}(X)$ are isomorphisms.

By Lemmas 4.12 and 4.14, our Theorem 4.9 is a generalization of Theorem 4.15 in terms of local triviality.

4.10 Proof of Theorem 1.3

By above preparations, we immediately get a proof of our main theorem.

Proof of Theorem 1.3

Let X be an LLC metric space. By Lemma 4.12, all the precosheaves $H_k^\mathrm {sing}$, ${\tilde{H}}_k^\mathrm {Lip}$ and $H_k^\mathrm {IC}$ on X are locally trivial, for every $k \ge 0$. By Lemma 3.15, the chain map $[\,\cdot \,]:S_0^\mathrm {Lip}(X) \rightarrow {\mathbf {I}}_0^\mathrm {c}(X)$ is an isomorphism. Due to Sect. 4.7, the functor ${\mathfrak {S}}_\bullet ^\mathrm {Lip}:{\mathsf {O}}(X) \rightarrow {\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$ is a flabby cosheaf and ${\mathfrak {S}}_0^\mathrm {Lip}= S_0^\mathrm {Lip}$ by the definition. Therefore, Theorem 1.3 follows from those things and Lemmas 3.16 and 4.10 and Theorem 4.9. $\square$

Proof of Corollary 1.4

Let H denote one of $H_*^\mathrm {Lip}$ and $H_*^\mathrm {IC}$. Let us set $\iota :H \rightarrow H_*^\mathrm {sing}$ the natural isomorphism obtained in Theorem 1.3. Let X and Y be LLC metric spaces, and $f:X \rightarrow Y$ a continuous map. Then, we define a homomorphism $H(f):H(X) \rightarrow H(Y)$ by $H(f) = \iota _Y^{-1} \circ H_*^\mathrm {sing}(f) \circ \iota _X$. Furthermore, for another continuous map $g:Y \rightarrow Z$ to an LLC metric space Z, we obtain

$$\begin{aligned} H(g) \circ H(f)= & {} \iota _Z^{-1} \circ H_*^\mathrm {sing}(g) \circ \iota _Y \circ \iota _Y^{-1} \circ H_*^\mathrm {sing}(f) \circ \iota _X \\= & {} \iota _Z^{-1} \circ H_*^\mathrm {sing}(g) \circ H_*^\mathrm {sing}(f) \circ \iota _X \\= & {} \iota _Z^{-1} \circ H_*^\mathrm {sing}(g \circ f) \circ \iota _X \\= & {} H(g \circ f). \end{aligned}$$

This shows that H is extended as a functor on the category of LLC metric spaces and continuous maps such that H is naturally isomorphic to the functor $H_*^\mathrm {sing}$. Furthermore, if $h:X \times [0,1] \rightarrow Y$ is a continuous homotopy, then we have

$$\begin{aligned} H(h_0) = \iota _Y^{-1} \circ H_*^\mathrm {sing}(h_0) \circ \iota _X = \iota _Y^{-1} \circ H_*^\mathrm {sing}(h_1) \circ \iota _X = H(h_1). \end{aligned}$$

This implies the homotopy invariance of H. This completes the proof of Corollary 1.4. $\square$

5 Several remarks

5.1 Relative homologies

A relative version of the singular Lipschitz homology is defined in a similar way to define the relative singular homology. For a subset A in a metric space X, the inclusion $i:A \rightarrow X$ induces an injective morphism $i_\#:S_\bullet ^\mathrm {Lip}(A) \rightarrow S_\bullet ^\mathrm {Lip}(X)$. So, we have a chain complex $S_\bullet ^\mathrm {Lip}(X,A)$ as the quotient of $S_\bullet ^\mathrm {Lip}(X)$ modulo $S_\bullet ^\mathrm {Lip}(A)$. Its homology $H_*(S_\bullet ^\mathrm {Lip}(X,A))$ is called the relative singular Lipschitz homology and is denoted by $H_*^\mathrm {Lip}(X,A)$.

Let (X, A) be as above. The pushforward $i_\#:{\mathbf {I}}_\bullet ^\mathrm {c}(A) \rightarrow {\mathbf {I}}_\bullet ^\mathrm {c}(X)$ is injective by Lemma 2.12. Hence, we obtain a chain complex ${\mathbf {I}}_\bullet ^\mathrm {c}(X,A) = {\mathbf {I}}_\bullet (X) / i_\# {\mathbf {I}}_\bullet ^\mathrm {c}(A)$. Its homology $H_*({\mathbf {I}}_\bullet ^\mathrm {c}(X,A))$ is called the relative homology of integral currents with compact support and is denoted by $H_*^\mathrm {IC}(X,A)$.

Let (Y, B) be another pair of metric spaces. A map f from (X, A) to (Y, B) is a map $f:X \rightarrow Y$ with $f(A) \subset B$. We say that a map $f:(X,A) \rightarrow (Y,B)$ is (locally) Lipschitz, if $f:X \rightarrow Y$ is (locally) Lipschitz. Obviously, all pairs of metric spaces and all locally Lipschitz maps give a category. We also have

Theorem 5.1

On the category of all pairs of LLC metric spaces and all locally Lipschitz maps,  the functors $H_*^\mathrm {sing},$$H_*^\mathrm {Lip}$ and $H_*^\mathrm {IC}$ are naturally isomorphic,  where $H_*^\mathrm {sing}$ denotes the usual relative singular homology.

In particular,  $H_*^\mathrm {Lip}$ and $H_*^\mathrm {IC}$ can be extended functors on the category of pairs of LLC spaces and continuous maps.

Proof

This follows from Theorem 1.3 and the five lemma. $\square$

5.2 Reduced homology of metric currents

Let X be a metric space. We consider a map

$$\begin{aligned} \varepsilon :{\mathbf {I}}_\bullet ^\mathrm {c}(X) \ni T \mapsto T(1) \in {\mathbb {Z}}. \end{aligned}$$

This is an augmentation of the complex ${\mathbf {I}}_\bullet ^\mathrm {c}(X)$. Actually, for $S \in {\mathbf {I}}_1^\mathrm {c}(X)$, we have

$$\begin{aligned} \varepsilon \partial S = S(1,1) = 0. \end{aligned}$$

So, we obtain the reduced homology of ${\mathbf {I}}_\bullet ^\mathrm {c}(X)$ augmented by $\varepsilon$, which is denoted by ${\tilde{H}}_*^\mathrm {IC}(X)$. As Theorem 1.3, we have

Theorem 5.2

${\tilde{H}}_*^\mathrm {IC}$ is actually a functor on the category of LLC metric spaces and locally Lipschitz maps. On that category,  the functors ${\tilde{H}}_*^\mathrm {sing},$${\tilde{H}}_*^\mathrm {Lip}$ and ${\tilde{H}}_*^\mathrm {IC}$ are naturally isomorphic. Here,  ${\tilde{H}}_*^\mathrm {sing}$ denotes the usual reduced singular homology.

In particular,  ${\tilde{H}}_*^\mathrm {Lip}$ and ${\tilde{H}}_*^\mathrm {IC}$ can be extended to functors on the category of LLC metric spaces and continuous maps.

Proof

Let $X_0$ be a set of a single point. Then, all the homologies ${\tilde{H}}_*^\mathrm {sing}(X)$, ${\tilde{H}}_*^\mathrm {Lip}(X)$ and ${\tilde{H}}_*^\mathrm {IC}(X)$ are represented as the kernels of $\pi _*$ between correspondence non-reduced homologies induced by the canonical map $\pi :X \rightarrow X_0$. By the naturality of the non-reduced homologies (Theorem 1.3), we obtain the conclusion of the theorem. $\square$

5.3 A remark on the axiom of finite mass

Remark 5.3

Our definition (Definition 3.1) and the original definition given in [1] of currents are slightly different. The main different point is the finite mass axiom. Furthermore, in [1], it was supposed that all sets satisfy some set-theoretical axiom about the cardinalities.

The original metric currents were defined only on complete metric spaces assuming the set-theoretical axiom [1]. The set-theoretical axiom implies that any finite Borel measure on every complete metric space is automatically tight. Therefore, the original definition did not impose that the mass measures of currents are tight. On the other hands, the LLC-condition is an open property (Proposition 2.4). Therefore, if we employ the original definition of metric currents, then an area which is applicable to our theory is very small. For instance, if a metric space is complete and LLC, then its open set is LLC, but is not complete, in general. Furthermore, we want to ignore an additional set-theoretical axiom.

Fortunately, as mentioned in [1], if one deal with only metric currents having tight mass measures, then such currents satisfy all the same results obtained there, further, they can be defined on all metric spaces without the set-theoretical axiom. This is the reason why we used currents with tight mass measures.

5.4 Alexandrov spaces revisited

As mentioned in Sect. 2.2, any finite dimensional Alexandrov space is SLLC. The proof of it was based on the theory of gradient flows of distance functions founded by Perelman and Petrunin [11, 13]. Actually, in [8], we proved that any point x in an Alexandrov space X has a positive number r such that the distance function d from the metric sphere S(x, 2r) centered at x of radius 2r is regular on $U(x,r) {\setminus } \{x\}$ and further that the absolute gradient $|\nabla d|$ is uniformly bounded on $U(x,r) {\setminus } \{x\}$. Here, $|\nabla d|(y) = \limsup _{z \rightarrow y} \frac{|d(z)-d(y)|}{d(z,y)}$. Then, the gradient flow of d gives a strong Lipschitz contraction from U(x, r) to x. On the other hands, extremal subsets of X, introduced by Perelman and Petrunin [12], have well-behavior in the gradient flows of distance functions. Indeed, extremal subsets are characterized by the property that they are preserved under the gradient flow of any distance functions (see [13]). This fact and the proof of the main result in [8] implies

Theorem 5.4

Any extremal subset in an Alexandrov space is strongly locally Lipschitz contractible. In particular,  the boundary of an Alexandrov space is strongly locally Lipschitz contractible.

Due to Theorems 5.1 and 5.4, we have

Corollary 5.5

Let X be a finite dimensional Alexandrov space and E a its subset. Suppose that E belongs to one of three classes of sets in the following :  open subsets,  discrete subsets,  and extremal subsets. Then,  we have natural isomorphisms $H_*(X,E) \cong H_*^\mathrm {Lip}(X,E) \cong H_*^\mathrm {IC}(X,E)$.

5.5 An LLC space not having the homotopy type of CW-complices

This subsection is devoted to prove

Theorem 5.6

(cf. [2, 15]) There is an LLC metric space such that it does not have the homotopy type of CW-complices.

Indeed, a space satisfying the topological property written in Theorem 5.6 was constructed by Borsuk [2]. We prove that such a space can admit an LLC metric. We also refer Chapter 6 of the book [15] for the construction and recall terminologies used there.

A metrizable space X is called an ANR (absolute neighborhood retract) if it is a neighborhood retract of an arbitrary metrizable space that contains X as a closed subset. Here, a closed subset A of a space Y is called a neighborhood retract if there exist a neighborhood U of A and a continuous map $r:U \rightarrow A$ such that $r |_A = \mathrm {id}_A$. For an open covering ${\mathcal {V}}$ of a space X, we say that two maps $f,g:Y \rightarrow X$ from a space Y are ${\mathcal {V}}$-close if for any $y \in Y$, there is a $V \in {\mathcal {V}}$ such that $f(y), g(y) \in V$. Let ${\mathcal {U}}$ be an open covering of X such that ${\mathcal {V}}$ is a refinement of ${\mathcal {U}}$. We say that ${\mathcal {V}}$ is an h-refinement of ${\mathcal {U}}$ if any two ${\mathcal {V}}$-close continuous maps $f,g:Y \rightarrow X$ from a metrizable space Y are ${\mathcal {U}}$-homotopic. Here, f and g are ${\mathcal {U}}$-homotopic if there is a continuous map $h:X \times [0,1] \rightarrow Y$ such that $h_0=f$, $h_1=g$ and that for any $x \in X$, there is a $U \in {\mathcal {U}}$ such that $h (\{x\} \times [0,1]) \in U$.

Lemma 5.7

Let X be a metrizable space. If the open cover $\{X\}$ consisting of only X has no h-refinement,  then X does not have the homotopy type of absolute neighborhood retracts. In particular,  X does not have the homotopy type of CW-complicies.

Proof

We suppose that there is an ANR Y such that X and Y are homotopic. Let $\phi :X \rightarrow Y$ be a homotopy equivalence. By Corollary 6.3.5 in [15], the cover $\{Y\}$ of Y has an h-refinement ${\mathcal {U}}$. Let us set ${\mathcal {V}} = \{\phi ^{-1}(U) \mid U \in {\mathcal {U}}\}$. Then, ${\mathcal {V}}$ is an h-refinement of $\{X\}$, which contradicts to the assumption. Indeed, we take ${\mathcal {V}}$-close maps $f,g:Z \rightarrow X$. Then, $\phi \circ f$ and $\phi \circ g$ are ${\mathcal {U}}$-close. Since ${\mathcal {U}}$ is an h-refinement of $\{Y\}$, there is a homotopy $h:Z \times [0,1] \rightarrow Y$ such that $h_0 = \phi \circ f$ and $h_1 = \phi \circ g$. Let $\psi :Y \rightarrow X$ be a homotopy inverse of $\phi$. By using a homotopy $\psi \circ h$ between $\psi \circ \phi \circ f$ and $\psi \circ \phi \circ g$, we obtain a homotopy between f and g. This completes the proof of the first statement. Since every CW-complex is an ANR, the latter statement follows. $\square$

For a metric space U and its subset $U_1$, we say that $U_1$ is a Lipschitz deformation retract ofU if there exists a Lipschitz homotopy $h:U \times [0,1] \rightarrow U$ such that $h_0(x)=x$, $h_1(x) \in U_1$ and $h_1(y) = y$ for every $x \in U$ and $y \in U_1$. Such a map h is called a Lipschitz deformation retraction fromUto$U_1$. A Lipschitz contraction is a special Lipschitz deformation retraction.

Lemma 5.8

Let V be a metric space and $V_1$ and $V_0$ its subsets. Suppose that $V_0 \subset V_1 \subset V$ and that $V_0$ is a Lipschitz deformation retract of $V_1$ and $V_1$ is a Lipschitz deformation retract of V. Then,  $V_0$ is a Lipschitz deformation retract of V.

Proof

Let us take Lipschitz deformation retractions h from V to $V_1$ and g from $V_1$ to $V_0$. Then, we define a map $k:V \times [0,1] \rightarrow V$ by

$$\begin{aligned} k(x,t) = \left\{ \begin{array}{ll} h(x,2 t) &{}\quad \text {if } t \le 1/ 2, \\ g(h_1(x), 2 t -1) &{}\quad \text {if } t \ge 1/2. \end{array} \right. \end{aligned}$$

This map is well-defined. Furthermore, it is Lipschitz. Indeed, for $x, y \in V$ and for $t,s \in [0,1]$, we have

$$\begin{aligned} d(k(x,t),k(y,t))\le & {} \max \{\mathrm {Lip}(g),1\} \max \{\mathrm {Lip}(h),1\} d(x,y), \\ d(k(x,t),k(x,s))\le & {} 2 \max \{\mathrm {Lip}(h), \mathrm {Lip}(g)\} |s-t|. \end{aligned}$$

By the construction, k is a deformation retraction from V to $V_0$ in the usual sense. Therefore, $V_0$ is a Lipschitz deformation retract of V. This completes the proof of the lemma. $\square$

Proof of Theorem 5.6

Let $\ell _2$ denote the standard Hilbert space of countably infinite dimension. Let Q be a subset of $\ell _2$ defined by

$$\begin{aligned} Q = \{(x_k)_{k =0}^\infty \in \ell _2 \mid 0 \le x_k \le 2^{-k} \}. \end{aligned}$$

We consider the following spaces.

$$\begin{aligned} X_0= & {} \{(x_k) \in Q \mid x_0 = 0 \}, \\ C_n= & {} \{(x_k) \in Q \mid (n+1)^{-1} \le x_0 \le n^{-1}, x_k = 0 \text { for } k > n \}, \\ X_n= & {} \partial C_n (\text {the boundary }n\text {-sphere of the }(n+1)\text {-cube } C_n), \end{aligned}$$

where $n \ge 1$. Then, we prove that $X = \bigcup _{n=0}^\infty X_n$ is the desired space. By the construction, this space is homeomorphic to the space in Theorem 6.3.8 in [15]. Therefore, the cover $\{X\}$ has no h-refinement. By Lemma 5.7, X does not have the homotopy type of CW-complices.

We prove that X is LLC. Since an open set $\bigcup _{n=1}^\infty X_n = X {\setminus } X_0$ in X is a locally finite simplicial complex, it is SLLC. Let $x \in X_0$. We denote by $p_m:X \rightarrow \prod _{k=0}^m [0,2^{-k}]$ the projection into the first $(m+1)$-coordinates. For any $r > 0$, there exist an $m \ge 1$ and a convex neighborhood W of $(x_1,\ldots , x_m)$ in $\prod _{k=1}^m [0,2^{-k}]$ such that $p_m^{-1}([0,m^{-1}] \times W) \subset U(x,r) \cap X$, where U(x, r) is the open ball in $\ell _2$ centered at x of radius r. Indeed, for $y \in p_m^{-1}([0,m^{-1}] \times W)$, we have

$$\begin{aligned} \Vert x-y\Vert _{\ell _2}^2 \le m^{-2} + \sum _{k=1}^m |x_k - y_k|^2 + \sum _{k>m} 4^{-k}. \end{aligned}$$

Hence, we can have such an m and a W. Then, we define a neighborhood V of x in X by

$$\begin{aligned} V = \left\{ \begin{array}{ll} p_{m+1}^{-1}([0,m^{-1}] \times W \times [0,2^{-(m+1)})) &{}\quad \text {if } x_{m+1} \le 4^{-(m+1)}, \\ p_{m+1}^{-1}([0,m^{-1}] \times W \times (0,2^{-(m+1)}]) &{}\quad \text {if } x_{m+1} > 4^{-(m+1)}. \end{array} \right. \end{aligned}$$

Furthermore, we consider the following sets.

$$\begin{aligned} V_0= & {} \left\{ \begin{array}{ll} p_{m+1}^{-1}(\{0\} \times W \times \{0\}) &{}\quad \text {if } x_{m+1} \le 4^{-(m+1)}, \\ p_{m+1}^{-1}(\{0\} \times W \times \{2^{-(m+1)}\}) &{}\quad \text {if } x_{m+1}> 4^{-(m+1)}, \end{array} \right. \\ V_1= & {} \left\{ \begin{array}{ll} p_{m+1}^{-1}([0,m^{-1}] \times W \times \{0\}) &{}\quad \text {if } x_{m+1} \le 4^{-(m+1)}, \\ p_{m+1}^{-1}([0,m^{-1}] \times W \times \{2^{-(m+1)}\}) &{}\quad \text {if } x_{m+1} > 4^{-(m+1)}. \end{array} \right. \end{aligned}$$

Note that all the sets V, $V_1$ and $V_0$ are contained in $U(x,r) \cap X$. As written in [15], $V_1$ is a strong deformation retract of V by a deformation $h:V \times [0,1] \rightarrow V$ sliding along the $(m+2)$-th coordinate, and $V_0$ is a strong deformation retract of $V_1$ by a deformation $g:V_1 \times [0,1] \rightarrow V_1$ sliding along the first coordinate. From the constructions, h and g are Lipschitz homotopies. Furthermore, $V_0$ is Lipschitz contractible in itself. Hence, V is Lipschitz contractible in V to some point, by Lemma 5.8. Therefore, we conclude that X is WLLC. By Lemma 2.3, X is LLC. This completes the proof of Theorem 5.6. $\square$

5.6 Remark on the homology of normal currents

By Lemma 4.10, the functor ${\mathbf {N}}_\bullet ^\mathrm {c}:{\mathsf {O}}(X) \rightarrow {\mathsf {C}}\mathrm {(}\mathsf {Ab}\mathrm {)}$ is known to be a flabby cosheaf on each metric space X. Hence, we may apply Theorem 4.9 to this cosheaf ${\mathbf {N}}_\bullet ^\mathrm {c}$. Now, we note that the space ${\mathbf {N}}_0^\mathrm {c}(X)$ is identified with the space ${\mathcal {M}}(X)$ of all finite signed Borel measures on X with compact support. Therefore, when X is LLC, we can guess that the homology $H_*({\mathbf {N}}_\bullet ^\mathrm {c}(X))$ coincides with the homology of some chain complex ${\mathbf {C}}_\bullet (X)$ such that its 0-th group ${\mathbf {C}}_0(X)$ is ${\mathcal {M}}(X)$. Furthermore, the homology $H_*({\mathbf {C}}_\bullet (X))$ should be a topological invariant. Such a chain complex actually exists, called the measure chain complex, introduced by Thurston [16]. However, there is no canonical map between ${\mathbf {C}}_\bullet (X)$ and ${\mathbf {N}}_\bullet ^\mathrm {c}(X)$. We discuss such a difficult point in another paper.

5.7 Localizations

Lemma 5.9

Let $A_\bullet$ and $B_\bullet$ be chain complices of indexed by integers and $f:A_\bullet \rightarrow B_\bullet$ a chain map. Suppose that $H_m(f): H_m(A_\bullet ) \rightarrow H_m(B_\bullet )$ is surjective and $H_{m-1}(f): H_{m-1}(A_\bullet ) \rightarrow H_{m-1}(B_\bullet )$ is injective. Then,  for any $b \in B_m$ and $a \in A_{m-1}$ with $\partial b = f(a)$ and $\partial a = 0,$ there are ${\bar{a}} \in A_m$ and ${\bar{b}} \in B_{m+1}$ such that $\partial {\bar{b}} = b+f({\bar{a}})$.

Proof

Let us take $b \in B_m$ and $a \in A_{m-1}$ with $\partial b = f(a)$ and $\partial a = 0$. Since $H_{m-1}(f)$ is injective, there is ${\bar{a}} \in A_m$ such that $\partial {\bar{a}} = a$. Then, we have $\partial (b- f({\bar{a}}))=0$. Since $H_m(f)$ is surjective, there exist $\bar{{\bar{a}}} \in A_m$ and ${\bar{b}} \in B_{m+1}$ such that $\partial \bar{{\bar{a}}}=0$ and $f(\bar{{\bar{a}}}) + f({\bar{a}}) = b + \partial {\bar{b}}$. This completes the proof. $\square$

As a corollary to the proof of Theorem 4.9, we have

Corollary 5.10

Let $m \ge 1$ and $\epsilon > 0$. Let X be a metric space which is ${\tilde{H}}_j^\mathrm {Lip}$-locally trivial for $0 \le j \le m-1$.

• (1) If X is $H_k^\mathrm {sing}$-locally trivial for $1 \le k \le m,$ then for $c \in S_m(X)$ with $\partial c \in S_{m-1}^\mathrm {Lip}(X),$ there exist finitely many elements $c_1,\ldots , c_N \in S_m(X),$$c_1^L,\ldots , c_N^L \in S_m^\mathrm {Lip}(X)$ and ${\bar{c}}_1,\ldots , {\bar{c}}_N \in S_{m+1}(X)$ such that

  • (1-a) there is $n \ge 0$ such that $\sum _i c_i = {\mathrm {sd}}^n (c)$ and $\partial {\bar{c}}_i = c_i - c_i^L;$

  • (2-a) $\mathrm {im}({\bar{c}}_i) \subset U(\mathrm {im}(c), \epsilon )$ and $\mathrm {diam}\, \mathrm {im}({\bar{c}}_i) < \epsilon$.

• (2) If X is $H_k^\mathrm {IC}$-locally trivial for $1 \le k \le m,$ then for $T \in {\mathbf {I}}_m^\mathrm {c}(X)$ and for $c \in S_{m-1}^\mathrm {Lip}(X)$ satisfying $\partial T = [c]$ and $\partial c = 0,$ there exist finitely many elements $T_1,\ldots , T_N \in {\mathbf {I}}_m^\mathrm {c}(X),$$c_1^L,\ldots , c_N^L \in S_m^\mathrm {Lip}(X)$ and $S_1,\ldots , S_N \in {\mathbf {I}}_{m+1}^\mathrm {c}(X)$ such that

  • (2-a) $\sum _i T_i = T$ and $\partial S_i = T_i - [c_i^L];$

  • (2-b) there is $n \ge 0$ such that $\sum _i c_i^L = {\mathrm {sd}}^n c;$

  • (2-c) $\mathrm {spt}(S_i) \cup \mathrm {im}(c_i^L) \subset U(\mathrm {spt}(T) \cup \mathrm {im}(c), \epsilon )$ and $\mathrm {diam}\, \mathrm {spt}(S_i) < \epsilon$.

Proof

Let X be as in the assumption. Let us prove (1). We suppose that X is $H_k^\mathrm {sing}$-locally trivial for $1 \le k \le m$ and take $c \in S_m(X)$ with $\partial c \in S_{m-1}^\mathrm {Lip}(X)$. Let us take a finite open covering ${\mathcal {U}} = \{U_i\}_{i=1}^N$ of $\mathrm {im}(c)$ such that $\mathrm {diam}(U_i) < \epsilon /2$. We set $U = \bigcup _{i=1}^N U_i$. Then, we may regard c as an element in $c \in S_m(U)$. Furthermore, the same symbol c denotes the element in ${\mathfrak {S}}_m(U)$ represented by c. Since $\partial c \in {\mathfrak {S}}_{m-1}^\mathrm {Lip}(U)$, by Lemma 4.6, there are elements $c_{k, m-k-1}^L \in {\check{C}}_k({\mathcal {U}}, {\mathfrak {S}}_{m-k-1}^\mathrm {Lip})$ such that

$$\begin{aligned} \varepsilon c_{0,m-1}^L = \partial c \quad \text {and}\quad \Phi (c_{k,m-1}^L) = \partial c_{k-1,m-k}^L \end{aligned}$$

for $1 \le k \le m-1$. On the other hands, since $\varepsilon :{\check{C}}_0({\mathcal {U}}, {\mathfrak {S}}_m) \rightarrow {\mathfrak {S}}_m(U)$ is surjective, there is $c_{0,m} \in {\check{C}}_0({\mathcal {U}},{\mathfrak {S}}_m)$ such that $\varepsilon c_{0,m} = c$. By the choice, we have

$$\begin{aligned} \varepsilon (\partial c_{0,m}-c_{0,m-1}^L) = 0. \end{aligned}$$

Therefore, we obtain $c_{1,m-1} \in {\check{C}}_1({\mathcal {U}},{\mathfrak {S}}_{m-1})$ satisfying

$$\begin{aligned} \Phi c_{1,m-1} = \partial c_{0,m} - c_{0,m-1}^L. \end{aligned}$$

By repeating such an argument, we have a sequence of elements $c_{k,m-k} \in {\check{C}}_k({\mathcal {U}},{\mathfrak {S}}_{m-k})$ such that

$$\begin{aligned} \Phi c_{k,m-k} = \partial c_{k-1,m-k+1} + (-1)^k c_{k-1, m-k}^L \end{aligned}$$

from all $1 \le k \le m$. Then, since ${\mathfrak {S}}_0 = {\mathfrak {S}}_0^\mathrm {Lip}$, the element $c_{m,0}$ belongs to ${\check{C}}_m({\mathcal {U}},{\mathfrak {S}}_0^\mathrm {Lip})$. Furthermore, we have $\partial c_{m-1,1} \in {\check{C}}_{m-1}({\mathcal {U}},{\mathfrak {S}}_0^\mathrm {Lip})$. Since $H_1^\mathrm {sing}\cong H_1^\mathrm {Lip}$, by Lemma 5.9, there are elements $c_{m-1,1}^L \in {\check{C}}_{m-1}({\mathcal {U}},{\mathfrak {S}}_1^\mathrm {Lip})$ and $c_{m-1.2} \in {\check{C}}_{m-1}({\mathcal {U}},{\mathfrak {S}}_2)$ such that

$$\begin{aligned} \partial c_{m-1,2} = c_{m-1,1} - c_{m-1,1}^L. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \partial (\Phi c_{m-1,2} - c_{m-2,2}) = (-1)^{m-1} c_{m-2,1}^L - \Phi c_{m-1,1}^L \end{aligned}$$

which is an element of ${\check{C}}_{m-2}({\mathcal {U}}, {\mathfrak {S}}_1^\mathrm {Lip})$. By Lemma 5.9, there are elements $c_{m-2,3} \in {\check{C}}_{m-2}({\mathcal {U}},{\mathfrak {S}}_3)$ and $c_{m-2,2}^L \in {\check{C}}_{m-2}({\mathcal {U}},{\mathfrak {S}}_2)$ such that

$$\begin{aligned} \partial c_{m-2,3} = \Phi c_{m-1,2} - c_{m-2,2} - c_{m-2,2}^L. \end{aligned}$$

By using Lemma 5.9 repeatedly, we have sequences of elements $c_{m-k,k+1} \in {\check{C}}_{m-k}({\mathcal {U}},{\mathfrak {S}}_{k+1})$ and $c_{m-k,k}^L \in {\check{C}}_{m-k}({\mathcal {U}},{\mathfrak {S}}_k^\mathrm {Lip})$ such that

$$\begin{aligned} \partial c_{m-k,k+1} = \Phi c_{m-k+1,k} + (-1)^{k+1} c_{m-k,k} - c_{m-k,k}^L \end{aligned}$$

for $2 \le k \le m$. Let us set

$$\begin{aligned} {\tilde{c}}_{0,m} := (-1)^{m+1} \Phi c_{1,m} + c_{0,m} \in {\check{C}}_0({\mathcal {U}}, {\mathfrak {S}}_m). \end{aligned}$$

Then, we have

$$\begin{aligned} \left\{ \begin{aligned}&\varepsilon {\tilde{c}}_{0,m} = c, \\&\partial c_{0,m+1} = (-1)^{m+1} {\tilde{c}}_{0,m} - c_{0,m}^L. \end{aligned} \right. \end{aligned}$$

(5.1)

Furthermore, let us set ${\tilde{c}}_{0,m} = (c_i)_{i=1}^N$, $(-1)^{m+1} c_{0,m+1} = ({\bar{c}}_i)_{i=1}^N$ and $(-1)^{m+1} c_{0,m}^L = (c_i^L)_{i=1}^N$, where $c_i \in {\mathfrak {S}}_m(U_i)$, ${\bar{c}}_i \in {\mathfrak {S}}_{m+1}(U_i)$ and $c_i^L \in {\mathfrak {S}}_m^\mathrm {Lip}(U_i)$. Then, the relation (5.1) is translated as

$$\begin{aligned}&\sum _{i=1}^N c_i = c \in {\mathfrak {S}}_m(U), \\&\partial {\bar{c}}_i = c_i - c_i^L \in {\mathfrak {S}}_m(U_i) \end{aligned}$$

for every $i \in \{1,\ldots , N\}$. We consider representatives of $c_i$ in $S_m(U_i)$, $c_i^L$ in $S_m^\mathrm {Lip}(U_i)$ and ${\bar{c}}_i$ in $S_{m+1}(U_i)$. Taking subdivision of them sufficiently many times, we obtain the conclusions of the statement (1).

The statement (2) can be proved by an argument similar to the proof of (1). This completes the proof of Theorem 5.10$\square$

Remark that Corollary 5.10 is a generalization of statements in [14] in terms of local triviality of homology theories.