On Bertelson-Gromov Dynamical Morse Entropy

Abstract

In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose M is compact oriented connected $C^\infty$ manifold of finite dimension. Assume that $f_0 :M \rightarrow [0,1]$ is a surjective Morse function. For a given natural number n, consider the set $M^n$ and for $x=(x_0,x_1,\ldots ,x_{n-1}) \in M^n$, denote $f_n (x) = \frac{1}{n} \, \sum _{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\subset [0,1]$ the asymptotic growth of the number of critical points of $f_n$ in I, when $n \rightarrow \infty$. The part related to the Betti number entropy does not requires the differentiable structure. One can describe generic properties of potentials defined in the XY model of Statistical Mechanics with this machinery.

Instituto de Matematica—UFRGS—Brasil A. O . Lopes was partially supported by CNPq and INCT.

1 Introduction

We follow the main guidelines and notation of [1].

A Morse function is a smooth function such all critical points are not degenerate (see [2]).

Suppose M is compact oriented $C^\infty$ manifold of dimension $q \ge 1.$ Assume that $f_0 :M \rightarrow [0,1]$ is a surjective Morse function and $\varGamma$ is a free group with basis $\gamma _1,\ldots ,\gamma _n$. We assume that $f_0$ has p critical points ($p\ge 2$).

Suppose $\varOmega \subset \varGamma$ is a finite non-empty set. If $x \in M^\varOmega$ we denote $x_\gamma \in M$, $\gamma \in \varOmega$, the corresponding coordinate.

Then, we define $f_\varOmega : M^\varOmega \rightarrow [0,1]$ by the expression

$$ f_\varOmega (x) = \frac{1}{| \varOmega |} \, \sum _{\gamma \in \varOmega } f_0 (x_\gamma ),$$

where $|\varOmega |$ is the cardinality of $\varOmega$. This function $f_\varOmega$ is also a surjective Morse function.

2 The $X\,Y$ Model

As a particular case we can consider $\varGamma =\mathbb {Z}$, the set $M^\mathbb {Z}$ and for $x=(x_j)_{j \in \mathbb {Z}} \in M^\mathbb {Z}$, $n>0$, $f_0: M \rightarrow \mathbb {R}$, and

$$ f_n (x) = - \frac{1}{n} \, \sum _{j=0}^{n-1} f_0 (x_j).$$

We mention this case because it is a more well known model in the literature and we want to trace a parallel to what will be done here.

The question about the minus sign in front of the sum is not important but if we want that $f_0$ represents a kind of Hamiltonian (energy) we will keep the—(at least in this section).

In this model it is natural to consider that adjacent molecules in the lattice interact via a potential (an Hamiltonian) which is described by the smooth function of two variables $f_0$. The mean energy up to position n is described by $f_n$. The points $x\in M^n$ where the mean n-energy is lower or higher are of special importance. We are interested here, among other things, in the growth of the number of critical values, when $n \rightarrow \infty$. The critical points are called the stationary states (see [1]).

Denote by $\text {Cri}_n (I)$ the number of critical points of $f_n$ in a certain interval $f^{-1}(I)$. Roughly speaking the purpose of [1] is to provide for a fixed value $c\in [0,1]$ a topological lower bound for

$$ \lim _{\delta \rightarrow 0} \lim _{n \rightarrow \infty } \frac{\log (\text {Cri}_n (I) )}{n},\,\,\text {where}\,\, I=(c-\delta ,c+\delta ) ,$$

in terms of a certain strictly positive concave function (a special kind of entropy). This is done by taking into account the homological behavior of the functions $f_n$.

The so called classical XY model consider the case where $M=S^1$ (see for instance [3,4,5,6,7,8,9] or [10]). A function $A: (S^1)^\mathbb {Z} \rightarrow \mathbb {R}$ describes an interaction between sites on the lattice $\mathbb {Z}$ where the spins are on $S^1$. One is interested in equilibrium probabilities $\hat{\mu }$ on $(S^1)^\mathbb {Z}$ which are invariant for the shift $\hat{\sigma }:(S^1)^\mathbb {Z} \rightarrow (S^1)^\mathbb {Z}.$ A point x on $(S^1)^\mathbb {Z}$ is denoted by $x=(\ldots ,x_{-2},x_{-1}\,|\,x_0,x_1,x_2,\ldots ).$

In the case the potential A depend just on the first coordinate $x_0\in S^1$, that is $A(x)=f_0(x_0)$, then the setting described above applies.

In the case the potential A depend just on the two first coordinate $x_0,x_1\in S^1$, that is $A(x)=f_0(x_0,x_1)$, then, we claim that the setting described above in the introduction applies. This is the case when $f_0: S^1 \times S^1 \rightarrow \mathbb {R}$. Indeed, in this case one can take $M = S^1 \times S^1$ and consider that $f_0$ acts on M. In this case we can say that $f_0$ depends just in the first coordinate on $M^\mathbb {Z}=(S^1 \times S^1)^\mathbb {Z}$ and adapt the general formalism we describe here.

Therefore, we will state all results for $f_0 : M \rightarrow \mathbb {R}$, that is, the case the potential on $M^\mathbb {Z}$ depends just on the first coordinate.

In the case $\hat{\mu }$ is ergodic the sequence $f_n$ describes Birkhoff means which are $\hat{\mu }$ almost everywhere constant. We are here interested more in the topological and not in the measure theoretical point of view.

In the measure theoretical (or Statistical Mechanics) point of view, if one is interested in equilibrium states at positive temperature $T=1/\beta$, then, is natural to consider expressions like $\int \,e^{ \sum _{j=0}^{n-1} -\beta \, f_0 (x_j)}\, dx_0\,dx_1 \ldots d_{x_{n-1}}$ (or, when the set of spins is finite: $\sum e^{ \sum _{j=0}^{n-1} - \beta \, f_0 (x_j)}$) and its normalization (see [11,12,13]) which defines the partition function.

By the other hand if one is interested in the zero temperature case (see for instance [14]), then, expressions like $-\sum _{j=0}^{n-1} f_0 (x_j)$ are the main focus. For instance, if $f_0$ has a unique point of minimum $x^{-} \in S^1$, then $\delta _{(x^{-})^\infty }$ defines the ground state (maximizing probability). In the generic case the function $f_0$ has indeed a unique point of minimum.

Given $f_0: M \times M \rightarrow \mathbb {R}$ and n one can also consider periodic conditions. In this case we are interested in sums like

$$ \tilde{f}_n (x) = -\frac{1}{n} \, ( f_0 (x_0)+ f_0 (x_1)+ \cdots + f_0 (x_{n-2})+ f_0 (x_0)),$$

or

$$ -( f_0 (x_0)+ f_0 (x_1)+ \cdots + f_0 (x_{n-2})+ f_0 (x_0)).$$

In the case we want to get Gibbs states via the Thermodynamic Limit (see for instance [11] or [13]), given a natural number n, we have to look for the probability $\mu$ on $M^n$ (absolutely continuous with respect to Lebesgue probability) which maximizes

$$\int \,e^{-\, \sum _{j=0}^{n-1} \beta \, f_0 (x_j)}\, d\, \mu (dx_0,\,dx_1,\ldots ,d_{x_{n-1}}),$$

or, at zero temperature the periodic probability $\mu$ on $M^n$ which maximizes

$$-\,\int \sum _{j=0}^{n-1} \, f_0 (x_j)\,\, d\, \mu (dx_0,\,dx_1,\ldots ,d_{x_{n-1}}).$$

One can easily adapt the reasoning of [15] to show that for a generic $f_0$ we get that $\tilde{f}_n$ is a Morse function for all n.

When $f_0$ is not generic several pathologies can occur (see for instance [3, 5, 10]).

Suppose the case when there is a unique point $x^{-}$ of minimum for $f_0$. For each $\beta >0$ and n denote by $\mu _{n,\beta }$ the absolutely continuous with respect to Lebesgue probability which maximizes

$$\int \,e^{ - \,\sum _{j=0}^{n-1} \beta \, f_0 (x_j)}\, d\, \mu (dx_0,\,dx_1,\ldots ,d_{x_{n-1}}).$$

By the Laplace method (adapting Proposition 3 in [7] or Lemma 4 in [8]) we get that when $\beta \rightarrow \infty$ and $n \rightarrow \infty$ the probability $\mu _{n,\beta }$ converges to the Dirac delta on $(x^{-})^\infty$. Therefore, in the generic case this last probability is the ground state (zero temperature limit).

3 The General Model—The Dynamical Morse Entropy

From now we forget the—sign in front of $f_0$. For instance, $f_n (x) = \frac{1}{n} \, \sum _{j=0}^{n-1} f_0 (x_j, x_{j+1}).$

Given $c \in [0,1]$ and $\delta >0$, take $N_\varOmega (c, \delta )$ the number of critical points of $f_\varOmega$ in $f^{-1}_\varOmega [c-\delta , c+ \delta ]$. Note that if $f_0$ has p critical points then $f_\varOmega$ has $p^{| \varOmega |}$ critical points.

Consider the cylinder sets

$$ \varOmega _i \,=\, \{a_1\, \gamma _1+ \cdots + a_n \gamma _n\,;\, |a_j| \le i,\,1\le \,j\le n\,\},\,\, i=1,2,\ldots .,$$

where $a_j$ are integers.

Denote $N_i (c, \delta )= N_{\varOmega _i}(c, \delta )$. Then, of course, $N_i(c, \delta )$ for c fixed decrease with $\delta$.

For a fixed $0\le c \le 1$, we denote the entropy by

$$ \varepsilon (c) = \lim _{\delta \rightarrow 0} \,\left( \,\liminf _{i \rightarrow + \infty }\, \frac{\log ( N_i(\,c, \delta )\,)}{| \varOmega _i|} \,\right) .$$

The above limit exists and it is bounded by $\log p$ but in principle could take the value $-\infty$. We call $\varepsilon (c)$ the dynamical Morse entropy on the value c.

In the case $\varGamma =\mathbb {Z}$ as we mentioned before $\varepsilon (c)$ is described by

$$ \varepsilon (c) = \lim _{\delta \rightarrow 0} \,\left( \,\liminf _{n \rightarrow + \infty }\, \frac{\log (\text {number of critical points of} \,\,f_n \,\,\text {in}\,\,f^{-1}_n [c-\delta ,c+\delta ] \,)}{n} \,\right) .$$

Later we introduce a function b(c) (see Definition 3 and also Definition 2), which will be a topological invariant of $f_0$. The function b(c) is defined in terms of rank of linear operators and Cohomology groups.

We will show later that

(1) $0 \le b(c) \le \varepsilon (c),$ $\,0\le c \le 1$;

(2) b(c) is continuous and concave;

(3) b(c) is not constant equal to 0.

Finally, in the case $M=S^1$ (the unitary circle) and $f_0$ has just two critical points, we show in Sect. 7 that

$$\varepsilon (c) = b(c) = - c \log c - (1-c) \log (1-c). $$

b(c) is sometimes called the Betti entropy of $f_0$.

Our definition of b(c) is different from the one in [1] but we will show later (see Sect. 8) that is indeed the same.

A key result in the understanding of the main reasoning of the paper is Lemma 6 which claims that for any Morse function f, given $a,b\in \mathbb {R}$, $a<b$, the number of critical points of f in $f^{-1} [a,b]$ is bigger or equal to the dimension of the vector space

$$\,\, \frac{H^* ( f^{-1} (\infty ,b)\,)}{H^* (f^{-1} (-\infty , a)\,)} ,$$

where $H^*$ denotes the corresponding cohomology groups which will be defined in the following paragraphs (see also [16] for basic definitions and properties).

$H^* (X,\mathbb {R})$ denotes the usual cohomology. Note that $H^*$ will have another meaning (see Definition 1).

4 Cohomology

Suppose X is a metrizable, compact, oriented topological manifold $C^\infty$ manifold. We will consider the singular homology. Suppose $U \subset X$ is an open set and $a \in H^* (X, \mathbb {R}).$ The meaning of the statement supp $a\, \subset U$ is: there exist an open set $V\subset X$, such that, $X= U \cup V$, and $a|_V =0.$

Definition 1

$H^*_X (U)= \{\,a\in H^* (X,\mathbb {R}) \,:\, \text {supp } \,\,a \subset U\},$ where U is an open subset of X. When X is fixed we denote $H^*_X(U) = H^* (U)$.

Remember (see for instance [16]) that when $U\subset X$ is open we get the exact cohomology sequence:

$$\begin{aligned} ... \rightarrow H^{k-1} (X - U,\mathbb {R}) \rightarrow H^k_c (U,\mathbb {R})\rightarrow H^k (X,\mathbb {R}) \rightarrow H^k (X-U,\mathbb {R}) \rightarrow H^{k+1}_c (U,\mathbb {R}) \rightarrow ... \end{aligned}$$

(1)

where $H_c^*$ denotes the compact support cohomology.

Lemma 1

If U is an open set, then

$$ H^* (U) = \text {Im} (\,\,H^*_c (U,\mathbb {R}) \rightarrow H^* (X,\mathbb {R})\,\,)=\,\text {Ker}\,(\,\, H^* (X,\mathbb {R}) \rightarrow H^* (X-U,\mathbb {R})\,\,) .$$

Proof

The second equality follows from the fact that the above sequence is exact.

We will prove that

$$ \text {Im} (\,\,H^*_c (U,\mathbb {R}) \rightarrow H^* (X,\mathbb {R})\,\,)\subset H^* (U) \subset \,\text {Ker}\,(\,\, H^* (X,\mathbb {R}) \rightarrow H^* (X-U,\mathbb {R})\,\,) .$$

Let $a\in \text {Im} (\,\,H^*_c (U,\mathbb {R}) \rightarrow H^* (X,\mathbb {R})\,\,)$. Then, a is represented by a cocycle $\alpha$ with compact support $K\subset U$. Therefore, $a\,|(X-K)=0$.

Defining $V=X-K$ we have that $U \cup V=X$ and $a\,|V=0$. Then, $a \in H^* (U)$.

Let be $\alpha \in H^* (U)$. Let $V\subset X$ be an open set such that $U \cup V=X$ and $\alpha \,| V=0$.

Since $X-U\subset V$, we have $\alpha \,|(X-U)=0$.

Then, $\alpha \in$ Ker $(\,H^* (X, \mathbb {R}) \rightarrow H^* (X-U, \mathbb {R}) \,).$ $\square$

Lemma 2

If U is an open set then $H^* (U)$ is a graded ideal of the ring of cohomology of X.

Proof

This follows at once from Lemma 1. $\square$

Now we consider a continuous function $f:X \rightarrow \mathbb {R}.$

Definition 2

Given $\delta >0$ and $c \in \mathbb {R}$ we define

$$ b_{c, \delta } ' = Dim \, \left( \,\frac{H^* (f^{-1} (- \infty , c +\delta )\,)\,) }{H^* (f^{-1} (-\infty , c- \delta )\, )} \,\right) .$$

Proposition 1

Suppose X and Y are metrizable compact, oriented topological manifolds, moreover take $f:X \rightarrow \mathbb {R}$, $g:Y \rightarrow \mathbb {R}$ continuous functions. If we define $f \oplus g: X \times Y \rightarrow \mathbb {R}$, by $(f \oplus g)(x,y) = f(x) + g(y)$, then, if $c,c'\in \mathbb {R}$, $\delta , \delta '>0$, we get

$$\begin{aligned} b_{c,\,\delta } ' (f) \, \, b_{c ',\,\delta '} '(g) \,\le b_{c+ c', \,\delta + \delta '} ' ( f \oplus g). \end{aligned}$$

(2)

Before the proof of this important proposition we need two more lemmas.

As it is known (see [16]) the cup product $\vee$ defines an isomorphism

$$ \mu : H^* (X,\mathbb {R}) \otimes H^* (Y,\mathbb {R}) \,\rightarrow \, H^* (X \times Y,\mathbb {R}).$$

Lemma 3

If $U\subset X$ and $V\subset Y$ are open sets, then

$$ \mu (\, H^*_X (U) \otimes H^* (Y,\mathbb {R}) + H^* (X,\mathbb {R}) \otimes H^*_Y (V)\,)\, =H^*_{X \times Y}(\,( U \times Y) \cup (X \times V)\,) .$$

Proof

By Lemma 1 we get

$$H^*_{X \times Y}(\,( U \times Y) \cup (X \times V)\,) = \text {Ker}\, (\,H^* (X \times Y,\mathbb {R}) \rightarrow H^* ((X-U) \times (Y - V),\mathbb {R} ).$$

Then,

$$ H^*_{X \times Y} (\,(U\times Y) \cup (X \times V)\,) = $$

$$\mu (\,\, \text {Ker}\, (\,H^* (X,\mathbb {R}) \otimes H^* ( Y ,\mathbb {R})\, \rightarrow H^* (X-U,\mathbb {R}) \otimes H^* (Y - V,\mathbb {R}) \,)\,\,).$$

From simple Linear Algebra arguments the claim follows from Lemma 1. $\square$

Lemma 4

If $U \subset X$ and $V \subset Y$ are open sets then

$$ \mu ( \,H^*_X (U) \otimes H^*_Y (V)\,) = H^*_{X \times Y} (U \times V).$$

Proof

The $\vee$ product defines a natural isomorphism

$$ H^* (X, X-U,\mathbb {R}) \otimes H^* (Y, Y-V,\mathbb {R}) \, \rightarrow H^*(X \times Y, (X \times (Y-V) \cup (X-U) \times Y,\mathbb {R})= $$

$$ H^*(\,X \times Y, (X \times Y ) - (U \times V,\mathbb {R})\,). $$

By Lemma 1 and the exact relative cohomology sequence we get:

$$ H^*_X ( U) =\text {Im} \,(\, H^* (X, X-U,\mathbb {R}) \,\rightarrow H^*(X,\mathbb {R})\,), $$

$$ H^*_Y (V) =\text {Im} \,(\, H^* (Y, Y-V,\mathbb {R}) \,\rightarrow H^*(Y,\mathbb {R})\,), $$

and

$$ H^*_{X\times Y} (U \times V) =\text {Im} \,(\, H^* (X \times Y, (X \times Y) - (U \times V,\mathbb {R}))\,\rightarrow \, H^* (X \times Y,\mathbb {R})\,) . $$

From this the claims follows at once. $\square$

Now we will present the proof of Proposition 1.

Take $h= f \oplus g$ and denote

$$ A^{-} = f^{-1} (-\infty , c-\delta ), \,\,B^{-} = g^{-1} (-\infty , c'-\delta '), \,\,C^{-} = h^{-1} (-\infty , (c+c')-(\delta + \delta ')) \,\,,$$

and

$$ A^{+} = f^{-1} (-\infty , c+\delta ), \,\,B^{+} = g^{-1} (-\infty , c'+\delta '), \,\,C^{+} = h^{-1} (-\infty , (c+c')+(\delta + \delta ')) \,\,.$$

Note that

$$ A^+ \times B^+ \subset C^+ \subset (A^+ \times Y) \cup (X \times B^+)$$

$$ A^{-} \times B^{-} \subset C^{-} \subset (A^{-} \times Y) \cup (X \times B^{-}).$$

Consider the commutative diagram

$$H^* (X,\mathbb {R}) \otimes H^* (Y) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\rightarrow \,(\text {using} \, \mu \,)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, H^* (X \times Y,\mathbb {R})$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\cup \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\cup \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$

$$H^*_X (A^{+}) \otimes H^*_Y (B^{+}) \rightarrow H^*_{X \times Y} (C^{+}) \subset H^*_{X \times Y} (\, (A^{+} \times Y)\cup (X \times B^{+})\,)$$

$$ \cup \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\cup $$

$$ H^*_X (A^{+}) \otimes H^*_Y (B^{-}) + H^*_X (A^{-}) \otimes H^*_Y (B^{+})\rightarrow H^*_{X \times Y} (\, (A^{-} \times Y)\cup (X \times B^{-})\,)$$

$$ \,\,\,\,\,\, \cup \,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\cup $$

$$ H^*_ X (A^{-}) \otimes H^*_Y (B^{-}) \rightarrow H^*_{X\times Y}(C^{-}).$$

From this follows the linear transformation

$$ \tilde{\mu }: \, \, \frac{H^*_X (A^{+}) \otimes H^*_Y (B^{+})}{ H^*_X (A^{-}) \otimes H^*_Y (B^{-})}\, \rightarrow \frac{ H^*_{X\times Y}(C^{+})}{ H^*_{X\times Y}(C^{-})}.$$

By the other hand

$$ (\,H^*_X (A^{+}) \otimes H^*_Y (B^{+})\,\cap \mu ^{-1} ( \,H^*_{X\times Y}(C^{-})\,)\subset $$

$$(\, H^*_X (A^{+}) \otimes H^*_Y (B^{+})\,\cap \mu ^{-1} ( \,H^*_{X\times Y}(\, (A^{-} \times Y) \cup (X \times B^{-})\,)\,)=$$

$$ (\,H^*_X (A^{+}) \otimes H^*_Y (B^{+})\,) \cap (\, \,H^*_{X} (A^{-}) \otimes H^* (Y,\mathbb {R}) + H^*(X,\mathbb {R}) \otimes H^*_Y (B^{-})\,)=$$

$$ \,H^*_X (A^{-}) \otimes H^*_Y (B^{+})\, + \,H^*_X (A^{+}) \otimes H^*_Y (B^{-})\, .$$

The first equality above follows from Lemma 3; the second follows from Linear Algebra; namely, if $E_2\subset E_1 \subset E$ and $F_2\subset F_1 \subset F$, then

$$ (E_1 \otimes F_1) \cap (E_2 \otimes F + E \otimes F_2) = E_2 \otimes F_1 + E_1 \otimes F_2.$$

From the above it follows that

$$ \text {Ker}\, \tilde{\mu }\, \subset \frac{ H^*_X (A^{-}) \otimes H^*_Y (B^{+})\, + \,H^*_X (A^{+}) \otimes H^*_Y (B^{-}) }{ H^*_X (A^{-}) \otimes H^*_Y (B^{-})} .$$

Therefore,

$$b_{c + c', \delta + \delta '} ' \,=\, \text {dim}\, \frac{H^*_{X \times Y} (C^+) }{H^*_{X \times Y} (C^{-}) }\ge \text {dim}\, (\text {Im}\,\tilde{\mu })\ge $$

$$ \text {dim}\, \frac{ H^*_X (A^{+}) \otimes H^*_Y (B^{+})}{ H^*_X (A^{-}) \otimes H^*_Y (B^{+})\, + \,H^*_X (A^{+}) \otimes H^*_Y (B^{-}) } = $$

$$ \text {dim}\,\left( \,\frac{ H^*_X (A^{+})}{ H^*_Y (A^{-})}\, \otimes \frac{\,H^*_Y (B^{+})}{ H^*_Y (B^{-})}\,\right) \,= b_{c,\delta } '(f) \, \,b_{c', \delta '} '(g). $$

$\square$

5 Critical Points

In what follows X is a compact, oriented $C^\infty$ manifold and $f:X \rightarrow \mathbb {R}$ is a Morse function.

Lemma 5

Suppose X is a compact, oriented $C^\infty$ manifold and $U\subset X$ is an open set. If $a \in H^* (X, \mathbb {R}),$ then, supp $a\subset U$, if and only if, there exists a closed $C^\infty$ differentiable form w such that supp $w \subset U$, and a is the de Rham cohomological class of w.

Proof

If there exists $w \in a$, such that supp $w \subset U$, then

$$ a|_{(X- \text {supp}\,\, w)}=0 \, \text {and}\,\, U \, \cup \, (X- \text {supp}\,\, w)\,=\,X. $$

If there exists an open set $V\subset X$ such that $U \cup V=X$ and $a|_V =0$, then, there exist a $C^\infty$ form $\eta$ on V such that $d \eta = w|_V$ where $w \in a$.

Let W be an open set such that $\overline{W}\subset V$ and $W\cup U=X$. Take a $C^\infty$ function $\varphi : X \rightarrow [0,1]$ such that $\varphi |_{\overline{W}} =1$ and $\varphi |_{X-K} =0$, where K is compact set such that $\overline{W} \subset K \subset V$. Then, $\varphi \, \eta$ has an extension to X and $(w- d \,(\varphi \, \eta ) \,)\in a$. But,

$$ \text {supp}\,\,\,(w- d \,(\varphi \, \eta ) \,)\subset X-W \subset U.$$

$\square$

Lemma 6

Given $a,b\in \mathbb {R}$, $a<b$, then, the number of critical points of f in $f^{-1} [a,b]$ is bigger or equal that

$$ dim \,\, \frac{H^* ( f^{-1} (\infty ,b)\,)}{H^* (f^{-1} (-\infty , a)\,)} .$$

Proof

Without lost of generality we can assume that a and b are regular values of f (decrease a and increase b a little bit).

Given $c_1<c_2< \cdots <c_m$, the critical values of f in (a, b), take

$$ a=d_0<c_1<d_1<c_2<d_2< \cdots<d_{m-1}< c_m< d_m=b.$$

By Proposition 3 and Lemma 8, the number of critical points in $f^{-1} (c_i)$, $i=1=,2,\ldots ,m$, is bigger or equal to

$$ dim \,\, \frac{H^* ( f^{-1} (\infty ,d_i)\,)}{H^* (f^{-1} (-\infty , d_{i-1})\,)} .$$

Finally consider the filtration

$$ \,\, H^* ( f^{-1} (\infty ,a)\,=\,H^* (f^{-1} (-\infty , d_0)\,) \subset \,H^* (f^{-1} (-\infty , d_1)\,) \subset ...$$

$$ \,\subset \, H^* ( f^{-1} (-\infty ,d_{m-1})\,\subset \,H^* (f^{-1} (-\infty , d_m)\,) = \,H^* (f^{-1} (-\infty , b)\,).$$

$\square$

Now we denote $b_\varOmega '(c,\delta )=b_{c,\delta }'(f_\varOmega )$ and $b_i '(c,\delta )=b_{\varOmega _i}'(c,\delta )$, $0\le c \le 1,$ $\delta >0$.

Corollary 1

$b_i '(c,\delta )\le N_i(c, \delta )$ for all $i=1,2,3,\ldots$ and $0\le c\le 1$, $\delta >0$.

Now we define the function b using Proposition 3(a)

Definition 3

$$b(c)= \lim _{\delta \rightarrow 0} \, \liminf _{i \rightarrow \infty } \,\,\frac{\log ( b_i ' (c, \delta ))}{| \,\varOmega _i\,|}, \,\, 0\le c \le 1 .$$

We will show that in above definition we can change the $\liminf$ by $\lim$.

Lemma 7

$$b(c) \le \varepsilon (c)\le \log \,(\text {the number of critical points of }\,\,f_0).$$

Proof

The first inequality follows from Corollary 1. From the definition is easy to see that $\varepsilon (c)$ is smaller than $\log$ of the number of critical points of $f_0.$ $\square$

We denote $B(\varGamma )$ a family of finite subsets of $\varGamma$ and $B_N (\varGamma )$, $N \in \mathbb {N}$, the family of sets $\varOmega \in B(\varGamma )$ such that $|\varOmega | >N.$

Proposition 2

Suppose $\varOmega ', \varOmega ''\in B(\varGamma )$ are disjoint not empty sets. Then,

$$ b_{\varOmega \cup \varOmega ''} ' (\alpha c_1 + (1- \alpha ) c_2,\delta ) \ge b_{\varOmega '} ' (c_1,\delta )\, b_{\varOmega ''} ' (c_2, \delta ), $$

where $0 \le c_1,c_2\le 1$, $\delta >0$ and $\alpha = \frac{| \,\varOmega '\,|}{|\,\varOmega ' \,| + | \,\varOmega ''\,|}.$

Proof

By definition

$$ f_{\varOmega ' \cup \varOmega ''} = \alpha f_{\varOmega '} \oplus (1- \alpha ) f_{\varOmega ''}.$$

By Proposition 1, as $\delta = \alpha \, \delta + (1- \alpha ) \delta ,$ then

$$ b_{\alpha \,c_1 + (1- \alpha )\, c_2,\delta } ' (f _{\varOmega '\cup \varOmega ''}) \ge b_{\alpha \, c_1,\alpha \, \delta } ' (\alpha \,f _{\varOmega '})\,b_{(1- \alpha ) \,c_2,(1-\alpha )\,\delta } ' ((1-\alpha )\, f _{ \varOmega ''})= $$

$$b_{c_1 , \delta } ' (f_{\varOmega '})\, \, \, b_{c_2, \delta } ' (f_{\varOmega ''}) . $$

$\square$

Lemma 8

Suppose the interval [a, b] does no contains critical values of f. Then,

$$H^* ( \,f^{-1} (\,-\infty , a\,)\,)= H^* (\, f^{-1} (\,-\infty , b\,)\,).$$

Proof

This follows from Lemma 1 and the fact that $f^{-1} [b,\,\infty )\,$ is a deformation retract of $f^{-1} [a,\,\infty \,).$ $\square$

Definition 4

Given $c \in \mathbb {R}$ we define

$$ \tilde{b}_c (f) = \lim _{\delta \rightarrow 0} b_{c,\delta } ' (f).$$

Proposition 3

For a fixed c we have

(a) $b_{c,\delta } ' (f)$ decreases with $\delta$ and $b_{c,\delta } ' (f) = \tilde{b}_c (f)$ for all $\delta$ small enough.

(b) $\tilde{b}_c (f)=0$ if c is not a critical value of f

(c) $\tilde{b}_c (f)$ is smaller than the number of critical points of f in $f^{-1}(c)$

(d) $\sum _c \, \tilde{b}_c (f)=$ Dim $H^* (X).$

Proof

(a) follows from the above definitions and Lemma 8.

(b) follows from Lemma 8

For the proof of (c) consider the exact diagram

$$\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,H^* (X,\mathbb {R}) $$

$$\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\, \,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\downarrow \, r_1\,\,\,\,\,\,\,\, \,\,\,\, \,\,\,\,\,\,\,\,\,\,\, \,\,\,r_2 \searrow $$

$$H^* ( f^{-1}[c-\delta ,\infty )) ,f^{-1}(c+\delta ,\infty ),\mathbb {R}\,) \rightarrow H^* [ f^{-1}(c-\delta ,\infty ),\mathbb {R})\rightarrow H^* ( f^{-1}[c+\delta ,\infty ),\mathbb {R}),$$

where $r_1$ and $r_2$ are the restriction homomorphisms.

By Lemma 1

$$ H^* ( f^{-1}(-\infty ,c+\delta ))=\text {Ker}\, r_2\, \,\,\,\text {and}\,\,\,H^* ( f^{-1}(-\infty , c-\delta ))=\, \text {Ker}\, r_1.$$

From this follows that

$$ b_{c,\delta } ' (f) = \,\,\text {Dim}\,\, ( r_1 (\text {Ker} \, (r_2)\, )\,\,\le \,\,\text {Dim}\,(\, H^* ( f^{-1}(c-\delta ,\infty )) ,f^{-1}(c+\delta ,\infty )\,),\mathbb {R}\, ) $$

because the above sequence is exact.

In order to finish the proof we apply Morse Theory (see [2]) with $\delta$ small enough.

For the proof of (d) suppose $c_1<c_2< \cdots <c_m$ are the critical values of f. Now, consider

$$ d_0<c_1<d_1<c_2<d_2< \cdots<d_{m-1}<c_m<d_m.$$

Now, from (a) and Lemma 8 we have

$$ \tilde{b}_{c_i} (f) = \text {Dim}\, ( \frac{H^* ( \,f^{-1}(-\infty , d_i)\,) }{H^* (\,f^{-1}(-\infty ,d_{i-1})\,)}\, ),\,\,\,\, i=1,2,\ldots ,m.$$

Finally, note that

$$ 0=H^* ( \,f^{-1}(-\infty , d_0))\subset H^* ( \,f^{-1}(-\infty , d_1))\subset ...\subset $$

$$ H^* ( \,f^{-1}(-\infty , d_m)) =H^* (X).$$

$\square$

Lemma 9

Given $\delta >0$, there exists an integer N such that: $b_\varOmega ' (c, \delta )\ge 1$ for all $c \in [0,1]$ and all $\varOmega \in B_N (\varGamma )$. Therefore, $b(c)\ge 0$, for all $0\le c\le 1$.

Before the Proof of Lemma 9 we need two more lemmas.

Lemma 10

Suppose X is a compact oriented $C^\infty$ manifold and $f:X \rightarrow \mathbb {R}$ is a Morse function. Then, for all $\delta >0$

$$ b_{a_1, \delta } ' (f) \ge 1\,\, \,\text {and}\,\,\,\, b_{a_2, \delta } ' (f) \ge 1, $$

where $a_1$ and $a_2$ are respectively the maximum and minimum of f.

Proof

If $\delta$ is small enough, $f^{-1} (-\infty , a_2 + \delta )$ is the disjoint union of a finite number of open discs and $f^{-1} (-\infty , a_2 -\delta )= \emptyset$.

If n is the dimension of X, then, it follows from Lemma 1 that

$$ H^n (X,\mathbb {R}) \subset H^* (f^{-1} (-\infty , a_2 + \delta )\,)\ne 0 $$

and

$$ H^* (f^{-1} (-\infty , a_2 - \delta )\,)=0. $$

Then, $b_{a_2, \delta } ' (f) \ge 1$, if $\delta >0$ is small enough. Therefore, this claim is also true for any $\delta >0$ by Proposition 3(a).

In a similar way we have that for small $\delta >0$

$$ H^0 (X,\mathbb {R}) \subset H^* (f^{-1} (-\infty , a_1 + \delta )\,) $$

and

$$ H^0 (X,\mathbb {R}) \,\,\text {is not contained}\,\,\, H^* (f^{-1} (-\infty , a_1 - \delta )\,). $$

From this the final claim is proved. $\square$

Lemma 11

Consider $\varOmega \in B (\varGamma )$ where $|\varOmega |=m\ge 1$, then, $b_\varOmega ' (k/m, \,\delta ) \ge 1,$ for all $\delta > 0$ and $k=0,1,2,\ldots ,m$.

Proof

If $k=0$, or m, the claim follows from Lemma 10 with $X=M^\varOmega$, $f=f_\varOmega$.

Given 0, k, m, $0<k<m$, take $\varOmega = \varOmega ' \cup \varOmega ''$, where $\varOmega ', \varOmega ''$ are disjoints and $k=|\varOmega '|$.

By Proposition 2 with $c_1=1$ and $c_2=0$ we get

$$ b_\varOmega ' (k/m, \,\delta )\ge b_{\varOmega '} ' (1, \,\delta )\, b_{\varOmega ''} ' (0, \,\delta )\ge 1. $$

Yet from last lemma. $\square$

Now we will prove Lemma 9.

Proof

Take $N> \frac{2}{\delta }$, $\varOmega \in B_N (\varGamma ) , |\varOmega |=m >N$ and k such that $\frac{k}{m} \le c < \frac{k+1}{m},$

By definition,

$$ b_{c, \delta } ' (f_\varOmega ) \ge b_{k/m, \, \delta /2} ' (f_\varOmega ),$$

since $c-\delta <k/m-\delta /2$ and $c+\delta >k/m + \delta /2$.

Therefore, $b_\varOmega ' (c, \delta ) \ge b_\varOmega ' (k/m,\,\delta /2)\ge 1$ by Lemma 11. $\square$

Proposition 4

$$ 0 \le b(c) \le \varepsilon (c) \le \, \log (\text {number of critical points of}\, f_0\,\mathrm{)}, \,\, 0\le c\le 1.$$

Proof

This follows from Lemmas 7 and 9$\square$

Lemma 12

Given $c\in [0,1]$ and $\delta >0$, consider a non-empty set $\varOmega \in B(\varGamma )$ and $\gamma \in \varGamma$. Then,

$$ b_{\varOmega } ' (c,\delta ) = b_{\varOmega + \gamma } '(c,\delta ).$$

In the case $\varGamma =\mathbb {Z}$ we have that for any $\varOmega =\{1,2,\ldots ,k\}$

$$ b_{\varOmega } ' (c,\delta ) = b_{\hat{\sigma }(\varOmega )} '(c,\delta ),$$

where $\hat{\sigma }$ is the shift acting on $M^\mathbb {Z}$.

Proof

For fixed $\gamma$ consider the transformation $x\in M^\varOmega \rightarrow y\in M^{\varOmega + \gamma }$, such that $y_w =x_{w-\gamma }$, which is a diffeomorphism which commutes $f_{\varOmega + \gamma }$ with $f_\varOmega$.

The result it follows from this fact. $\square$

We will show now that indeed one can change $\liminf$ by $\inf$ in Definition 3. In order to do that we need the following proposition which describes a kind of subadditivity.

Proposition 5

Given an integer number $N>0$ take $h: B_N (\varGamma ) \rightarrow \mathbb {R}$, $h\ge 0$, which is invariant by $\varGamma$ and such that

$$ h( \varOmega ' \cup \varOmega '') \ge h(\varOmega ') + h (\varOmega ''),$$

if $\varOmega ', \,\varOmega ''\,\in B_N (\varGamma ),$ are disjoint. Then, the limiit

$$ \lim _{i\rightarrow \infty } \frac{h(\varOmega _i)}{|\varOmega _i|}\ge 0\,\,\,\,\text {exists: finite or}\, +\infty \,.$$

From this follows:

Corollary 2

For $c\in [0,1]$ and $\delta >0$,

(a) there exist the limit

$$ \lim _{i \rightarrow \infty } \frac{ \log b_i ' (c, \delta )}{|\,\varOmega _i\,|}\,=\, b ' (c,\delta ).$$

(b) $0 \le b ' (c,\delta ) \le \log$ (number of critical points of  $f_0$),

(c) $b(c)= \lim _{\delta \rightarrow 0} b' (c,\delta )$

Proof

The claim (a) follows from last proposition applied to $h(\varOmega ) = \log b_\varOmega ' (c, \delta )$, by Lemma 9, Proposition 2 taking $c_1=c_2=c$ and also by Lemma 12.

Item (b) follows from Lemma 2 and Corollary 1.

Item (c) follows from item (a) and the definition of b(c). $\square$

Before the proof of Proposition 5 we need two lemmas.

Lemma 13

Given an integer positive number k, then for each $i > (3\, k\,+1)$ there exists $\varOmega _{k,i} \in B(\varGamma )$ such that: (a) $\varOmega _{k,i} \subset \varOmega _i$;   (b) $\varOmega _{k,i}$ is a disjoint union of a finite number of translates of $\,\varOmega _k$ ;  (c) $\lim _{i \rightarrow \infty } \frac{|\,\varOmega _{k,i}|\, }{|\,\varOmega _i\,|} =1$; (d) $|\varOmega _i| - |\varOmega _{k,i}|\ge (2k+1)^n$, where n is the number of generators of $\varGamma$.

Proof

For the purpose of the proof we can assume that $\varGamma = \underbrace{\mathbb {Z} \oplus \mathbb {Z}\oplus \cdots \oplus \mathbb {Z}}_n$ and take $\gamma _1,\gamma _2, \ldots ,\gamma _n$ the canonical basis.

Take $m \ge 1$ an integer such that

$$ k + \,m\, (2 \,k + \,1) \le i < k = \, (m+1) \, (2\, k + 1),$$

and

$$ \varOmega _{k,i} = \cup \,\,\{\,\,\varOmega _k + (\,j_1 (2 k +1),\ldots ,j_n (2 k+1)\,)\,|$$

$$\, -m\le j_1, \ldots ,j_n\le m, \, (j_1,\ldots ,j_n)\ne (0,\ldots ,0) \,\,\}.$$

It is easy to see that the sets $\varOmega _{k,i}$ satisfy all the above claims. $\square$

Lemma 14

Given real numbers $x_i\ge 0$ $i=1,2,3,\ldots$, suppose that for each k and each $\varepsilon >0$ there exist $N_{k,\varepsilon }$ such that

$$ x_i \ge x_k (1-\varepsilon ) \,\,\text {if}\,\, \, i\ge N_{k,\varepsilon }.$$

Then, the $\lim _{i \rightarrow \infty } x_i$ exists (can finite or $+\infty$).

Proof

Take $L= \limsup _{i \rightarrow \infty } x_i$ and $a\in \mathbb {R}$, $a<L$. Then, there exists $x_k>a$. Therefore, $x_i \ge a$, if i is very large. Then, $\liminf _{i \rightarrow \infty } x_i \ge a$. From this follows the claim. $\square$

Now we will prove Proposition 5.

Proof

Suppose k is such that $(2 k +1)^n>N$. Take $i> 3 \, k +1$, then, $|\,\varOmega _{k,i}\,| \ge (2 k + 1)^n>\,N$ and $|\varOmega _i - \varOmega _{k,i}| \ge (2 k +1)^n >N$.

Then, $h(\varOmega _i) = h ( \,\varOmega _{k,i} \,\cup \, (\,\varOmega _i - \varOmega _{k,i} )\,) \ge h(\varOmega _{k,i}).$

Moreover, each translate of $\varOmega _k$ has cardinality $(2k+1)^n$. Therefore,

$$ h( \varOmega _{k,i} )\,\ge \, \frac{|\,\varOmega _{k,i} \,|}{|\, \varOmega _k \,|} h(\varOmega _k).$$

From this follows that

$$ \frac{ h(\varOmega _i)}{|\, \varOmega _i\,|}\ge \frac{ h(\varOmega _{k,i})}{|\, \varOmega _{k,i}\,|}\,\frac{|\,\varOmega _{k,i}\,|}{|\, \varOmega _i\,|} \ge \frac{ h(\varOmega _k)}{|\, \varOmega _{k}\,|}\,\frac{|\,\varOmega _{k,i}\,|}{|\, \varOmega _i\,|}, $$

and the claim is a consequence of Lemmas 13 and 14. $\square$

The next lemma will be used later

Lemma 15

Under the hypothesis of Proposition 5 consider

$$ \varOmega _i ' = ( \,\varOmega _i + (2 i +1) \, \gamma _1\,)\,\cup \varOmega _i,\,\,\,i=1,2,3,\ldots $$

Then,

$$ \lim _{i \rightarrow \infty } \frac{h( \varOmega _i ')}{|\, \varOmega _i '\,|}= \lim _{i \rightarrow \infty } \frac{h( \varOmega _i )}{|\, \varOmega _i \,|}.$$

Proof

If $i >N$, then $|\varOmega _i|>N$. Therefore,

$$ h(\varOmega _i ') \ge h (\,\varOmega _i + \,(\,2\, i +1\,)\, \gamma _1\,)\, +\, h(\varOmega _i) = 2\,h(\varOmega _i).$$

From this follows

$$ \frac{h(\varOmega _i ')}{|\,\varOmega _i '\,|}\,\ge \,\frac{h(\varOmega _i)}{|\,\varOmega _i \,|} .$$

Therefore,

$$ \liminf _{i \rightarrow \infty } \frac{h(\varOmega _i ')}{|\,\varOmega _i '\,|}\,\ge \,\liminf _{i \rightarrow \infty } \frac{h(\varOmega _i)}{|\,\varOmega _i \,|} .$$

We assume that $\varGamma = \underbrace{\mathbb {Z} \oplus \mathbb {Z}\oplus ... \oplus \mathbb {Z}}_n$ and $\gamma _1,\gamma _2,..,\gamma _n$ is the canonical basis.

Take k such that $(2 k +1)^n >N$. For $i> 5 \,k + 2$, take $m>1$ such that $k + m\, (2 k +1) \le i \le k + (m+1)\, (2\, k +1).$

Consider

$$ \varOmega _{k,i} ' = \cup \,\,\{\,\,\varOmega _k ' + (\,j_1 (2 k +1),\ldots ,j_n (2 k+1)\,)\,|\, j_1 \,\text {is even}\,, -m\le j_1\le m -1,$$

$$\, -m\le j_2,..,j_n\le m, \, (j_1,j_2,\ldots ,j_n)\ne (0,\ldots ,0) \,\,\}.$$

Then, $\varOmega _{k,i}' \subset \varOmega _i$, and $\varOmega _{k,i} '$ is a finite union of disjoints translates of $\varOmega _k '$. Moreover $\lim _{i \rightarrow \infty } \frac{|\varOmega _{k,i} '|}{|\varOmega _i|}=1$,

$$ | \,\varOmega _{k,i} '\,|\ge 2\, (2 \, k +1)^n>N\,\,\text {and}\,\,\, |\, \varOmega _i - \varOmega _{k,i} '\,|\ge 2\, (2 \, k +1)^n >N\,\,.$$

From this follows that

$$h( \varOmega _i ) = h( \varOmega _{k,i} ' \, \cup \, (\,\varOmega _i - \varOmega _{k,i} ' )\,)\ge h (\,\varOmega _{k,i} '\,), $$

By the other hand, all translate of $\varOmega _k '$ has cardinality bigger than N.

Therefore,

$$ h( \varOmega _{k,i} ' ) \ge \frac{|\, \varOmega _{k,i} '\,|}{|\, \varOmega _k '\,|}\, h(\varOmega _k ') .$$

Then,

$$ \frac{\, h(\varOmega _{i} \,)}{|\, \varOmega _i \,|}\,\ge \frac{\, h(\varOmega _{k,i}' \,)}{|\, \varOmega _i \,|}\,\ge \frac{1}{ |\varOmega _i|}\,\frac{\, |\,\varOmega _{k,i} '\,|\,h(\varOmega _{k} ' \,)}{|\, \varOmega _k ' \,|}\,=\frac{|\, \varOmega _{k,i} ' \,|}{|\, \varOmega _i\,|}\,\,\frac{\, h(\varOmega _{k} ' \,)}{|\, \varOmega _k '\,|}\,.$$

Now, for a fixed k, taking $i \rightarrow \infty$ in the above inequality we get

$$ \lim _{i \rightarrow \infty } \frac{\, h(\varOmega _{i} \,)}{|\, \varOmega _i \,|}\,\ge \frac{\, h(\varOmega _{k} ' \,)}{|\, \varOmega _k '\,|}\,.$$

From this follows that

$$ \lim _{i \rightarrow \infty } \frac{\, h(\varOmega _{i} \,)}{|\, \varOmega _i \,|}\,\ge \limsup _{k \rightarrow \infty }\,\frac{\, h(\varOmega _{k} ' \,)}{|\, \varOmega _k '\,|}\,.$$

$\square$

6 Properties of b(c)

Lemma 16

There exists $c\in [0,1]$ such that

$$ b(c) \ge \log ( \,\text {dim}\, H^* (M,\mathbb {R})\,)>0$$

Proof

Note that dim  $(\,H^* (M)\,)\ge 2$ because dim $M\ge 1$. Let q be the number of connected components of M.

If $|\varOmega _i|=m_i$, take $0=t_0<t_1< \cdots <t_{m_i} =1$, a partition of [0, 1] in $m_i$ intervals of the same size. By Lemma 1

$$ H^* ( f_{\varOmega _i}^{-1} (-\infty , t_{m_i})) = \oplus _{r>0} H^r (M^{\varOmega _i},\mathbb {R}),$$

Denote $A_{ij}$ a supplement of $H^* ( f_{\varOmega _i}^{-1} (-\infty , t_{j-1}))$ in $H^* ( f_{\varOmega _i}^{-1} (-\infty , t_{j}))$, $1\le j\le m_{i}$. Then,

$$ \sum _{j=1}^{m_i} \text {dim} \, A_{ij} =\, \text {dim} \, H^* ( f_{\varOmega _i}^{-1} (-\infty , t_{m_i})) =\, \text {dim}\, H^* (M^{\varOmega _i},\mathbb {R})-q. $$

Therefore, there exists a certain $A_{i \, j_i}=A_i$, such that,

$$ \text {dim}\, A_i\, \ge \, \frac{(\text {dim} \,H^* (M,\mathbb {R}))^{m_i} -q}{m_i}.$$

Denote $s_i$ the middle point of $(t_{j_i -1}, t_{j_i}]$ and $\delta _i = \frac{1}{2\, m_i}.$

Then, by definition of $b_i ' (s_i,\delta _i) =$ dim $A_i$.

There exists a subsequence $s_{i_k}\rightarrow c\in [0,1]$, when $k \rightarrow \infty$.

Given $\delta >0$, there exists a $K>0$ such that $\delta _{i_k} < \delta /2$ and $|s_{i_k} - c| < \delta /2$, if $k>K$.

This means $c-\delta < s_{i_k} - \delta _{i_k}$ and $s_{i_k} + \delta _{i_k} < c + \delta .$

From this follows that $b_{i_k} ' (c,\delta )\ge b_{i_k} ' (s_{i_k},\delta _{i_k})=$ dim $A_{i_k}$.

Finally, we get

$$ \frac{ \log (\,b_{i_k} ' (c,\delta )\,)}{|\, \varOmega _{i_k}\,|} \ge \frac{1}{m_{i_k} }\, \log \frac{(\text {dim}\, H^* (M,\mathbb {R}))^{m_{i_k}}-q }{m_{i_k}}. $$

Now, taking limit in $k \rightarrow \infty$ in the above expression we get

$$ b ' (c, \delta ) \ge \log ( \text {dim}\, (H^* (M,\mathbb {R})).$$

$\square$

Lemma 17

The function b(c) is upper semicontinuous.

Proof

Suppose $c_k$, $k \in \mathbb {N}$ is a sequence of points in [0, 1] such that, $c_k \rightarrow c$.

Given $\varepsilon >0$, take $\delta >0$, such that, $b ' (c,\delta ) < b(c) + \varepsilon$. There exists a $N>0$ such that $|c-c_k|< \delta /2$, if $k \ge N.$ Then, $c- \delta < c_k - \delta /2$ and $c_k + \delta /2< c + \delta$, if $k \ge N.$

Then, $b_i ' (c, \delta ) \ge b_i ' (c_k, \delta /2)$, if $k\ge N$, for all $i=1,2,3,\ldots$

From this follows that $b' (c, \delta ) \ge b ' (c_k, \delta /2)$. Therefore,

$$ b(c) + \varepsilon > b ' (c, \delta ) \ge b ' (c_k , \delta /2) \ge b(c_k) ,\, \text {if}\,\, k\ge N.$$

Therefore

$$ \limsup _{k \rightarrow \infty } b(c_k) \le b(c) + \varepsilon ,$$

for any $\varepsilon >0$. From this it follows the claim. $\square$

Lemma 18

The function b(c) is concave.

Proof

Consider $0\le c_1< c_2\le 1$ and $0\le t\le 1$, we will show that

$$ b(\, t\, c_1 + (1-t)\, c_2) \ge t\, b(c_1) + (1-t)\, b (c_2).$$

First we will show the claim for $t=1/2$. Denote $\tilde{\varOmega }_i = \varOmega _i + (2\, i +1) \gamma _1$ and $\varOmega _i ' = \varOmega _i \cup \tilde{\varOmega }_i$.

By Proposition 2 and Lemma 12 we get:

$$ b_{\varOmega _i '} '(1/2\, c_1 + \,1/2\, c_2, \, \delta ) \ge b_{\varOmega _i} ' (c_1, \delta )\, b_{ \tilde{\varOmega }_i} ' (c_2, \delta )= b_{i} ' (c_1, \delta ) \, b_i ' (c_2, \delta ),$$

for all $\delta >0$.

Now, applying Lemma 15 to $h(\varOmega ) = \log b_{\varOmega } ' ( 1/2\, c_1 + 1/2\, c_2, \delta )$, we get $b ' ( 1/2\, c_1 + 1/2\, c_2, \delta )\ge 1/2\, b ' (\, c_1, \delta )\, +\,1/2\,b ' ( \, c_2, \delta )$.

Now, taking $\delta \rightarrow 0$, we get $b ( 1/2\, c_1 + 1/2\, c_2)\ge 1/2 b (\, c_1)\,+\, 1/2\, b ( \, c_2)$.

The inequality we have to prove is true for a dense set of values of t in [0, 1]. Then, by Lemma 17 is true for all $t\in [0,1]$. $\square$

Corollary 3

The function b(c) is continuous for $c \in [0,]$.

Proof

This follows from Lemmas 17 and 18. $\square$

We summarize the above results in the following theorem.

Theorem 1

(a) $0 \le b(c) \le \varepsilon (c) \le \log ($ number of critical points of $f_0$), for all $0 \le c \le 1$.

(b) b(c) is continuous on [0, 1]

(c) b(c) is concave, that is, its graph is always above the cord

(d) b(c) is not constant equal zero. Moreover, there exists a point c where $b(c)\ge \log$ ( dim $H^* (M,\mathbb {R})\,)>0$.

7 An Example

The next example shows that the item (d) in the above theorem can not be improved.

Take $M=S^n$, $n \ge 1$, and a Morse function $f_0: M \rightarrow [0,1]$ which is surjective with only two critical points. Suppose $x_{-}$ is the minimum and $x_{+}$ the maximum of $f_0$. We will compute b(c) and $\varepsilon (c)$.

Take $\varOmega \in B(\varGamma )$ with $|\varOmega |=m\ge 1$. For each $\varOmega ' \subset \varOmega$ consider the canonical projection $p_{\varOmega '}: M^\varOmega \rightarrow M^{\varOmega '}.$ Now, take

$$ \mu ^{\varOmega '} = p_{\varOmega '}^* (\,[\,M^{\varOmega '}\,]\, ) \in H^{n\, |\,\varOmega '\,|}(M^{\varOmega },\mathbb {R}),$$

where $[\,\,\,]$ represents fundamental class. Then,

$$ \{\, \mu ^{\varOmega '} \, :\, \varOmega ' \subset \varOmega \} $$

is a $\mathbb {R}$-homogeneous basis of $H^* (M^{\varOmega },\mathbb {R}).$

For $0\le d\le 1$ denote

$$ L_d =\{ x \in M^\varOmega \, : \, f_\varOmega (x) <d\, \} \subset M^\varOmega .$$

For $x \in M^\varOmega$ we denote by $x_\gamma$ the corresponding coordinate, where $\gamma \in \varGamma$.

Lemma 19

If $0\le d\le 1$, where d is not rational, then

$$\{ \mu ^{\varOmega '} \, :\, |\varOmega '|> m\, (1-d)\,\}$$

is a basis of $H^* (L_d)$.

Proof

Take $K_d = M^\varOmega - L_d$. By Lemma 1

$$ H^* (L_d) = \text {Ker} (\,H^* (M^\varOmega ,\mathbb {R}) \,\rightarrow \, H^* ( K_d,\mathbb {R})\,)\,\,(\text {natural restriction}). $$

The claim follows from

(1) $H^k (M^\varOmega ,\mathbb {R}) \rightarrow H^k (K_d,\mathbb {R})$ is zero if $k> m \,(1-d)\, n$, and

(2) $H^k (M^\varOmega ,\mathbb {R}) \rightarrow H^k (K_d,\mathbb {R})$ is injective if $k< m \,(1-d)\, n$.

Now we prove (1) and (2).

(1)   Suppose $\varOmega ' \subset \varOmega$ is such that $\mu ^{\varOmega '} \in H^k (M^\varOmega )$ where $k> m\, (1-d)\, n$. Then, $|\varOmega '| > m\, (1-d)$. Suppose

$$ F_{\varOmega '} = \{ x\in M^\varOmega \, :\, x_\gamma = x_{-}\, ,\,\,\text {if}\, \gamma \in \varOmega '\} .$$

If $x \in F_{\varOmega '}$, then $f_\varOmega (x) \le \frac{1}{m}\, (m- | \varOmega '|)< d$. Then, $F_{\varOmega '} \cap K_d=\emptyset .$ This means that: if $x \in K_d\, \rightarrow \, x_\gamma \ne x_{-}$ for some $\gamma \in \varOmega '.$ Then, $K_d \subset p^{-1}_{\varOmega '} (M^{\varOmega '} -\,\{z\})$ where $z_\gamma =x_{-}$ for all $\gamma \in \varOmega '$.

From this follows

$$ \mu ^{\varOmega '}\, |\, K_d\,=\, p^{*}_{\varOmega '}\,(\,[\,M^{\varOmega '} \,]\,)\,\,|\,\, K_d\,=0,\,\,\text {because}\,[\,M^{\varOmega '} \,]\,\,|\, \, (\,[\,M^{\varOmega '} \,]\,-\{z\})=0.$$

(2)  Denote $T=\{x \in M^\varOmega \,:\, \text {cardinality} (\, \{\gamma \,:\, x_\gamma = x^{+} \,\})\,>\, m\, d\,\}.$ The set T is closed.

If $x \in T$, then $f_\varOmega (x) > \frac{1}{m}\, m\, d =d$. Then, $T\subset K_d$.

We have to show that

$$ H^k (M^\varOmega ,\mathbb {R}) \rightarrow H^k (T,\mathbb {R})\,\,\text {is injective if}\,\, k<m\, (1-d)\, n.$$

As we had seen before $H^k (M^\varOmega ,\mathbb {R})=0$ if k is not multiple of n. Then, we can assume that $k=q\, n$, if $q=0,1,2,\ldots$. The claim follows from the next lemma, taking s the integer part of $m\, d$, by the exact sequence of homology, given that $U= U_s (\varOmega ).$

Lemma 20

Suppose $s=0,1,2,..,m$. Suppose

$$U_s (\varOmega ) \,=\, \{x \in M^\varOmega \,:\, \text {card}\, ( \,\{\gamma \,:\, x_\gamma = x^{+} \}\,) \,\,\le s \},$$

then, $H^k_c ( U_s ( \varOmega ),\mathbb {R}) =0$, if $k <(m-s)\,n$.

Proof

The claim is trivial for $s=0$ or $s=m$ ($U_0(\varOmega )$ is homeomorphic to $(\,\mathbb {R}^n\,)^m$).

The proof is by induction in m. The claim for $m=1$ is trivial. Suppose is true for $m-1\ge 1$. Take $0<s<m$. Fix $w\in \varOmega$ and take $\varOmega ' = \varOmega -\{w\}$.

Consider $\varphi : M^{\varOmega '} \rightarrow M^\varOmega$ and $\psi : M^{\varOmega '} \times (M-\{\,x^{+}\,\}) \, \rightarrow \, M^\varOmega$, where for a given x we define $\varphi (x)$ by $x_\omega = x^{+}$ if $x \in M^{\varOmega ' },$ and $\psi (x,u)$ is defined by $x_w =u$ if $x \in M^{\varOmega '}$ and $u \in M,$ $u \ne x^{+}.$

$\psi$ identifies $U_s (\varOmega ') \times (\,M - \{x^{+}\}\,)$ with an open set A contained in $U_s (\varOmega )$.

Moreover, $\varphi$ identifies $U_{s-1} (\varOmega ')$ with the complement of this open set A in $U_s (\varOmega )$.

As $M - \{x^{+}\}$ is homeomorphic to $\mathbb {R}^n$ and by recurrence we get that

$$ H^k_c ( \,U_s (\varOmega ')\, \times \, (\,M - \{x^{+}\}\,)\,,\mathbb {R})=0, $$

if $k< (m -1 -s)\, n +n = (m-s)\, n$ and, moreover, $H^k_c (\,U_{s-1} (\varOmega ',\mathbb {R})\,)=0$, if $k< (\, (m-1) - (s-1)\,) \, n= (m-s)\, n$.

Now, using the exact sequence of homology we finish the proof. $\square$

Now we fix irrationals $d_1,d_2$, $0< d_1<d_2<1$. Denote $a_m = m\, (1-d_1)$, $b_m = m (1-d_2),$ and, $c_m =$ dim $(\, H^* (L_{d_2})/ H^* (L_ {d_1})\,)$.

By Lemma 19 we get

$$ c_m = \sum \,\{ \left( \begin{array}{cc} m\\ j \end{array} \right) :\, b_m< j<a_m \}.$$

Assume m is much bigger than $(d_2-d_1)$.

Take an integer $j_m$, such that $b_m<j_m<a_m$,

$$ \left( \begin{array}{cc} m\\ j_m \end{array} \right) \, =\, \sup \, \{ \left( \begin{array}{cc} m\\ j \end{array} \right) :\, b_m< j<a_m \}. $$

Then,

$$ \left( \begin{array}{cc} m\\ j_m \end{array} \right) \, \le c_m \le (a_m - b_m +1)\, \left( \begin{array}{cc} m\\ j_m \end{array} \right) \, .$$

By Stirling formula:

$$ \frac{1}{m} \log \left( \begin{array}{cc} m\\ j \end{array} \right) \, \sim \frac{1}{m} \, \log \left( \, \frac{m^{m + \,\,\,1/2}}{j^{j + \,\,\,1/2}\, (m-j)^{\,m-j + \,\,\,1/2} }\,\right) =$$

$$\frac{1}{m} \log \left( \,m^{-1/2} \,\left( \frac{j}{m}\right) ^{-1/2} \,\left( 1- \frac{j}{m}\right) ^{-1/2}\, \left( \frac{j}{m}\right) ^{-j}\, \left( 1-\frac{j}{m}\right) ^{ -m + j} \,\right) . $$

Therefore,

$$ \frac{1}{m} \log \left( \begin{array}{cc} m\\ j_m \end{array} \right) \, \sim \frac{1}{m} \log \left( \, \left( \frac{j_m}{m}\right) ^{-j_m}\, \left( 1-\frac{j_m}{m}\right) ^{ -m + j_m} \,\right) = $$

$$ - \frac{j_m}{m} \, \log \left( \frac{j_m}{m}\right) \,-\, \left( 1 - \frac{j_m}{m} \right) \, \log \left( 1- \frac{j_m}{m}\right) ,$$

when $m \sim \infty$.

As $1- d_2< \frac{j_m}{m}< 1- d_1$, then (changing x by $(1-x)$) we get

$$ \limsup _{m \rightarrow \infty } \frac{1}{m} \log \left( \begin{array}{cc} m\\ j_m \end{array} \right) \,\le \, \sup _{d_1<x<d_2} \,(\,-\, x \log (x) - (1-x)\,\log (1-x)\,) ,$$

and

$$ \liminf _{m \rightarrow \infty } \frac{1}{m} \log \left( \begin{array}{cc} m\\ j_m \end{array} \right) \,\ge \, \inf _{d_1<x<d_2} \,(\,-\, x \log (x) - (1-x)\,\log (1-x)\,) .$$

From this follows

$$ \limsup _{m \rightarrow \infty } \frac{\log c_m}{m} \,\le \, \sup _{d_1<x<d_2} \,(\,-\, x \log (x) - (1-x)\,\log (1-x)\,) ,$$

and

$$ \liminf _{m \rightarrow \infty } \frac{\log c_m}{m} \,\ge \, \inf _{d_1<x<d_2} \,(\,-\, x \log (x) - (1-x)\,\log (1-x)\,) .$$

Proposition 6

$$\varepsilon (c) = b(c) = -\,c\, \log c -\, (1-c)\, \log (1-c), \,\,0 \le c\le 1.$$

Proof

Given $0<c<1$, there exists small $\delta >0$ such that

$$ 0< c - \delta<c< c+ \delta <1\,\,\,\text {and}\,\,\ c-\delta , c+\delta \,\,\text {are not in }\,\,\mathbb {Q}.$$

From the above for $d_1= c -\delta$ and $d_2= c + \delta$ we get

$$ \inf _{d_1<x<d_2} \,(\,-\, x \log (x) - (1-x)\,\log (1-x)\,)\, \le b ' (c, \delta ) \le $$

$$\sup _{d_1<x<d_2} \,(\,-\, x \log (x) - (1-x)\,\log (1-x)\,) .$$

Now, taking $\delta \rightarrow 0$, we get

$$ b(c)\,=\,(\,-\, c \log (c) - (1-c)\,\log (1-c)\,) .$$

For $c=0$ or $c=1$ the result follows from continuity.

Now we will estimate $\varepsilon (c).$

The critical values of $f_\varOmega$ are $0,\frac{1}{m},\frac{2}{m},\ldots , 1$.

To the critical values $\frac{j}{m}$ ($j=0,1,2, \ldots ,m$) corresponds $\left( \begin{array}{cc} m\\ j \end{array} \right) \,$ critical points.

Therefore, given $d_1,d_2 \in \mathbb {R}$ $d_1<d_2$, the number $c_m '$ of critical points of $f_\varOmega$ in $f^{-1}_\varOmega (d_1,d_2)$ is

$$ c_m ' = \sum \, \{ \left( \begin{array}{cc} m\\ j \end{array} \right) \, :\, d_1<\, \frac{j}{m}\, < d_2\, \}=$$

$$\sum \, \{ \left( \begin{array}{cc} m\\ j \end{array} \right) \, :\, m\, (1-d_2)< \,j \,< m\,(1-d_1)\, \} .$$

The computation of $\varepsilon (c)$ is analogous to the one for b(c). This also follows from the last Theorem and the fact that $H^* (M)$ = number critical points of $f_0$ in the present case. $\square$

8 About the Definition of b(c)

We will show that the definition of b(c) presented here coincides with the one in [1].

Suppose X is a compact connected oriented $C^\infty$ manifold.

Lemma 21

Given an open set V in X consider $\alpha \in H^* (X,\mathbb {R})$ such that $\alpha |_V \ne 0.$ Then, there exists $\beta \in H^* (V)$ such that $\alpha \wedge \beta \ne 0.$

Proof

Take $w \in \alpha$. As $\alpha |_V \ne 0$, then there exists a cycle z on V such that $\int _z w \ne 0.$

Suppose $w'$ is a closed form with compact support on V such that its cohomology class in $H^*_c (V,\mathbb {R})$ is the Poincare dual of the homology class of z in $H_* (V,\mathbb {R}).$

$w'$ can be extended to a closed form on X (putting 0 where needed) and by Poincare duality:

$$ 0 \ne \int \limits _z w\,=\,\int \limits _V w \wedge w'\, = \int \limits _X w\, \wedge w'.$$

Therefore, $w \wedge w'$ is not exact on X.

Denote $\beta \in H^* (X,\mathbb {R})$ the cohomology class of $w'$. By Lemma 1 we have that $\beta \in H^*(V)$. As $w \wedge w'$ is not exact we get that $\alpha \wedge \beta \ne 0.$ $\square$

Notation: if $S \subset X$, then $\mathscr {H}^* (S)=\cap \,\{ H^* (W) \, : W \subset X$ is an open set and $S \subset W \}$.

Lemma 22

Suppose $U,V \subset X$ are open sets and $X=U \cup V$. Take $K=U-V$ and $\alpha \in H^* (U).$ Then, $\alpha \wedge \beta =0$ for all $\beta \in H^* (V)$, if and only if, $\alpha \in \mathscr {H}^* (K).$

Proof

Suppose $\alpha \in \mathscr {H}^* (K)$ and take $\beta \in H^* (V)$. By Lemma 5 there exists $w \in \beta$ such that supp $w \subset V$.

Take $W= X-$ supp w (which contains K). By definition we get that $\alpha \in H^* (W).$ Then, by Lemma 5, there exists $w ' \in \alpha$ such that supp $w' \subset W$. Therefore, $w \wedge w'=0,$ and finally it follows that $\alpha \wedge \beta =0.$

Reciprocally, suppose that $\alpha \wedge \beta =0$ for all $\beta \in H^*(V)$. By Lemma 21 we have that $\alpha \,|V =0$. Take $W \supset K$, then $V \cup W=X$. Therefore, by definition $\alpha \in H^* (W).$ $\square$

Lemma 23

Take $K \subset X$ a compact submanifold with boundary such that $K- \delta K$ is an open subset of X.

Then,

$$ \mathscr {H} (K) = \, \text {Ker}\, \,(\,H^* (X,\mathbb {R}) \rightarrow H^* (X-K,\mathbb {R})\,)\,\,\text {restriction}.$$

WE leave the rest of the proof for the reader.

Proof

Take W an open set by adding a necklace to K. Then, $X-K$ can be retracted by deformation over $X-W$.

Then, if $\alpha \in H^* (X,\mathbb {R})$, we get that $\alpha |_{X-K} =0$ is equivalent to $\alpha |_{X-W}=0.$

Now, the claim follows from Lemma 1 and by the definition of $\mathscr {H}(K).$ $\square$

Corollary 4

Under the same hypothesis of last lemma it also follows that $\mathscr {H} (K) = H^* ($ int $(K)\,)$.

Proof

This follows from the fact that $H^* ( X -$ int $(K)\,,\mathbb {R}) \rightarrow H^* (X-K,\mathbb {R})$ is an isomorphism. $\square$

Proposition 7

Suppose U, V are open sets such that $X= U\cup V$ and moreover that $\overline{U},\overline{V}$ are submanifolds with boundary of X.

Consider the linear transformation L such that

$$L: H^* (U) \rightarrow \text {Hom}\, (\,H^* (V) , H^* (U \cap V)\,),$$

where, $a \rightarrow \,(\, b \rightarrow a \wedge b\,).$

Then, the rank of L is dim $( \,H^*(U)/ H^* (M - \overline{V})\,).$

Proof

By Lemma 22 we get that Ker $L= H^* (X-V)$. Finally, by the last corollary $H^* (X-V)= H^* (M-\overline{V}).$ $\square$

Consider now a Morse function $f:X \rightarrow \mathbb {R}$ and $c\in \mathbb {R}$, $\delta >0.$

Definition 5

$b_{c,\delta } (f)$ is the rank of the linear transformation

$$ H^* (\,f^{-1} (-\infty , c + \delta )\,)\, \rightarrow \, \text {Hom}\, (\,H^* (f^{-1} (\, c - \delta , \infty \,) ,H^* (f^{-1} (c-\delta , c + \delta )\,)\, ) ,$$

where $a \rightarrow (b\, \rightarrow a \wedge b).$

Note that $b_{c,\delta } (f)$ decreases with $\delta$.

Lemma 24

If $c-\delta$ and $c+ \delta$ are regular values of f, then

$$ b_{c,\delta } (f)= b_{c,\delta }' (f).$$

Proof

Just apply Proposition 7 to $U= f^{-1} (-\infty , c + \delta )$ and $V= f^{-1} (\, c - \delta , \infty \,).$ $\square$

Note that $b_\varOmega (c,\delta )= b_{c, \delta } ( f_\varOmega )$, where $\varOmega \in B(\varGamma )$ and $\varOmega \ne \emptyset$, and moreover that $b_i (c,\delta ) = b_{\varOmega _i} (c, \delta ).$ The next limit exists (see [1]).

Definition 6

$$ b(c,\delta ) = \lim _{i \rightarrow \infty } \, \frac{\log (\,b_i (c, \delta )\,)}{|\, \varOmega _i\,|}. $$

The set $S\subset [0,1]$ of all critical values of all $f_\varOmega$ is countable. By Lemma 24 we get that $b_i ' (c, \delta )=b_i (c, \delta )$ if $c-\delta \notin S$ and $c+\delta \notin S$. Therefore, $b'(c,\delta ) = b(c,\delta )$ if $c-\delta \notin S$ and $c+\delta \notin S$.

Finally,

$$ \lim _{\delta \rightarrow 0} b ' (c,\delta ) = \lim _{\delta \rightarrow 0} b(c,\delta )$$

because both limits exist.

Therefore the function b(c) we define coincides with the one presented in [1].