On Equivariantly Formal 2-Torus Manifolds

Abstract

A 2-torus manifold is a closed connected smooth n-manifold with a non-free effective smooth $\mathbb Z^n_2$-action. In this paper, we prove that a 2-torus manifold is equivariantly formal if and only if the $\mathbb Z^n_2$-action is locally standard and every face of its orbit space (including the whole orbit space) is mod 2 acyclic. Our study is parallel to the study of torus manifolds with vanishing odd-degree cohomology by M. Masuda and T. Panov in (2006). As an application, we determine when such kind of 2-torus manifolds can have regular $\textrm{m}$-involutions (i.e., involutions with only isolated fixed points of the maximum possible number).

1 Introduction

Let G be a compact Lie group and BG be the classifying space of G. For a G-space X, the G-equivariant cohomology of X with coefficients in a field $\textbf{k}$ is the singular cohomology of the Borel construction $X_G$ (see [6])

$$\begin{aligned} H^*_G(X;\textbf{k}):=H^*(X_G;\textbf{k}). \end{aligned}$$

There is a natural fibration $X\rightarrow X_G \rightarrow BG$ associated with $X_G$ called the Borel fibration. If the inclusion of the fiber $\iota _X: X\rightarrow X_G$ induces a surjection on cohomology $\iota _X^*: H^*_G(X;\textbf{k})\rightarrow H^*(X;\textbf{k})$, X is called (cohomologically) equivariantly formal over $\textbf{k}$. This term was coined in 1998 in Goresky-Kottwitz-MacPherson [18]. But this condition had already been studied by A. Borel in  [5, § 4] and [6, Ch. XII] where X is called totally non-homologous to zero in $X_G$ (also, see [7, Ch. VII]).

For some special groups G shown below, the equivariant formality of a G-action can be interpreted in some other ways (see [5, § 4], [1, Ch. 3], and [2, Sec. 4]).

• When BG is simply connected (e.g., G is a torus $T^r=(S^1)^r$), X is equivariantly formal if and only if the Serre spectral sequence of the Borel fibration of X degenerates at the $E_2$ stage.

• When G is the p-torus $\mathbb Z^r_p$ (p is prime), X being equivariantly formal is equivalent to either one of the following conditions.

  • (i) The Serre spectral sequence with $\mathbb Z_p$-coefficients of the Borel fibration of X degenerates at the $E_2$ stage and the induced action of $\mathbb Z^r_p$ on $H^*(X;\mathbb Z_p)$ is trivial.

  • (ii) $H^*_{\mathbb Z^r_p}(X;\mathbb Z_p)\cong H^*(X;\mathbb Z_p)\otimes H^*(B\mathbb Z^r_p;\mathbb Z_p)$ is a free $H^*(B\mathbb Z^r_p;\mathbb Z_p)$-module.

Due to the above fact, we call a $\mathbb Z^r_p$-action on X weakly equivariantly formal if we only assume that the Serre spectral sequence (with $\mathbb Z_p$-coefficients) of the Borel fibration of X degenerates at the $E_2$ stage. So an equivariantly formal $\mathbb Z^r_p$-action is always weakly equivariantly formal.

When $G=T^r$ or $\mathbb Z^r_2$ and $\textbf{k}=\mathbb Q$ or $\mathbb Z_2$ respectively, there is another equivalent description of equivariantly formal G-actions given by the so called “Atiyah-Bredon sequence” (see Bredon [8] and Franz-Puppe [16] for the $T^r$ case, and Allday-Franz-Puppe [2] for the $\mathbb Z^r_2$ case). In addition, there are many sufficient conditions for a $T^r$-action to be equivariantly formal (for example: vanishing of odd-degree cohomology, all homology classes being representable by $T^r$-invariant cycles, etc.).

Equivariantly formal G-spaces provide many nice examples in geometry and topology. Some of them are as follows:

• Smooth compact toric varieties.

• Hamiltonian G-actions on symplectic manifolds which have moment maps (see Atiyah-Bott [3] and Jeffrey-Kirwan [22]).

• Quasitoric manifolds and small covers defined in Davis-Januszkiewicz [14].

• Torus manifolds with vanishing odd degree cohomology (see Masuda-Panov [27]).

In addition, when $G=T^r$ or $(\mathbb Z_p)^r$, the following theorem gives us an easy way to recognize equivariantly formal G-actions.

Theorem 1.1

(see Theorem (3.10.4) in Allday-Puppe [1]) Let $G=T^r$ or $(\mathbb Z_p)^r$ where p is a prime and $\textbf{k}=\mathbb Q$ or $\mathbb Z_p$ respectively. Let X be a paracompact G-space with only finitely many orbit types and $\dim _{\textbf{k}} H^*(X;\textbf{k})<\infty$. Then, the fixed point set $X^G$ always satisfies

$$\begin{aligned} \dim _{\textbf{k}} H^*(X^G;\textbf{k}) \le \dim _{\textbf{k}} H^*(X;\textbf{k}) \end{aligned}$$

where the equality holds if and only if X is equivariantly formal over $\textbf{k}$. Here $\dim _{\textbf{k}} H^*(X;\textbf{k})$ denotes the sum of the rank of the cohomology groups of X in all dimensions over $\textbf{k}$.

A very special case is when $G=\mathbb Z_2$ and $X^{\mathbb Z_2}$ consists only of isolated points. By Theorem 1.1, we have

$$\begin{aligned} |X^{\mathbb Z_2}| = \dim _{\mathbb Z_2}H^*(X^{\mathbb Z_2};\mathbb Z_2) \le \dim _{\mathbb Z_2} H^*(X;\mathbb Z_2) \end{aligned}$$

(1)

Such a $\mathbb Z_2$-action on X is equivariantly formal if and only if the number of the fixed points reaches the maximum, i.e., $|X^{\mathbb Z_2}| = \dim _{\mathbb Z_2}H^*(X;\mathbb Z_2)$. In this case, the involution determined by the $\mathbb Z_2$-action is called an $\textrm{m}$-involution on X (this term was named by Puppe [28]).

There is an interesting relation between $\textrm{m}$-involutions on closed manifolds and binary codes. It was shown in [28] that one can obtain a self-dual binary code from any $\textrm{m}$-involution on an odd-dimensional closed manifold. This motivates the study in Chen-Lü-Yu [12] on the $\textrm{m}$-involutions on a special kind of closed manifolds called small covers (see [14]). In this paper, we want to study a more general type of closed manifolds with 2-torus actions defined below.

Definition 1.2

(see Lü-Masuda [25]) A 2-torus manifold is a closed connected smooth n-manifold W with a non-free effective smooth action of $\mathbb Z^n_2$. For such a manifold W, since $\dim (W)=n=\textrm{rank}(\mathbb Z^n_2)$ and the $\mathbb Z^n_2$-action is effective, the fixed point set $W^{\mathbb Z^n_2}$ must be discrete. Then, since W is compact, $W^{\mathbb Z^n_2}$ is a finite set of isolated points (if not empty). Note that we require all 2-torus manifolds to be connected in this paper.

• For brevity, we call a 2-torus manifold W equivariantly formal or weakly equivariantly formal if the $\mathbb Z^n_2$-action on W is so, respectively.

• We call W locally standard if for every point $x \in W$, there is a $\mathbb Z^n_2$-invariant neighborhood $V_x$ of x such that $V_x$ is equivariantly homeomorphic to an invariant open subset of a real n-dimensional faithful linear representation space of $\mathbb Z^n_2$. An equivalently way to describe such a neighborhood $V_x$ is: $V_x$ is weakly equivariantly homeomorphic to an invariant open subset of $\mathbb R^n$ under the standard $\mathbb Z^n_2$-action defined by: for any $(x_1,\cdots , x_n)\in \mathbb R^n$ and $(g_1,\cdots , g_n)\in \mathbb Z^n_2$,

  $$\begin{aligned} \quad \ (g_1,\cdots , g_n) \cdot (x_1,\cdots , x_n)\longmapsto \big ((-1)^{g_1}x_1,\cdots , (-1)^{g_n}x_n \big ). \end{aligned}$$

• Every non-zero element $g\in \mathbb Z^n_2$ determines a nontrivial involution $\tau _g$ on W, called a regular involution on W.

We will prove in Theorem 3.3 that if a 2-torus manifold is equivariantly formal, then it must be locally standard.

For an n-dimensional locally standard 2-torus manifold W, the orbit space $Q\!=\!W/\mathbb Z^n_2$ naturally becomes a connected smooth nice n-manifold with corners and with non-empty boundary (since the $\mathbb Z^n_2$-action is non-free). Moreover,

• The $\mathbb Z^n_2$-action on W determines a characteristic function

  $$\begin{aligned} \lambda _W: \{F_1,\cdots , F_m\}\rightarrow \mathbb Z^n_2 \end{aligned}$$

  where $F_1,\cdots , F_m$ are all the facets (codimension-one faces) of Q.

• The free part of the $\mathbb Z^n_2$-action on W determines a principal $\mathbb Z^n_2$-bundle $\xi _W$ over Q.

It is shown in Lü-Masuda [25, Lemma 3.1] that W can be recovered from the data $(Q,\lambda _W, \xi _W)$ up to equivariant homeomorphism. In addition, let $\pi : W\rightarrow Q$ denote the orbit map. If f is a codimension-k face of Q, then $W_f:=\pi ^{-1}(f)$ is a codimension-k submanifold of W called a facial submanifold of W. Let $G_f$ denote the isotropy subgroup of $W_f$. Then, $W_f$ is also a 2-torus manifold with respect to the induced action of $\mathbb Z^n_2/G_f$. In the following, when we say $W_f$ is equivariantly formal, we always consider $W_f$ being equipped with the induced $\mathbb Z^n_2/G_f$-action from W.

The main purpose of this paper is to answer the following two questions:

Question-1: What kind of 2-torus manifolds are equivariantly formal?

Question-2: What kind of locally standard 2-torus manifolds have regular $\textrm{m}$-involutions?

Generally speaking, it is very hard to compute the equivariant cohomology of a locally standard 2-torus manifold W directly from its orbit space Q and the data $(\lambda _W, \xi _W)$. So it is difficult to judge whether W is equivariantly formal by directly verifying the condition in the definition. Meanwhile, it was proved by Masuda-Panov [27] that a smooth $T^n$-action on a connected smooth 2n-manifold with non-empty fixed points is equivariantly formal if and only if the $T^n$-action is locally standard and every face of its orbit space is acyclic (also see Goertsches-Töben [19, Theorem 10.19] for a reformulation of this result). This result is also implied by Franz [15, Theorem 1.3]. The arguments in [27] inspire us to prove the following parallel result for 2-torus manifolds.

Theorem 1.3

Let W be a 2-torus manifold with orbit space Q.

• (i) W is equivariantly formal if and only if W is locally standard and Q is mod 2 face-acyclic.

• (ii) W is equivariantly formal and $H^*(W;\mathbb Z_2)$ is generated by its degree-one part as a ring if and only if W is locally standard and Q is a mod 2 homology polytope.

The definitions of “mod 2 face-acyclic” and “mod 2 homology polytope” are given in Definition 2.1.

The main strategy in our proof of Theorem 1.3 is very similar to the strategy used in [27] for equivariantly formal torus manifolds. Besides, our proof uses the mod 2 GKM theory introduced in Biss-Guillemin-Holm [4] which allows us to observe the equivariant cohomology of an equivariantly formal 2-torus manifold by restricting to its fixed point set (see Sect. 2.3).

Remark 1.4

If a 2-torus manifold W is assumed to be locally standard in the first place, Theorem 1.3 (i) can also be derived from Chaves [11, Theorem 1.1] whose proof uses the theory of syzygies in the mod 2 equivariant cohomology (see Allday-Franz-Puppe [2, Theorem 10.2]) and the mod 2 “Atiyah-Bredon sequence”. But we will use a completely different approach in our proof here.

Using Theorem 1.3, we can easily derive the following theorem which gives an answer to Question-2.

Theorem 1.5

Let W be an n-dimensional locally standard 2-torus manifold with orbit space Q. Then, there exists a regular $\textrm{m}$-involution on W if and only if Q is mod 2 face-acyclic (or equivalently W is equivariantly formal) and the values of the characteristic function $\lambda _W$ on all the facets of Q consist exactly of a linear basis of $\mathbb Z^n_2$.

A nice manifold with corners Q is called k-colorable if we can assign k different colors to all the facets of Q so that no two adjacent facets are of the same color. Clearly, there exists a 2-torus manifold over Q whose characteristic function takes value in a linear basis of $\mathbb Z^n_2$ if and only if Q is n-colorable.

Remark 1.6

By Theorem 1.5 and the construction in Puppe [28], we can obtain a self-dual binary code $\mathcal {C}_Q$ from an n-colorable mod 2 face-acyclic nice smooth n-manifold with corners Q when n is odd. This generalizes the self-dual binary codes from n-colorable simple convex n-polytopes in Chen-Lü-Yu [12]. Moreover, we can write down $\mathcal {C}_Q$ explicitly in the same way as the self-dual binary code obtained in [12, Corollary 4.5].

The paper is organized as follows. In Sect. 2, we review the definitions and some basic facts of locally standard 2-torus manifolds and quote some well known results that are useful for our proof. In Sect. 3, we study various properties of equivariantly formal 2-torus manifolds. Since the philosophy of our study is very similar to the study of torus manifolds with vanishing odd degree cohomology in Masuda-Panov [27], many lemmas in this paper are parallel to those in [27]. In Sect. 4, we prove some special properties of equivariantly formal 2-torus manifolds whose mod 2 cohomology rings are generated by their degree-one part. Then finally, in Sect. 5, we prove Theorem 1.3 and Theorem 1.5.

2 Preliminaries

2.1 Manifolds with Corners and Locally Standard 2-Torus Manifolds

Recall a (smooth) n-dimensional manifold with corners Q is a Hausdorff space together with a maximal atlas of local charts onto open subsets of $\mathbb R_{\ge 0}^n$ such that the transition functions are (diffeomorphisms) homeomorphisms which preserve the codimension of each point. Here, the codimension c(x) of a point $x=(x_1,\cdots ,x_n)$ in $\mathbb R_{\ge 0}^n$ is the number of $x_i$ that is 0. So we have a well defined map $c: Q\rightarrow \mathbb Z_{\ge 0}$ where c(q) is the codimension of a point $q\in Q$. An open face of Q of codimension k is a connected component of $c^{-1}(k)$. A (closed) face is the closure of an open face. A face of codimension one is called a facet of Q. When Q is connected, we also consider Q itself as a face (of codimension zero).

• For any $k\in \mathbb Z_{\ge 0}$, the k-skeleton of Q is the union of all the faces of Q with dimension $\le k$.

• The face poset of Q, denoted by $\mathcal {P}_Q$, is the set of faces of Q ordered by reversed inclusion (so Q is the initial element).

A manifold with corners Q is said to be nice if either its boundary $\partial Q$ is empty or $\partial Q$ is non-empty and any codimension-k face of Q is a component of the intersection of k different facets in Q. If Q is nice, $\mathcal {P}_Q$ is a simplicial poset. But in general $\mathcal {P}_Q$ may not be the face poset of a simplicial complex. Indeed, $\mathcal {P}_Q$ is the face poset of a simplicial complex if and only if all non-empty multiple intersections of facets of Q are connected (see [27, Sec. 5.2]).

Definition 1.1

Let Q be a nice manifold with corners.

• We call Q mod 2 face-acyclic if every face of Q (including Q itself) is a mod 2 acyclic space.

• We call Q a mod 2 homology polytope if Q is mod 2 face-acyclic and $\mathcal {P}_Q$ is the face poset of a simplicial complex.

A topological space B is called mod 2 acyclic if $H^*(B;\mathbb Z_2)\cong H^*(pt;\mathbb Z_2)$.

It is not difficult to prove the following lemma (see [27, p.743 Remark] for a short argument).

Lemma 1.2

If Q is mod 2 face-acyclic, then every face of Q has a vertex and the 1-skeleton of Q is connected.

In the following, let W be an n-dimensional locally standard 2-torus manifold with orbit space Q. Then, Q is a smooth nice manifold with corners with $\partial Q eq \varnothing$. Let $\pi : W\rightarrow Q$ denote the projection, and let the set of facets of Q be

$$\begin{aligned} \mathcal {F}(Q) =\{F_1,\cdots , F_m \}. \end{aligned}$$

Then, $\pi ^{-1}(F_1),\cdots ,\pi ^{-1}(F_m)$ are embedded codimension-one closed connected submanifolds of W, called the characteristic submanifolds of W. Moreover, the $\mathbb Z^n_2$-action on W determines a characteristic function on Q which is a map

$$\begin{aligned} \lambda _W: \mathcal {F}(Q)\rightarrow \mathbb Z^n_2 \end{aligned}$$

(2)

where $\lambda _W(F_i)\in \mathbb Z^n_2$ is the generator of the $\mathbb Z_2$ subgroup that pointwise fixes the submanifold $\pi ^{-1}(F_i)$, $1\le i \le m$. Since the $\mathbb Z^n_2$-action is locally standard, the function $\lambda _W$ satisfies the following linear independence condition:

$$\begin{aligned} \quad \begin{array}{l} \text {whenever the intersection of }k\text { different facets }F_{i_1},\cdots , F_{i_k}\text { is non-empty,} \\ \text {the elements }\lambda _W(F_{i_1}),\cdots , \lambda _W(F_{i_k})\text { are linearly independent when viewed} \\ \text {as vectors of }\mathbb Z^n_2\text { over the field }\mathbb Z_2. \end{array} \end{aligned}$$

For a codimension-k face f of Q, let $F_{i_1},\cdots , F_{i_k}$ be all the facets containing f. Then, the isotropy subgroup of the facial submanifold $W_f$ is

$$\begin{aligned} G_f = \text {the subgroup generated by }\{\lambda _W(F_{i_1}),\cdots , \lambda _W(F_{i_k})\} \subseteq \mathbb Z^n_2. \end{aligned}$$

(3)

By the linear independence condition of $\lambda _W$, $G_f\cong \mathbb Z^k_2$. Hence $W_f$ is also a 2-torus manifold with respect to the induced action of $\mathbb Z^n_2/G_f \cong \mathbb Z^{n-k}_2$.

In addition, W determines a principal $\mathbb Z^n_2$-bundle over Q as follows. We take a small invariant open tubular neighborhood for each characteristic submanifold of W and remove their union from W. Then, the $\mathbb Z^n_2$-action on the resulting space is free, and its orbit space can naturally be identified with Q, which gives a principal $\mathbb Z^n_2$-bundle over Q, denoted by $\xi _W$. It is shown in Lü-Masuda [25] that W can be recovered (up to equivariant homeomorphism) from ($Q, \xi _W,\lambda _W$). For example, when $\xi _W$ is a trivial $\mathbb Z^n_2$-bundle, W is equivariantly homeomorphic to the following “canonical model” determined by $(Q,\lambda _W)$.

$$\begin{aligned} M_Q(\lambda _W) := Q\times \mathbb Z^n_2 /\sim \end{aligned}$$

(4)

where $(q,g)\sim (q',g')$ if and only if $q=q'$ and $g-g'\in G_{f(q)}$ where f(q) is the unique face of Q that contains q in its relative interior. This canonical model is a generalization of a result of Davis-Januszkiewicz [14, Prop. 1.8]. We will see that the canonical model plays an important role in our proof of Theorem 1.3 in Sect. 5.

2.2 Borel Construction and Equivariant Cohomology

For a topological group G, there exists a contractible free right G-space EG called the universal G-space. The quotient $BG= EG /G$ is called the classifying space for free G-actions. For example, when $G=\mathbb Z_2^n$, we can choose

$$\begin{aligned} E\mathbb Z^n_2 = (E\mathbb Z_2)^n = (S^{\infty })^n, \ \ B\mathbb Z_2^n = (B\mathbb Z_2)^n = (\mathbb RP^{\infty })^n. \end{aligned}$$

Let X be a topological space with a left G-action (we call X a G-space for brevity). The Borel construction of X is denoted by

$$ EG\times _G X = EG \times X /\sim $$

where $(e,x)\sim (eg,g^{-1}x)$ for any $e\in EG$, $x\in X$ and $g\in G$.

The equivariant cohomology of X with coefficients in a field $\textbf{k}$ is defined as

$$\begin{aligned} H^*_G(X;\textbf{k}):= H^*(EG\times _G X;\textbf{k}). \end{aligned}$$

Convention: The term “cohomology” of a space in this paper, always mean singular cohomology if not specified otherwise.

The Borel construction determines a canonical fibration called Borel fibration:

$$\begin{aligned} X \rightarrow EG\times _G X \rightarrow BG. \end{aligned}$$

(5)

The map $\rho$ collapsing X to a point induces a homomorphism

$$\begin{aligned} \rho ^* : H^*_G(pt ;\textbf{k}) = H^*(BG;\textbf{k}) \rightarrow H^*_G(X;\textbf{k}) \end{aligned}$$

(6)

which defines a canonical $H^*(BG;\textbf{k})$-module structure on $H^*_G(X;\textbf{k})$. A useful fact is: when X is a paracompact space with finite cohomology dimension, and $G=T^r$ or $(\mathbb Z_p)^r$ where p is a prime and $\textbf{k}=\mathbb Q$ or $\mathbb Z_p$ respectively, $\rho ^*$ is injective if and only if the fixed point set $X^G$ is non-empty (see [21, Ch. IV]).

In general, $H^*_G(X;\textbf{k})$ may not be a free $H^*(BG;\textbf{k})$-module. The following localization theorem due to A. Borel (see [21, p. 45]) says that we can compute the free $H^*(BG;\textbf{k})$-module part of $H^*_G(X;\textbf{k})$ by restricting to the fixed point set.

Theorem 1.3

(Localization Theorem) Let $G=T^r$ or $(\mathbb Z_p)^r$ where p is a prime and $\textbf{k}=\mathbb Q$ or $\mathbb Z_p$ respectively. For a paracompact G-space X with finite cohomology dimension, the following localized restriction homomorphism is an isomorphism:

$$\begin{aligned} S^{-1} H^*_G(X;\textbf{k}) \rightarrow S^{-1} H^*_G(X^G;\textbf{k}) = H^*(X^G;\textbf{k})\otimes _{\textbf{k}} (S^{-1} H^*(BG;\textbf{k})) \end{aligned}$$

where $S=R-\{0\}$ where R is the polynomial subring of $H^*(BG;\textbf{k})$. So the kernel of the restriction $H^*_G(X;\textbf{k}) \rightarrow H^*_G(X^G;\textbf{k})$ lies in the $H^*(BG;\textbf{k})$-torsion of $H^*_G(X;\textbf{k})$. In particular if X is equivariantly formal, $H^*_G(X;\textbf{k})\rightarrow H^*_G(X^G;\textbf{k})$ is injective.

The Borel construction can also be applied to a G-vector bundle $\pi : E\rightarrow X$ (i.e., both E and X are G-spaces and the projection $\pi$ is G-equivariant). In this case, the Borel construction $E_G$ of E is a vector bundle over $X_G$ whose mod 2 Euler class, denoted by $e^G(E)$, lies in $H^*_G(X;\mathbb Z_2)$. Note that using $\mathbb Z_2$-coefficients allows us to ignore the orientation of a vector bundle.

2.3 Mod 2 GKM-Theory

Let W be an n-dimensional equivariantly formal 2-torus manifold. Then, the fixed point set $W^{\mathbb Z^n_2}$ is a finite non-empty set (by Theorem 1.1), and $H^*_{\mathbb Z^n_2}(W;\mathbb Z_2)$ is a free module over $H^*(B\mathbb Z^n_2;\mathbb Z_2)$. Moreover, $H^*_{\mathbb Z^n_2}(W;\mathbb Z_2)$ can be computed by the so called Mod 2 GKM-theory (see Biss-Guillemin-Holm [4]) which is an extension of the GKM-theory in [18] to 2-torus actions. In this section, we briefly review some results related to our study. The reader is referred to [4] and [24] for more details.

For each $1\le i \le n$, let $\rho _i\in \textrm{Hom}(\mathbb Z^n_2,\mathbb Z_2)$ be the homomorphism defined by

$$ \rho _i((g_1,\cdots , g_n))=g_i,\ \forall (g_1,\cdots , g_n)\in \mathbb Z^n_2. $$

By a canonical isomorphism $\textrm{Hom}(\mathbb Z^n_2,\mathbb Z_2)\cong H^1(B\mathbb Z^n_2;\mathbb Z_2)$, we can identify $H^*(B\mathbb Z^n_2;\mathbb Z_2)$ with the graded polynomial ring $\mathbb Z_2[\rho _1,\cdots ,\rho _n]$ where $\textrm{deg}(\rho _i)=1$, $1\le i \le n$.

Let $Q=W/\mathbb Z^n_2$ be the orbit space of W. By our Theorem 3.3 proved later, a 2-torus manifold W being equivariantly formal implies that it is locally standard. Hence Q is a nice manifold with corners. Then, the 1-skeleton of Q, consisting of vertices (0-faces) and edges (1-faces) of Q, is an n-valent graph denoted by $\Gamma (Q)$. Let V(Q) and E(Q) denote the set of vertices and edges of Q, respectively.

Convention: We will not distinguish a vertex of Q and the corresponding fixed point in $W^{\mathbb Z^n_2}$ in the rest of the paper.

• Let $\pi : W\rightarrow Q$ be the quotient map.

• For each edge $e\in E(Q)$, $\pi ^{-1}(e)$ is a circle whose isotropy subgroup $G_e$ is a rank $n-1$ subgroup of $\mathbb Z^n_2$. Then, we obtain a map

  $$\begin{aligned} \alpha : E(Q) \rightarrow \textrm{Hom}(\mathbb Z^n_2,\mathbb Z_2)\cong H^1(B\mathbb Z^n_2;\mathbb Z_2) \end{aligned}$$

  where for each edge $e\in E(Q)$, $\ker (\alpha (e))=G_e$.

• For each vertex $p\in V(Q)$, let $\alpha _p = \{ \alpha (e)\,|\, p\in e \}\subset \textrm{Hom}(\mathbb Z^n_2,\mathbb Z_2)$.

Such a map $\alpha$ is called an axial function which has the following properties:

• (i) For every vertex $p\in V(Q)$, $\alpha _p$ is a linear basis of $\textrm{Hom}(\mathbb Z^n_2,\mathbb Z_2)$.

• (ii) For every edge $e\in E(Q)$, $\alpha _p \equiv \alpha _{p'}$ mod $\alpha (e)$ where $p,p'$ are the two vertices of e.

By [4, Theorem C] and [4, Remark 5.9], we have the following theorem which is a consequence of the $\mathbb Z_2$-version Chang-Skjelbred theorem (see [4, Theorem 4.1] and [10]).

Theorem 1.4

(see [4]) Let W be an n-dimensional equivariantly formal 2-torus manifold. If we choose an element $\eta _p \in H^*_{\mathbb Z^n_2}(W^{\mathbb Z^n_2};\mathbb Z_2)$ for each $p\in W^{\mathbb Z^n_2}$, then

$$(\eta _p)\in \bigoplus _{p\in W^{\mathbb Z^n_2}} H^*(B\mathbb Z^n_2;\mathbb Z_2) \cong H^*_{\mathbb Z^n_2}(W^{\mathbb Z^n_2};\mathbb Z_2)$$

is in the image of the restriction homomorphism $r: H^*_{\mathbb Z^n_2}(W;\mathbb Z_2) \rightarrow H^*_{\mathbb Z^n_2}(W^{\mathbb Z^n_2};\mathbb Z_2)$ if and only if for every edge $e\in E(Q)$ with vertices p and $p'$, $\eta _p - \eta _{p'}$ is divisible by $\alpha (e)$.

Moreover, we can understand the above axial function $\alpha$ in the following way. For brevity, we use the following notations for an n-dimensional locally standard 2-torus manifold W in the rest of this section.

• Let $G=\mathbb Z^n_2$.

• Let $W_i:=W_{F_i}=\pi ^{-1}(F_i)$, $1\le i \le m$, be all the characteristic submanifolds of W where $F_1,\cdots , F_m$ are all the facets of Q.

• Let $G_i:=\langle \lambda _W(F_i)\rangle \cong \mathbb Z_2$ be subgroup of G that fixes $W_i$ pointwise.

• Let $\nu _i$ be the (equivariant) normal bundle of $W_i$ in W. So we have the equivariant Euler class of $\nu _i$, denoted by $e^G(\nu _i)\in H^1_G(W_i;\mathbb Z_2)$.

• For any fixed point $p\in W^{\mathbb Z^n_2}$, let $I(p):=\{ i \,|\, p\in W_i \}$. We have the decomposition of tangent space $T_p W$ as

  $$\begin{aligned} T_p W = \bigoplus _{i\in I(p)} \nu _i|_p. \end{aligned}$$

  where $\nu _i|_p$ denotes the restriction of $\nu _i$ to p. So $\nu _i|_p$ is a 1-dimensional linear representation of G whose equivariant Euler class

  $$\begin{aligned} e^G(\nu _i|_p) = e^G(\nu _i)|_p\in H^1(B\mathbb Z^n_2;\mathbb Z_2). \end{aligned}$$

The inclusion map $\psi _i: W_i\hookrightarrow W$ defines an equivariant Gysin homomorphism $\psi _{i_{!}}: H^*_{G}(W_i;\mathbb Z_2)\rightarrow H^{*+1}_{G}(W;\mathbb Z_2)$ (see [1, §5.3] for example). For brevity, let

$$\begin{aligned} \tau _i =\tau _{F_i}= \psi _{i_{!}}(1)\in H^1_{G}(W;\mathbb Z_2) \end{aligned}$$

be the image of the identity $1 \in H^0_{G}(W_i;\mathbb Z_2)$. The element $\tau _i$ can be thought of as the Poincaré dual of the Borel construction of $W_i$ in $H^*_{G}(W;\mathbb Z_2)$ and is called the equivariant Thom class of $\nu _i$. A standard fact is

$$\begin{aligned} \tau _i|_p\text { agrees with the equivariant Euler class of }\nu _i|_p. \end{aligned}$$

Note that the elements of $\textrm{Hom}(\mathbb Z^n_2,\mathbb Z_2)$ are in one-to-one correspondence with all the 1-dimensional linear representations of $\mathbb Z^n_2$. So the canonical isomorphism between $\textrm{Hom}(\mathbb Z^n_2,\mathbb Z_2)$ and $H^1(B\mathbb Z^n_2;\mathbb Z_2)$ is given by the equivariant Euler class of a 1-dimensional representations of $\mathbb Z^n_2$. Then, we have the following identification:

$$\begin{aligned} \alpha _p = \{ \alpha (e)\,|\, p\in e \} \longleftrightarrow \{ e^G(\nu _i)|_p=\tau _i|_p\,;\, i\in I(p) \}. \end{aligned}$$

(7)

where an edge e containing p corresponds to the unique index $i\in I(p)$ so that the facet $F_i$ intersects e transversely (or equivalently $e \nsubseteq F_i$).

• For a codimension-k face f of Q, let $\nu _f$ denote the (equivariant) normal bundle of $W_f$ in W. Denote by $\tau _f \in H^k_G(W;\mathbb Z_2)$ the equivariant Thom class of $\nu _f$. Then, the restriction of $\tau _f$ to $H^k_G(W_f;\mathbb Z_2)$ is the equivariant Euler class of $\nu _f$, denoted by $e^G(\nu _f)$. In particular, if $f=Q$, $W_f=W$ and so $\tau _f$ is the identity element of $H^0_G(W_f;\mathbb Z_2)$.

Let $r_p: H^*_G(W;\mathbb Z_2)\rightarrow H_G^*(p;\mathbb Z_2)\cong H^*(BG;\mathbb Z_2)$ denote the restriction map at a fixed point $p\in W^G$. Then,

$$\begin{aligned} r=\bigoplus _{p\in W^G} r_p : H^*_G(W;\mathbb Z_2)\rightarrow H^*_G(W^G;\mathbb Z_2) = \bigoplus _{p\in W^G} H^*(BG;\mathbb Z_2). \end{aligned}$$

(8)

By Theorem 2.3, the kernel of r is the $H^*(BG;\mathbb Z_2)$-torsion subgroup of $H^*_G(W;\mathbb Z_2)$.

Clearly, $r_p(\tau _f) =0$ unless $p\in (W_f)^G$ (i.e., p is a vertex of f). It follows from (7) that for any $p\in W^G$,

$$\begin{aligned} r_p(\tau _f)= {\left\{ \begin{array}{ll} \underset{p\in e,\, e\nsubseteq f}{\prod }\ \alpha (e), &{} \text {if }p \in f; \\ \ \ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(9)

In addition, define

$$\begin{aligned} \widehat{H}^*_G(W;\mathbb Z_2):= H^*_G(W;\mathbb Z_2) \big /H^*(BG;\mathbb Z_2)\text {-torsion}. \end{aligned}$$

(10)

By the localization theorem (Theorem 2.3), the restriction homomorphism r induces a monomorphism $\widehat{H}^*_G(W;\mathbb Z_2) \rightarrow H^*_G(W^G;\mathbb Z_2)$, still denoted by r.

The following proposition is parallel to [27, Proposition 3.3].

Proposition 1.5

Let W be an n-dimensional locally standard 2-torus manifold.

• (i) For each characteristic submanifold $W_i$ with $(W_i)^{G}\ne \varnothing$ where $G=\mathbb Z^n_2$, there is a unique element $a_i\in H_1(BG;\mathbb Z_2)$ such that

  $$\begin{aligned} \rho ^*(t) = \sum _{i}\langle t,a_i\rangle \tau _i \ \, \text {modulo}\ \, H^*(BG;\mathbb Z_2)\text {-torsion} \end{aligned}$$

  for any element $t\in H^1(BG;\mathbb Z_2)$. Here the sum is taken over all the characteristic submanifolds $W_i$ with $(W_i)^G\ne \varnothing$ and $\rho ^*$ is defined in (6).

• (ii) For each $W_i$ with $(W_i)^G\ne \varnothing$, the subgroup $G_i$ fixing $W_i$ coincides with the subgroup determined by $a_i\in H_1(BG;\mathbb Z_2)$ through the identification $H_1(BG;\mathbb Z_2)\cong \textrm{Hom}(\mathbb Z_2,G)$.

• (iii) If n different characteristic submanifolds $W_{i_1},\cdots , W_{i_n}$ have a G-fixed point in their intersection, then the elements $a_{i_1},\cdots ,a_{i_n}$ form a linear basis of $H_1(BG;\mathbb Z_2)$ over $\mathbb Z_2$.

Proof

The argument is completely parallel to the arguments for torus manifolds in the proof of [26, Lemma 1,3, Lemma. 1.5, Lemma 1.7]. Indeed, we can just replace the torus manifold M in [26] by our 2-torus manifold W and replace $T^n$ by $\mathbb Z^n_2$ and $H^2(M;\mathbb Z)$ by $H^1(W;\mathbb Z_2)$ to obtain our proof here. The details of the proof are left to the reader.$\square$

In addition, the following lemma is completely parallel to the torus manifold case [27, Lemma 6.2]. Its proof is also parallel to [27], hence omitted.

Lemma 1.6

Let W be a locally standard 2-torus manifold with orbit space Q. For any $\eta \in H^*_G(W;\mathbb Z_2)$ and any edge $e\in E(Q)$, $r_p(\eta )-r_{p'}(\eta )$ is divisible by $\alpha (e)$ where p and $p'$ are the endpoints of e.

2.4 Face Ring

A poset (partially ordered set) $\mathcal {P}$ is called simplicial if it has an initial element $\hat{0}$ and for each $x\in \mathcal {P}$ the lower segment $[\hat{0},x]$ is a boolean lattice (the face lattice of a simplex).

Let $\mathcal {P}$ be a simplicial poset. For each $x\in \overline{\mathcal {P}}:=\mathcal {P}-\{\hat{0}\}$, we assign a geometrical simplex whose face poset is $[\hat{0},x]$ and glue these geometrical simplices together according to the order relation in $\mathcal {P}$. The cell complex we obtained is called the geometrical realization of $\mathcal {P}$, denoted by $|\mathcal {P}|$. We may also say that $|\mathcal {P}|$ is a simplicial cell complex.

For any two elements $x, x'\in \mathcal {P}$, denote by $x\vee x'$ the set of their least common upper bounds, and by $x\wedge x'$ their greatest common lower bounds. Since $\mathcal {P}$ is simplicial, $x\wedge x'$ consists of a single element if $x\vee x'$ is non-empty.

Definition 1.7

(see Stanley [29]) The face ring of a simplicial poset $\mathcal {P}$ over a field $\textbf{k}$ is the quotient

$$\begin{aligned} \textbf{k}[\mathcal {P}]:= \textbf{k}[v_x:x\in \mathcal {P}] \big /\mathcal {I}_{\mathcal {P}} \end{aligned}$$

where $\mathcal {I}_{\mathcal {P}}$ is the ideal generated by all the elements of the form

$$\begin{aligned} v_x v_{x'} - v_{x\wedge x'}\cdot \sum _{x''\in x\vee x'} v_{x''}. \end{aligned}$$

Let Q be a nice manifold with corners. It is easy to see that the face poset of Q is a simplicial poset, denoted by $\mathcal {P}_Q$. We call $|\mathcal {P}_Q|$ the simplicial cell complex dual to Q.

We define the face ring of Q to be the face ring of $\mathcal {P}_Q$. Equivalently, we can write the face ring of Q as

$$\begin{aligned} \textbf{k}[Q]:=\textbf{k}[v_f: f \ \text {a face of }Q] \big /\mathcal {I}_Q. \end{aligned}$$

where $\mathcal {I}_Q$ is the ideal generated by all the elements of the form

$$\begin{aligned} v_f v_{f'} - v_{f\vee f'}\cdot \sum _{f''\in f\cap f'} v_{f''}. \end{aligned}$$

where $f\vee f'$ denotes the unique minimal face of Q containing both f and $f'$.

Convention: For any face f of Q, define the degree of $v_f$ to be the codimension of f. Then, $\textbf{k}[Q]=\textbf{k}[\mathcal {P}_Q]$ becomes a graded ring. Note that in the discussion of torus manifolds in [27], the degree of $v_f$ is defined to be twice the codimension of f to fit the study there.

The f-vector of Q is defined as $\textbf{f}(Q) =(f_0,\cdots , f_{n-1})$ where $n=\dim (Q)$ and $f_i$ is the number of faces of codimension $i+1$. The equivalent information is contained in the h-vector $\textbf{h}(Q)=(h_0,\cdots ,h_n)$ determined by the equation:

$$\begin{aligned} h_0 t^n +\cdots + h_{n-1}t+ h_n = (t-1)^n + f_0 (t-1)^{n-1}+\cdots + f_{n-1}. \end{aligned}$$

(11)

The Hilbert series of $\textbf{k}[Q]$ is $F(\textbf{k}[Q]):=\sum _i \dim _{\textbf{k}}\textbf{k}[Q]_{i} \cdot t^i$ where $\textbf{k}[Q]_{i}$ denotes the homogeneous degree i part of $\textbf{k}[Q]$. By [29, Proposition 3.8],

$$\begin{aligned} F(\textbf{k}[Q];t) = \frac{h_0 + h_1 t +\cdots + h_nt^n}{(1-t)^n}. \end{aligned}$$

(12)

The following construction is taken from [27, Sect. 5]. For any vertex (0-face) $p\in Q$, we define a map

$$\begin{aligned} s_p: \textbf{k}[Q] \rightarrow \textbf{k}[Q]\big /(v_f: p\notin f). \end{aligned}$$

(13)

If p is the intersection of n different facets $F_1,\cdots , F_n$, then $\textbf{k}[Q]\big /(v_f: p\notin f)$ can be identified with the polynomial ring $\textbf{k}[v_{F_1},\cdots ,v_{F_n}]$.

Lemma 1.8

(Lemma 5.6 in [27]) If every face of Q has a vertex, then the direct sum $s=\bigoplus _{p} s_p$ over all vertices $p\in Q$ is a monomorphism from $\textbf{k}[Q]$ to the sum of polynomial rings $\textbf{k}[Q]\big /(v_f: p\notin f)$.

A finitely generated graded commutative ring R over $\textbf{k}$ is called Cohen-Macaulay if there exists an h.s.o.p (homogeneous system of parameters) $\theta _1,\cdots ,\theta _n$ such that R is a free $\textbf{k}[\theta _1,\cdots ,\theta _n]$-module. Clearly, if $\textbf{k}[Q]=\textbf{k}[\mathcal {P}_Q]$ is Cohen-Macaulay, then it has a l.s.o.p (linear system of parameters).

A simplicial complex K is called a Gorenstein* complex over $\textbf{k}$ if its face ring $\textbf{k}[K]$ is Cohen-Macaulay and $H^*(K;\textbf{k})\cong H^*(S^{d};\textbf{k})$ where $d=\dim (K)$. The reader is referred to Bruns-Herzog [9] and Stanley [30] for more information of Cohen-Macaulay rings and Gorenstein* complexes.

The following proposition is parallel to [27, Lemma 8.2 (1)].

Proposition 1.9

If Q is an n-dimensional mod 2 homology polytope, then the geometrical realization $|\mathcal {P}_Q|$ of $\mathcal {P}_Q$ is a Gorenstein* simplicial complex over $\mathbb Z_2$. In particular, $\mathbb Z_2[\mathcal {P}_Q]$ is Cohen-Macaulay and $H^*(|\mathcal {P}_Q|;\mathbb Z_2)\cong H^*(S^{n-1};\mathbb Z_2)$.

Proof

The proof is almost identical to the proof in [27, Lemma 8.2] except that we use $\mathbb Z_2$-coefficients instead of $\mathbb Z$-coefficients when applying [30, II 5.1] in the argument.$\square$

3 Equivariantly Formal 2-Torus Manifolds

In this section, we study various properties of equivariantly formal 2-torus manifolds. One may find that many discussions on 2-torus manifolds here are parallel to the discussions in [27] on torus manifolds. The condition “vanishing of odd degree cohomology” on a torus manifold in [27] is now replaced by the equivariant formality condition on a 2-torus manifold and, the coefficients $\mathbb Z$ is replaced by $\mathbb Z_2$. Many arguments in [27] are transplanted into our proof here while some of them actually become simpler.

In Sect. 3.1, we prove some general results of equivariantly formal $\mathbb Z^r_2$-actions on compact manifolds. In particular, we prove that any equivariantly formal 2-torus manifold is locally standard, and the equivariant formality of a 2-torus manifold is inherited by all its facial submanifolds.

In Sect. 3.2, we explore the relations between the equivariant cohomology of a locally standard 2-torus manifold and the face ring of its orbit space.

In Sect. 3.3, we prove that the equivariant formality of a 2-torus manifold is preserved under real blow-ups along its facial submanifolds. Our proof uses a result from Gitler [17].

3.1 Equivariantly Formal $\Rightarrow$ Locally Standard

Lemma 1.10

Suppose M is a compact manifold whose connected components are $M_1,\cdots , M_k$. A $\mathbb Z_2^r$-action on M is equivariantly formal if and only if each $M_i$ is $\mathbb Z_2^r$-invariant and the restricted $\mathbb Z_2^r$-action on $M_i$ is equivariantly formal.

Proof

The “if” part is obvious. For the “only if” part, assume that $M_{1},\cdots , M_{s}$, $s\le k$, are all the components each of which is preserved under the $\mathbb Z_2^r$-action. Since the $\mathbb Z_2^r$-action on M is equivariantly formal, by Theorem 1.1 we have

$$\dim _{\mathbb Z_2}H^*(M^{\mathbb Z^r_2};\mathbb Z_2)= \dim _{\mathbb Z_2}H^*(M;\mathbb Z_2).$$

So in particular, $M^{\mathbb Z^r_2}$ is not empty. Clearly, $M^{\mathbb Z^r_2}$ must lie in $M_{1}\cup \cdots \cup M_{s}$, so $s>0$ and $M^{\mathbb Z^r_2}$ is the disjoint union of $M_1^{\mathbb Z^r_2},\cdots , M_s^{\mathbb Z^r_2}$. Then, by Theorem 1.1,

$$\begin{aligned} \dim _{\mathbb Z_2}H^*(M^{\mathbb Z^r_2};\mathbb Z_2)= & {} \sum ^s_{i=1} \dim _{\mathbb Z_2}H^*(M_i^{\mathbb Z^r_2};\mathbb Z_2) \le \sum ^s_{i=1} \dim _{\mathbb Z_2}H^*(M_i;\mathbb Z_2)\\\le & {} \dim _{\mathbb Z_2}H^*(M;\mathbb Z_2). \end{aligned}$$

By comparing this inequality with the previous equation, we can deduce that $s=k$ and on every component $M_i$, $\dim _{\mathbb Z_2}H^*(M_i^{\mathbb Z^r_2};\mathbb Z_2) = \dim _{\mathbb Z_2}H^*(M_i;\mathbb Z_2)$. So by Theorem 1.1 again, the $\mathbb Z_2^r$-action on $M_i$ is equivariantly formal.$\square$

Lemma 1.11

If a $\mathbb Z_2^r$-action on a compact manifold M is equivariantly formal, then for every subgroup H of $\mathbb Z_2^r$,

• (i) The action of H on M is equivariantly formal.

• (ii) The induced action of $\mathbb Z^r_2$ on $M^H$ and $\mathbb Z^r_2/H$ on $M^H$ are both equivariantly formal.

• (iii) The induced action of $\mathbb Z^r_2$ (or $\mathbb Z^r_2/H$) on every connected component N of $M^H$ is equivariantly formal, hence N has a $\mathbb Z^r_2$-fixed point.

Proof

• (i) By Theorem 1.1, it is equivalent to prove

  $$\begin{aligned} \dim _{\mathbb Z_2} H^*(M^H;\mathbb Z_2)=\dim _{\mathbb Z_2} H^*(M;\mathbb Z_2). \end{aligned}$$

  (14)

  Otherwise, assume $\dim _{\mathbb Z_2} H^*(M^H;\mathbb Z_2)<\dim _{\mathbb Z_2} H^*(M;\mathbb Z_2)$. Observe that the $\mathbb Z^r_2$-action on M induces an action of $\mathbb Z_2^r/H$ on $M^H$ and we have

  $$\begin{aligned} M^{\mathbb Z^r_2} = (M^H)^{\mathbb Z^r_2 /H}. \end{aligned}$$

  (15)

  So by Theorem 1.1, $\dim _{\mathbb Z_2} H^*(M^{\mathbb Z^r_2};\mathbb Z_2)\le \dim _{\mathbb Z_2} H^*(M^{H};\mathbb Z_2) <\dim _{\mathbb Z_2} H^*$$(M;\mathbb Z_2)$, which contradicts the assumption that the $\mathbb Z_2^r$-action on M is equivariantly formal. This proves (i).

• (ii) By (15) and the assumption that the $\mathbb Z^r_2$-action is equivariantly formal,

  $$\begin{aligned} \dim _{\mathbb Z_2} H^*\big ((M^H)^{\mathbb Z^r_2 /H};\mathbb Z_2\big )&= \dim _{\mathbb Z_2} H^*\big ( M^{\mathbb Z^r_2};\mathbb Z_2\big ) \\&= \dim _{\mathbb Z_2} H^*\big ( M;\mathbb Z_2\big )\overset{14}{=}\ \dim _{\mathbb Z_2} H^*(M^H;\mathbb Z_2). \end{aligned}$$

  Then, by Theorem 1.1, the action of $\mathbb Z^r_2/H$ on $M^H$ is equivariantly formal, so is the action of $\mathbb Z^r_2$ on $M^H$.

• (iii) By the conclusion in (ii) and Lemma 3.1, the induced action of $\mathbb Z^r_2$ (or $\mathbb Z^r_2/H$) on every connected component N of $M^H$ is equivariantly formal. So by Theorem 1.1, N must have a $\mathbb Z^r_2$-fixed point.

$\square$

Next, we prove a theorem that is parallel to [27, Theorem 4.1].

Theorem 1.12

If a 2-torus manifold W is equivariantly formal, then W must be locally standard.

Proof

Suppose $\dim (W)=n$. For a point $x\in W$, denote by $G_x$ the isotropy group of x.

• If $G_x$ is trivial, then x is in a free orbit of the $\mathbb Z_2^n$-action. So W is locally standard near x.

• Otherwise, let N be the connected component of $W^{G_x}$ containing x. By Lemma 3.2 (iii), the induced $\mathbb Z_2^n$-action on N has a fixed point, say $x_0$. Since $W^{\mathbb Z^n_2}$ is discrete, the tangential $\mathbb Z^n_2$-representation $T_{x_0}W$ is faithful. Then, since x and $x_0$ are in the same connected component fixed pointwise by $G_x$, the $G_x$-representation on $T_x W$ agrees with the restriction of the tangential $\mathbb Z^n_2$-representation $T_{x_0} W$ to $G_x$. This implies that W is locally standard near x.

The theorem is proved.$\square$

Proposition 1.13

Let W be an equivariantly formal 2-torus manifold with orbit space Q. For any face f of Q, the facial submanifold $W_f$ is also an equivariantly formal 2-torus manifold.

Proof

Suppose $\dim (W)=n$ and f is a codimension-k face of Q. By Theorem 3.3, W is locally standard. Then, $W_f$ is a connected $(n-k)$-dimensional embedded submanifold of W fixed pointwise by $G_f\cong \mathbb Z^k_2$ (see (3)). By Lemma 3.2 (iii), the induced action of $\mathbb Z^n_2/G_f \cong \mathbb Z^{n-k}_2$ on $W_f$ is equivariantly formal.$\square$

3.2 Equivariant Cohomology of Locally Standard 2-Torus Manifolds

Let W be an n-dimensional locally standard 2-torus manifold with orbit space Q. We explore the relation between $H^*_G(W;\mathbb Z_2)$ where $G=\mathbb Z^n_2$ and the face ring $\mathbb Z_2[Q]$ under some conditions on Q. In the following, we use the notations from Sect. 2.3.

First of all, we have a lemma that is parallel to [27, Lemma 6.3].

Lemma 1.14

For any faces f and $f'$ of Q, the relation below holds in $\widehat{H}^*_G(W;\mathbb Z_2)$:

$$\begin{aligned} \tau _f\tau _{f'} = \tau _{f\vee f'}\cdot \sum _{f''\in f\cap f'} \tau _{f''}. \end{aligned}$$

Here we define $\tau _{\varnothing }=0$.

Proof

The proof is parallel to the proof of [27, Lemma 6.3]. The idea is to use the monomorphism $r: \widehat{H}^*_G(W;\mathbb Z_2) \rightarrow H^*_G(W^G;\mathbb Z_2)$ to map both sides of the identity to the fixed points and then use the formula (9) to check that they are equal. $\square$

By Lemma 3.5, we obtain a well-defined homomorphism

$$\begin{aligned} \varphi : \mathbb Z_2[Q]&\longrightarrow \widehat{H}^*_G(W;\mathbb Z_2).\\ v_f\ {}&\longmapsto \tau _f \end{aligned}$$

The following lemma and its proof are parallel to [27, Lemma 6.4].

Lemma 1.15

The homomorphism $\varphi$ is injective if every face Q has a vertex.

Proof

According to the definitions of r and s (see (8) and (13)), we have $s=r\circ \varphi$ by identifying $H^*_G(p,\mathbb Z_2)$ with $\mathbb Z_2[Q]\big /(v_f: p\notin f)$ for every vertex p of Q. Then, by Lemma 2.8, s is injective if every face of Q has a vertex, so is $\varphi$.$\square$

The following lemma is parallel to [27, Proposition 7.4].

Lemma 1.16

If the 1-skeleton of every face of Q (including Q itself) is connected, then $\widehat{H}^*_G(W;\mathbb Z_2)$ is generated by the elements $\tau _{F_1},\cdots ,\tau _{F_m} \in H^1_G(W;\mathbb Z_2)$ as an $H^*(BG;\mathbb Z_2)$-module, where $F_1,\cdots , F_m$ are all the facets of Q.

Proof

The argument is a bit technical, but it is completely parallel to the proof of [27, Proposition 7.4]. The main idea of the proof is to consider the restriction of an element $\eta \in H^*_G(W;\mathbb Z_2)$ to the fixed point set $W^G$ via $r: H^*_G(W;\mathbb Z_2) \rightarrow H^*_G(W^G;\mathbb Z_2)$, and then use $\tau _{F_1},\cdots ,\tau _{F_m}$ and elements in $H^*(BG;\mathbb Z_2)$ to spell out $r(\eta )$ at each fixed point $p\in W^G$ (see Proposition 2.5). The details of the proof are left to the reader.$\square$

The following theorem is parallel to [27, Theorem 7.5].

Theorem 1.17

Let W be a locally standard 2-torus manifold with orbit space Q. If every face f of Q has a vertex and the 1-skeleton of f is connected, then the map $\varphi : \mathbb Z_2[Q] \rightarrow \widehat{H}^*_G(W;\mathbb Z_2)$ is an isomorphism of graded rings.

Proof

By Lemma 3.6, $\varphi$ is injective and, by Lemma 3.7, $\varphi$ is surjective.$\square$

Lemma 1.18

Let W be an equivariantly formal 2-torus manifold with orbit space Q. Then the 1-skeleton of every face of Q (including Q itself) is connected.

Proof

Since W is equivariantly formal, the localization theorem (Theorem 2.3) implies that the restriction homomorphism $r: H^*_{G}(W;\mathbb Z_2) \rightarrow H^*_{G}(W^{G};\mathbb Z_2)$ is injective. In addition, since W is connected, the image of $H^0_{G}(W;\mathbb Z_2)$ under the restriction homomorphism is isomorphic to $\mathbb Z_2$. So the “if” part of Theorem 2.4 implies that the 1-skeleton of Q must be connected.

For any proper face f of Q, the facial submanifold $W_f$ is also an equivariantly formal 2-torus manifold by Proposition 3.4. Then, by applying the above argument to $W_f$, we obtain that the 1-skeleton of f is also connected.$\square$

Corollary 1.19

If W is an equivariantly formal 2-torus manifold, then the map $\varphi : \mathbb Z_2[Q] \rightarrow H^*_G(W;\mathbb Z_2)$ is an isomorphism of graded rings.

Proof

Since W is equivariantly formal, its equivariant cohomology $H^*_G(W;\mathbb Z_2)$ is a free module over $H^*(BG;\mathbb Z_2)$. So by definition, $\widehat{H}^*_G(W;\mathbb Z_2) = H^*_G(W;\mathbb Z_2)$. For any face f of Q, the facial submanifold $W_f$ is also an equivariantly formal 2-torus manifold by Proposition 3.4. This implies that f has a vertex. Moreover, the 1-skeleton of f is connected by Lemma 3.9. Then, the corollary follows from Theorem 3.8.$\square$

When a 2-torus manifold W is equivariantly formal, Corollary 3.10 tells us that the equivariant cohomology ring of W is completely determined by the face poset of its orbit space (so independent on the characteristic function $\lambda _W$ or the principal bundle $\xi _W$). This suggests that the orbit space of W should be rather special.

The following corollary is parallel to [27, Corollary 7.8]. It generalizes the calculation of the mod 2 cohomology ring of a small cover in [14].

Corollary 1.20

If a 2-torus manifold W is equivariantly formal, then

$$\begin{aligned} H^*(W;\mathbb Z_2)\cong \mathbb Z_2[v_f: f\ \text {a face of }Q] \big /I \end{aligned}$$

where I is the ideal generated by the following two types of elements: $\mathrm {(a)}$ $v_fv_{f'} - v_{f\vee f'}\sum _{f''\in f\cap f'} v_{f''}$, $\mathrm {(b)}$ $\sum ^m_{i=1}\langle t,a_i\rangle v_{F_i}$, $t\in H^1(BG;\mathbb Z_2)$.

Here, $F_1,\cdots ,F_m$ are all the facets of Q, and the elements $a_i\in H_1(BG;\mathbb Z_2)$ are defined in Proposition 2.5.

Proof

Since W is equivariantly formal, $\iota _{W}^*: H^*_G(W;\mathbb Z_2)\rightarrow H^*(W;\mathbb Z_2)$ is surjective and its kernel is generated by all $\rho ^*(t)$ with $t\in H^1(BG;\mathbb Z_2)$ (see (6)). Then, the statement follows from Corollary 3.10 and Proposition 2.5.$\square$

3.3 Real Blow-up of a Locally Standard 2-Torus Manifold Along a Facial Submanifold

Let W be a locally standard 2-torus manifold with orbit space Q. For a codimension-k face f of Q, the facial submanifold $W_f$ is an embedded connected codimension-k submanifold of W. So the equivariant normal bundle $\nu _f$ of $W_f$ in W is a real vector bundle of rank k. If we replace $W_f \subset W$ by the real projective bundle $P(\nu _f)$, we obtain a new 2-torus manifold denoted by $\widetilde{W}^f$ called the real blow-up of W along $W_f$. This is analogous to the blow-up of a torus manifold in [27, Sec. 9] (also see [20, p. 605] and [15, Sec. 4]).

The orbit space of $\widetilde{W}^f$, denoted by $Q^f$, is the result of “cutting off” the face f from Q (see Fig. 1). So $\widetilde{W}^f$ is also locally standard. Correspondingly, the simplicial cell complex $|\mathcal {P}_{Q^f}|$ is obtained from $|\mathcal {P}_{Q}|$ by a stellar subdivision of the face dual to f.

Fig. 1

Cutting off a face from a nice manifold with corners

Proposition 1.21

Let W be a locally standard 2-torus manifold with orbit space Q and f be a proper face of Q with codimension-k. Then, $\widetilde{W}^f$ is equivariantly formal if and only if so is W.

Proof

(a) Let $\widetilde{\nu }_f$ denote the equivariant normal bundle of $P(\nu _f)$ in $\widetilde{W}^f$. Besides, let $\textrm{Th}(\nu _f)$ and $\textrm{Th}(\widetilde{\nu }_f)$ be the Thom space of $\nu _f$ and $\widetilde{\nu }_f$, respectively. Then, we have a natural commutative diagram of continuous maps:

where i and $\widetilde{i}$ are the inclusions; t and $\widetilde{t}$ are the Thom-Pontryagin maps; $p: \widetilde{W}^f\rightarrow W$ is the blow-down map; $p_0$ is the restriction of p to $P(\nu _f)$; and q is the induced map by p in the Thom spaces.

According to [17, §5] and [17, Theorem 3.7], there is a short exact sequence:

$$\begin{aligned} 0\! \longrightarrow \! H^*(\textrm{Th}(\nu _f);\mathbb Z_2)\!\overset{\alpha }{\longrightarrow }\! H^*(W;\mathbb Z_2)\oplus H^*(\textrm{Th}(\widetilde{\nu }_f);\mathbb Z_2)\! \overset{\beta }{\longrightarrow }\! H^*(\widetilde{W}^f;\mathbb Z_2)\!\longrightarrow \! 0. \end{aligned}$$

(16)

where $\alpha =(t^*, q^*)$ and $\beta =p^*-\widetilde{t}^*$. This implies:

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(\widetilde{W}^f;\mathbb Z_2) = \dim _{\mathbb Z_2} H^*(W;\mathbb Z_2) + \dim _{\mathbb Z_2} H^*(\textrm{Th}(\widetilde{\nu }_f);\mathbb Z_2) - \dim _{\mathbb Z_2} H^*(\textrm{Th}(\nu _f);\mathbb Z_2). \end{aligned}$$

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(\widetilde{W}^f;\mathbb Z_2) = \dim _{\mathbb Z_2} H^*(W;\mathbb Z_2) + \dim _{\mathbb Z_2} H^*(\textrm{Th}(\widetilde{\nu }_f);\mathbb Z_2) - \dim _{\mathbb Z_2} H^*(\textrm{Th}(\nu _f);\mathbb Z_2). \end{aligned}$$

By the Thom isomorphism, we have

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(\textrm{Th}(\nu _f);\mathbb Z_2)&= \dim _{\mathbb Z_2} H^*(W_f;\mathbb Z_2), \\ \dim _{\mathbb Z_2} H^*(\textrm{Th}(\widetilde{\nu }_f);\mathbb Z_2)&= \dim _{\mathbb Z_2} H^*(P(\nu _f);\mathbb Z_2). \end{aligned}$$

By Leray-Hirsch theorem, $H^*(P(\nu _f);\mathbb Z_2)\cong H^*(W_f;\mathbb Z_2)\otimes H^*(\mathbb RP^{k-1};\mathbb Z_2)$ (as $\mathbb Z_2$-vector spaces), which implies $\dim _{\mathbb Z_2} H^*(P(\nu _f);\mathbb Z_2) = k \cdot \dim _{\mathbb Z_2} H^*(W_f;\mathbb Z_2)$. So

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(\widetilde{W}^f;\mathbb Z_2) = \dim _{\mathbb Z_2} H^*(W;\mathbb Z_2) + (k-1) \cdot \dim _{\mathbb Z_2} H^*(W_f;\mathbb Z_2). \end{aligned}$$

(17)

If W is equivariantly formal, then W is locally standard and so Q is a nice manifold with corners. It is easy to see

$$ \# \text {vertices of }Q^f = \# \text {vertices of }Q + (k-1)\cdot \# \text {vertices of }f. $$

Since the fixed point set $W^{G}$ ($G=\mathbb Z^n_2$) corresponds to the vertex set of Q which is discrete, the number of fixed points of the G-action satisfies

$$\begin{aligned} |(\widetilde{W}^f)^G| = |W^G| + (k-1)\cdot |(W_f)^G|. \end{aligned}$$

(18)

By Proposition 3.4, $W_f$ is also equivariantly formal. So by Theorem 1.1,

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(W;\mathbb Z_2)=|W^G|, \ \ \dim _{\mathbb Z_2} H^*(W_f;\mathbb Z_2)=|(W_f)^G|. \end{aligned}$$

It follows from (17) and (18) that $|(\widetilde{W}^f)^G| = \dim _{\mathbb Z_2} H^*(\widetilde{W}^f;\mathbb Z_2)$. So we deduce from Theorem 1.1 that $\widetilde{W}^f$ is equivariantly formal.

Conversely, if $\widetilde{W}^f$ is equivariantly formal, we have

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(W;\mathbb Z_2)&\overset{17}{=}\ \dim _{\mathbb Z_2} H^*(\widetilde{W}^f;\mathbb Z_2) - (k-1) \cdot \dim _{\mathbb Z_2} H^*(W_f;\mathbb Z_2)\\ (\text {by Theorem~1.1}) \,&\le |(\widetilde{W}^f)^G| - (k-1)\cdot |(W_f)^G| \overset{17}{=}\ |W^G| = \dim _{\mathbb Z_2} H^*(W^G;\mathbb Z_2). \end{aligned}$$

But by Theorem 1.1, $\dim _{\mathbb Z_2} H^*(W^G;\mathbb Z_2) \le \dim _{\mathbb Z_2} H^*(W;\mathbb Z_2)$. So we must have $\dim _{\mathbb Z_2} H^*(W;\mathbb Z_2)= \dim _{\mathbb Z_2} H^*(W^G;\mathbb Z_2)$, which implies that W is equivariantly formal. The proposition is proved.$\square$

The following lemma is parallel to [27, Lemma 9.1]. Its proof is almost identical to the proof in [27], hence omitted.

Lemma 1.22

Let Q be a nice manifold with corners and f be a proper face of Q. Then, $Q^f$ is mod 2 face-acyclic if and only if so is Q.

4 Equivariantly Formal 2-Torus Manifolds with Mod 2 Cohomology Generated by Degree-One Part

In our study of equivariantly formal 2-torus manifolds, those manifolds whose mod 2 cohomology rings are generated by their degree-one part are of special importance. We will see in Sect. 5 that the study of general equivariantly formal 2-torus manifolds can be reduced to the study of these special 2-torus manifolds by a sequence of real blow-ups along facial submanifolds.

The following lemma is parallel to [27, Lemma 2.3].

Lemma 1.1

Suppose there is an equivariantly formal $\mathbb Z_2^r$-action on a compact manifold M where the cohomology ring $H^*(M;\mathbb Z_2)$ is generated by its degree-one part. Then, for any subgroup H of $\mathbb Z_2^r$ and every connected component N of $M^H$, the homomorphism $i^*: H^*(M;\mathbb Z_2)\rightarrow H^*(N;\mathbb Z_2)$ is surjective where $i: N\hookrightarrow M$ is the inclusion. In particular, $H^*(N;\mathbb Z_2)$ is also generated by its degree-one part.

Proof

First, we assume $H\cong \mathbb Z_2$. We have a commutative diagram as follows:

(19)

where $H^*_H(N;\mathbb Z_2)\cong H^*(N;\mathbb Z_2)\otimes H^*(BH;\mathbb Z_2)$ and $\widehat{i}^*_H$ is the homomorphism on equivariant cohomology induced by i. By our assumption, both $\iota ^*_M$ and $\iota ^*_N$ are surjective. The following argument is parallel to the proof of [27, Lemma 2.3].

By [7, Theorem VII.1.5], the inclusion $M^H \hookrightarrow M$ induces an isomorphism $H^k_H(M;\mathbb Z_2)\rightarrow H^k_H(M^H;\mathbb Z_2)$ for sufficiently large k, which implies that

$$\widehat{i}^*_H: H^k_H(M;\mathbb Z_2)\rightarrow H^k_H(N;\mathbb Z_2)$$

is surjective if k is sufficiently large.

Let $v_1,\cdots , v_d \in H^1(M;\mathbb Z_2)$ be a set of multiplicative generators of $H^*(M;\mathbb Z_2)$. For each $1\le l \le d$, let $\widehat{v}_l$ be a lift of $v_l$ in $H^*_H(M;\mathbb Z_2)$ and $w_l:= i^*(v_l)\in H^1(N;\mathbb Z_2)$. Let t be a generator of $H^1(BH;\mathbb Z_2)\cong \mathbb Z_2$. By the commutativity of the above diagram (19),

$$\begin{aligned} \widehat{i}^*(\widehat{v}_l) = b_l t+w_l \ \,\text {for some}\ \ b_l\in \mathbb Z_2. \end{aligned}$$

Then, for an arbitrary element $\zeta \in H^*(N;\mathbb Z_2)$, there exists a large enough integer $q\in \mathbb Z$ and a polynomial $P(x_1,\cdots , x_d)$ such that

$$ \widehat{i}^* \big ( P(\widehat{v}_1,\cdots , \widehat{v}_d) \big ) = \zeta \otimes t^{q}. $$

On the other hand, we have

$$\begin{aligned} \widehat{i}^* \big ( P(\widehat{v}_1,\cdots , \widehat{v}_d) \big ) = P(b_1 t+w_1,\cdots , b_d t+w_d)=\sum _{k\ge 0} P_k(w_1,\cdots ,w_d)\otimes t^k \end{aligned}$$

for some polynomials $P_k$, $k\ge 0$. Hence $\zeta = P_q(w_1,\cdots , w_d) = i^*(P(v_1,\cdots , v_d))$. Therefore, $i^*$ is surjective and $H^*(N;\mathbb Z_2)$ is generated by $w_1,\cdots , w_d \in H^1(N;\mathbb Z_2)$.

For the general case, suppose $H\cong \mathbb Z^s_2$, $1\le s\le r$. Then, we have a sequence:

$$\begin{aligned} \{0\}= H_0\subset H_1\subset H_2 \subset \cdots \subset H_s = H \end{aligned}$$

where $H_l\cong \mathbb Z^l_2$ for each $0\le l \le s$. Moreover, we have

$$\begin{aligned} M^H = \big ((M^{H_1})^{H_2/H_1})\cdots \big )^{H_s/H_{s-1}} ,\; H_l/H_{l-1}\cong \mathbb Z_2, \, l=1,\cdots , s. \end{aligned}$$

Repeating the above argument for each $H_l/H_{l-1}$ proves the lemma.$\square$

The following lemma is parallel to [27, Lemma 3.4].

Lemma 1.2

Let W be an equivariantly formal 2-torus manifold whose cohomology ring $H^*(W;\mathbb Z_2)$ is generated by its degree-one part. Then, all non-empty multiple intersections of the characteristic submanifolds of W are equivariantly formal 2-torus manifolds whose mod 2 cohomology rings are generated by their degree-one part as well.

Proof

Let $F_1,\cdots , F_m$ be all the facets of Q and $G=\mathbb Z^n_2$ where $n=\dim (W)$. In the following, we use the notations defined in Sect. 2.3. First of all, since the characteristic submanifold $W_i$ is a connected component of the fixed point set $X^{G_i}$, Lemma 4.1 implies that the restriction $H^*(W;\mathbb Z_2)\rightarrow H^*(W_i;\mathbb Z_2)$ is surjective. So the G-action on $W_i$ is equivariantly formal (by Proposition 3.4). Then, we have

$$\begin{aligned} H^*_G(W;\mathbb Z_2)&\cong H^*(W;\mathbb Z_2)\otimes H^*(BG;\mathbb Z_2), \\ H^*_G(W_i;\mathbb Z_2)&\cong H^*(W_i;\mathbb Z_2)\otimes H^*(BG;\mathbb Z_2). \end{aligned}$$

It follows that the restriction $H^*_G(W;\mathbb Z_2)\rightarrow H^*_G(W_i;\mathbb Z_2)$ is also surjective. In addition, by using Proposition 2.5 (i) and a completely parallel argument to the proof of [26, Prop. 3.4(2)], we can prove the following claim:

Claim: $H^*_G(W;\mathbb Z_2)$ is generated as a ring by all the equivariant Thom classes $\tau _1,\cdots ,\tau _m$ of the normal bundles of the characteristic submanifolds $W_1,\cdots , W_m$.

When $W_{j_1}\cap \cdots \cap W_{j_s} = \varnothing$, $\tau _{j_1}\cdots \tau _{j_s}$ clearly vanishes. So the above claim implies that for any $k\ge 0$, $H^k_G(W;\mathbb Z_2)$ is additively generated by the monomials $\tau ^{k_1}_{j_1}\cdots \tau ^{k_s}_{j_s}$ such that $W_{j_1}\cap \cdots \cap W_{j_s}\ne \varnothing$ and $k_1+\cdots +k_s =k$.

Let N be a connected component of $W_{i_1}\cap \cdots \cap W_{i_k}$, $1\le k \le n$. Then, N is the facial submanifold $W_f$ over some codimension-k face f of Q. So by Lemma 4.1, N is an equivariantly formal 2-torus manifold whose cohomology ring $H^*(N;\mathbb Z_2)$ is generated by its degree-one part. Moreover, by a completely parallel argument to the proof of [27, Lemma 3.4], we can show that N is the only connected component of $W_{i_1}\cap \cdots \cap W_{i_k}$ from the above discussion of $H^k_G(W;\mathbb Z_2)$. The lemma is proved.$\square$

The following proposition is parallel to [27, Lemma 8.2 (2)].

Proposition 1.3

Suppose W is an n-dimensional equivariantly formal 2-torus manifold with orbit space Q and the cohomology ring $H^*(W;\mathbb Z_2)$ is generated by its degree-one part. Then, the geometrical realization $|\mathcal {P}_Q|$ of the face poset $\mathcal {P}_Q$ of Q is a Gorenstein* simplicial complex over $\mathbb Z_2$. In particular, $\mathbb Z_2[\mathcal {P}_Q]=\mathbb Z[Q]$ is Cohen-Macaulay and $H^*(|\mathcal {P}_Q|;\mathbb Z_2)\cong H^*(S^{n-1};\mathbb Z_2)$.

Proof

By Lemma 4.2, all non-empty multiple intersections of the characteristic submanifolds of W are connected. This implies that $|\mathcal {P}_Q|$ is a simplicial complex. Moreover, by [30, II 5.1(d)], it is enough to verify the following three conditions to prove that $|\mathcal {P}_Q|$ is Gorenstein* over $\mathbb Z_2$:

• (a) $\mathbb Z_2[\mathcal {P}_Q]$ is Cohen-Macaulay;

• (b) Every $(n-2)$-simplex in $\mathcal {P}_Q$ is contained in exactly two $(n-1)$-simplices;

• (c) $\chi (\mathcal {P}_Q)=\chi (S^{n-1})$.

Since W is equivariantly formal, $H^*_G(W;\mathbb Z_2)$ is a free $H^*(BG;\mathbb Z_2)$-module and $\mathbb Z_2[\mathcal {P}_Q]=\mathbb Z_2[Q]$ is isomorphic to $H^*_G(W;\mathbb Z_2)$ (by Corollary 3.10) where $G=\mathbb Z^n_2$. This implies (a).

Note that each $(n-2)$-simplex of $\mathcal {P}_Q$ corresponds to a non-empty intersection of $n-1$ characteristic submanifolds of W. The latter intersection is an equivariantly formal 1-manifold by Lemma 4.2, so it is a circle with exactly two G-fixed points. This implies (b).

The proof of (c) is completely parallel to [27, Lemma 8.2 (2)], so we leave it to the reader. The proposition is proved.$\square$

Using the above proposition and the lemmas from Sect. 3, we obtain the following theorem that is parallel to [27, Theorem 7.7].

Theorem 1.4

Let W be a 2-torus manifold whose orbit space is Q. Then, W is equivariantly formal and the cohomology ring $H^*(W;\mathbb Z_2)$ is generated by its degree-one part if and only if the following three conditions are satisfied:

• (a) $H^*_G(W;\mathbb Z_2)$ is isomorphic to $\mathbb Z_2[Q]=\mathbb Z_2[\mathcal {P}_Q]$ as a graded ring.

• (b) $\mathbb Z_2[Q]$ is Cohen-Macaulay.

• (c) $|\mathcal {P}_Q|$ is a simplicial complex.

Proof

The argument is completely parallel to the proof of [27, Theorem 7.7]. We only need to replace $T^n$ by $\mathbb Z^2_n$ and $\mathbb {Q}$-coefficients by $\mathbb Z_2$-coefficients to obtain our proof here.$\square$

5 Proof of Theorem 1.3

In this section, we give a proof of Theorem 1.3. Our proof follows the proof of [27, Theorem 8.3, Theorem 9.3] almost step by step, while some arguments for 2-torus manifolds here are simpler than those for torus manifolds in [27].

5.1 Equivariant Cohomology of the Canonical Model

Let Q be a connected compact smooth nice n-manifold with corners. We call any function $\lambda : \mathcal {F}(Q)\rightarrow \mathbb Z^n_2$ that satisfies the linear independence relation in Sect. 2.1 a characteristic function on Q. By the same gluing rule in (4), we can obtain a space $M_Q(\lambda )$ from any characteristic function $\lambda$ on Q, called the canonical model determined by $(Q,\lambda )$. It is easy to see that $M_Q(\lambda )$ is a 2-torus manifold of dimension n.

Let $Q^{\vee }$ denote the cone of the geometrical realization of the order complex $\textrm{ord}(\overline{\mathcal {P}}_Q)$ of $\overline{\mathcal {P}}_Q=\mathcal {P}_Q-\{\hat{0}\}$. So topologically, $Q^{\vee }$ is homeomorphic to $\text {Cone}(|\mathcal {P}_Q|)$. Moreover, $Q^{\vee }$ is a “space with faces” (see Davis [13, Sec. 6]) where each proper face f of Q determines a unique “face” $f^{\vee }$ of $Q^{\vee }$ that is the geometrical realization of the order complex of the poset $\{ f' \,|\, f' \subseteq f \}$. More precisely, $f^{\vee }$ consists of all simplices of the form $f'_k\subsetneq \cdots \subsetneq f'_1\subsetneq f'_0=f$ in $\textrm{ord}(\overline{\mathcal {P}}_Q)$. The “boundary” of $Q^{\vee }$, denoted by $\partial Q^{\vee }$, is $\textrm{ord}(\overline{\mathcal {P}}_Q)$ which is homeomorphic to $|\mathcal {P}_Q|$. So we have homeomorphisms:

$$\begin{aligned} \partial Q^{\vee } \cong |\mathcal {P}_Q|, \ \ Q^{\vee }\cong \textrm{Cone}(|\mathcal {P}_Q|). \end{aligned}$$

(20)

Remark 1.1

When $|\mathcal {P}_Q|$ is a simplicial complex, the space $Q^{\vee }$ with the face decomposition was called in [14, p. 428] a simple polyhedral complex.

Suppose $F_1,\cdots , F_m$ are all the facets of Q. Let $\mathcal {F}(Q^{\vee }) = \{ F^{\vee }_1,\cdots , F^{\vee }_m\}$. Then, any characteristic function $\lambda : \mathcal {F}(Q)\rightarrow \mathbb Z^n_2$ induces a map $\lambda ^{\vee }: \mathcal {F}(Q^{\vee })\rightarrow \mathbb Z^n_2$ where $\lambda ^{\vee }(F^{\vee }_i) = \lambda (F_i)$, $1\le i \le m$. Then, by the same gluing rule in (4), we obtain a space $M_{Q^{\vee }}(\lambda ^{\vee })$ with a canonical $\mathbb Z^n_2$-action. By the same argument as in the proof of [27, Proposition 5.14], we can prove the following.

Proposition 1.2

There exists a continuous map $\phi : Q\rightarrow Q^{\vee }$ which preserves the face structure and induces an equivariant continuous map

$$\begin{aligned} \Phi : M_Q(\lambda ) \rightarrow M_{Q^{\vee }}(\lambda ^{\vee }). \end{aligned}$$

Here $\phi : Q\rightarrow Q^{\vee }$ is constructed inductively, starting from an identification of vertices and extending the map on each higher-dimensional face by a degree-one map. Since every face $f^{\vee }$ of $Q^{\vee }$ is a cone, there are no obstructions to such extensions.

In addition, by a similar argument to that in [14, Theorem 4.8], we can obtain the following result.

Proposition 1.3

$H^*_G(M_{Q^{\vee }}(\lambda ^{\vee });\mathbb Z_2)$ is isomorphic to $\mathbb Z_2[Q]$ where $G=\mathbb Z^n_2$.

On the other hand, $H^*_G(M_Q(\lambda );\mathbb Z_2)$ could be much more complicated. Indeed, it is shown in [31, Theorem 1.7] that $H^*_G(M_Q(\lambda );\mathbb Z_2)$ isomorphic to the so called topological face ring of Q over $\mathbb Z_2$ which involves the mod 2 cohomology rings of all the faces of Q.

5.2 Proof of Theorem 1.3 (ii)

Proof

We first prove the “if” part. Let Q be an n-dimensional mod 2 homology polytope and $G=\mathbb Z^n_2$. Since $H^1(Q;\mathbb Z_2)=0$ and W is locally standard, the principal G-bundle $\xi _W$ determined by W is a trivial G-bundle over Q. Then, by [25, Lemma 3.1], W is equivariantly homeomorphic to the canonical model $M_Q(\lambda _W)$ (see (4)). So by Proposition 5.3, there exists an equivariant continuous map

$$\begin{aligned} \Phi : W =M_Q(\lambda _W) \rightarrow M_{Q^{\vee }}(\lambda ^{\vee }_W):= W^{\vee }. \end{aligned}$$

Let $\pi : W\rightarrow Q$ and $\pi ^{\vee }: W^{\vee }\rightarrow Q^{\vee }$ be the projections, respectively. Let $F_1,\cdots , F_m$ be all the facets of Q. Since Q is a mod 2 homology polytope, so are $F_1,\cdots , F_m$. For brevity, let

$$\begin{aligned} W_i = \pi ^{-1}(F_i),\ \ W^{\vee }_i = (\pi ^{\vee })^{-1}(F^{\vee }_i), \ 1\le i \le m. \end{aligned}$$

It is easy to see that the $\mathbb Z^n_2$-actions on $W\backslash \bigcup _i W_i$ and $W^{\vee }\backslash \bigcup _i W^{\vee }_i$ are both free. Then, we have

$$ H^*_G\Big (\!W,\bigcup _i W_i;\mathbb Z_2 \Big ) \!\cong \! H^*(Q,\partial Q;\mathbb Z_2), \! \ H^*_G\Big (\!W^{\vee },\bigcup _i W^{\vee }_i;\mathbb Z_2\!\Big ) \!\cong \! H^*(Q^{\vee },\partial Q^{\vee };\!\mathbb Z_2). $$

So $\Phi : W \rightarrow W^{\vee }$ induces a map between the following two exact sequences:

(21)

Each $W_i$ is a 2-torus manifold over the homology polytope $F_i$. So using induction and a Mayer-Vietoris argument, we may assume that in the diagram (21), $\Phi ^*: H^*_G(\bigcup _i W^{\vee }_i;\mathbb Z_2)\rightarrow H^*_G(\bigcup _i W_i;\mathbb Z_2)$ is an isomorphism.

By Proposition 2.9, $H^*(|\mathcal {P}_Q|;\mathbb Z_2)\cong H^*(S^{n-1};\mathbb Z_2)$. Then, by (20), we obtain

$$\begin{aligned} H^*(Q^{\vee },\partial Q^{\vee };\mathbb Z_2)\cong H^*(D^n,S^{n-1};\mathbb Z_2). \end{aligned}$$

We also have $H^*(Q,\partial Q;\mathbb Z_2)\cong H^*(D^n,S^{n-1};\mathbb Z_2)$ since Q is an n-dimensional mod 2 homology polytope. By the construction of $\phi$, it is easy to see that the homomorphism $\phi ^*: H^*(Q^{\vee },\partial Q^{\vee };\mathbb Z_2)\rightarrow H^*(Q,\partial Q;\mathbb Z_2)$ is an isomorphism. Then, by applying the five-lemma to the diagram (21), we can deduce that $\Phi ^*: H^*_G(W^{\vee };\mathbb Z_2) \rightarrow H^*_G(W;\mathbb Z_2)$ is also an isomorphism. So by Proposition 5.3, $H^*_G(W;\mathbb Z_2)\cong \mathbb Z_2[Q]$.

Besides, we also know that $\mathbb Z_2[Q]$ is Cohen-Macaulay by Proposition 2.9. Then, since $|\mathcal {P}_Q|$ is a simplicial complex, all the three conditions in Theorem 4.4 are satisfied. Hence W is equivariantly formal and $H^*(W;\mathbb Z_2)$ is generated by its degree-one part as a ring. The “if” part is proved.

Next, we prove the “only if” part. By the assumption on W and Lemma 4.2, all non-empty multiple intersections of characteristic submanifolds of W are connected and their cohomology rings are generated by their degree-one elements. So we may assume by induction that all the proper faces of Q are mod 2 homology polytopes. In particular, the proper faces of Q are all mod 2 acyclic. From these assumptions, we need to prove that Q itself is mod 2 acyclic.

By Proposition 4.3, $|\mathcal {P}_Q|$ is a simplicial complex. So $|\mathcal {P}_Q|$ is the nerve simplicial complex of the cover of $\partial Q$ by the facets of Q. By a Mayer-Vietoris sequence argument, we can deduce that $H^*(\partial Q;\mathbb Z_2)\cong H^*(|\mathcal {P}_Q|;\mathbb Z_2)$. This together with Proposition 4.3 shows that

$$\begin{aligned} H^*(\partial Q;\mathbb Z_2)\cong H^*(S^{n-1};\mathbb Z_2). \end{aligned}$$

(22)

Claim: $H^1(Q;\mathbb Z_2)=0$.

Since W is equivariantly formal, $H^*_G(W;\mathbb Z_2)$ is a free $H^*(BG;\mathbb Z_2)$-module. On the other hand, $H^*(Q,\partial Q;\mathbb Z_2)$ is finitely generated over $\mathbb Z_2$ since Q is compact. So $H^*(Q,\partial Q;\mathbb Z_2)$ is a torsion $H^*(BG;\mathbb Z_2)$-module. It follows that the whole bottom row in the diagram (21) splits into short exact sequences:

$$\begin{aligned} 0\rightarrow H^k_G(W;\mathbb Z_2) \rightarrow H^k_G \Big (\bigcup _i W_i;\mathbb Z_2 \Big ) \rightarrow H^{k+1}(Q,\partial Q;\mathbb Z_2) \rightarrow 0, \ k\ge 0. \end{aligned}$$

(23)

Take $k=0$ above, we clearly have $H^0_G(W;\mathbb Z_2)\cong H^0_G \Big (\bigcup _i W_i;\mathbb Z_2 \Big )\cong \mathbb Z_2$. This implies $H^1(Q,\partial Q;\mathbb Z_2)=0$. So in the following exact sequence,

$$\begin{aligned} \cdots \rightarrow H^1(Q,\partial Q;\mathbb Z_2)\rightarrow H^1(Q;\mathbb Z_2) \rightarrow H^1(\partial Q;\mathbb Z_2) \rightarrow \cdots , \end{aligned}$$

$H^1(Q;\mathbb Z_2)$ is mapped injectively into $H^1(\partial Q;\mathbb Z_2)\cong H^1(S^{n-1};\mathbb Z_2)$. Note that if $n=1$, the claim is trivial. When $n=2$, we have $\partial Q =S^1$ and $H^1(Q;\mathbb Z_2) = 0$ or $\mathbb Z_2$. But by the classification of compact surfaces, the latter case is impossible. When $n\ge 3$, we have $H^1(\partial Q;\mathbb Z_2)=0$, so $H^1(Q;\mathbb Z_2)=0$. The claim is proved.

Now since $H^1(Q;\mathbb Z_2)=0$, by the above proof of the “if” part, there exists an equivariant homeomorphism $\Phi$ from W to the canonical model $M_Q(\lambda _W)$. In addition, by (20) and Proposition 4.3, we have

$$\begin{aligned} H^*(\partial Q^{\vee };\mathbb Z_2) \cong H^*( |\mathcal {P}_Q|;\mathbb Z_2) \cong H^*(S^{n-1};\mathbb Z_2). \end{aligned}$$

So we have an isomorphism

$$\begin{aligned} H^*(Q^{\vee },\partial Q^{\vee };\mathbb Z_2)\cong H^*(D^n,S^{n-1};\mathbb Z_2). \end{aligned}$$

(24)

Then, by the construction of $\phi$, the map $\phi ^*: H^*(Q^{\vee },\partial Q^{\vee };\mathbb Z_2) \!\rightarrow \! H^*(Q,\partial Q;\mathbb Z_2)$ is an isomorphism in degree n (since Q is connected) and thus is injective in all degrees. So by an extended version of the 5-lemma, we can deduce that in the diagram (21) the map $\Phi ^*: H^*_G(W^{\vee };\mathbb Z_2)\rightarrow H^*_G(W;\mathbb Z_2)$ is injective. Moreover,

• $H^*_G(W^{\vee };\mathbb Z_2)=H^*_G(M_{Q^{\vee }}(\lambda ^{\vee }_W);\mathbb Z_2)\cong \mathbb Z_2[Q]$ by Proposition 5.3, and

• $\mathbb Z_2[Q]\cong H^*_G(W;\mathbb Z_2)$ by Corollary 3.10.

So $H^*_G(W^{\vee };\mathbb Z_2)$ and $H^*_G(W;\mathbb Z_2)$ have the same dimension over $\mathbb Z_2$ in each degree. Therefore, the monomorphism $\Phi ^*: H^*_G(W^{\vee };\mathbb Z_2)\rightarrow H^*_G(W;\mathbb Z_2)$ is actually an isomorphism. Then, by the 5-lemma again, we can deduce from the diagram (21) that $\phi ^*: H^*(Q^{\vee },\partial Q^{\vee };\mathbb Z_2)\rightarrow H^*(Q,\partial Q;\mathbb Z_2)$ is an isomorphism. So by (24),

$$\begin{aligned} H^*(Q,\partial Q;\mathbb Z_2)\cong H^*(D^n,S^{n-1};\mathbb Z_2) \end{aligned}$$

which implies that Q is mod 2 acyclic by Poincaré-Lefschetz duality. This finishes the proof.$\square$

5.3 Proof of Theorem 1.3 (i)

Proof

We can reduce Theorem 1.3 (i) to Theorem 1.3 (ii) by real blow-ups of W along sufficient many facial submanifolds, which corresponds to doing some barycentric subdivisions of the face poset $\mathcal {P}_Q$ of Q (see Fig. 2). Indeed, after doing enough barycentric subdivisions to $\mathcal {P}_Q$, we can turn $|\mathcal {P}_Q|$ into a simplicial complex. Let $\widehat{W}$ be the 2-torus manifold obtained after these real blow-ups on W and $\widehat{Q}$ be its orbit space (with $|\mathcal {P}_{\widehat{Q}}|$ being a simplicial complex).

• Fact-1: $\widehat{W}$ is equivariantly formal if and only if so is W (by Proposition 3.12).

• Fact-2: $\widehat{Q}$ is mod 2 face-acyclic if and only if so is Q (by Lemma 3.13).

Fig. 2

Cutting a vertex and an edge

We first prove the “if” part. Suppose W is locally standard and Q is mod 2 face-acyclic. Then, $\widehat{W}$ is also locally standard and $\widehat{Q}$ is a mod 2 homology polytope by Fact-2. So by Theorem 1.3 (ii), $\widehat{W}$ is equivariantly formal, then so is W.

Next, we prove the “only if” part. If W is equivariantly formal, then so is $\widehat{W}$, and W is locally standard by Theorem 3.3. So by Corollary 3.10, we have a graded ring isomorphism $H^*_G(\widehat{W};\mathbb Z_2)\cong \mathbb Z_2[\widehat{Q}]$. Moreover, since $|\mathcal {P}_{\widehat{Q}}|$ is a simplicial complex, $\mathbb Z_2[\widehat{Q}]$ is generated by its degree-one elements, then so is $H^*_G(\widehat{W};\mathbb Z_2)$. In addition, since $\iota _{\widehat{W}}^*: H^*_G(\widehat{W};\mathbb Z_2)\rightarrow H^*(\widehat{W};\mathbb Z_2)$ is surjective, $H^*(\widehat{W};\mathbb Z_2)$ is also generated by its degree-one elements. Then, by Theorem 1.3 (ii), $\widehat{Q}$ is a mod 2 homology polytope. So by Fact-2, Q is mod 2 face-acyclic.$\square$

5.4 Proof of Theorem 1.5

Proof

We first prove the “if” part. Assume that there exists a regular $\textrm{m}$-involution $\tau$ on W. By definition the fixed point set $W^{\tau }$ of $\tau$ is discrete, then so is $W^{\mathbb Z^n_2}\subseteq W^{\tau }$. This implies that Q must have vertices. Let p be a vertex of Q and let $F_1,\cdots , F_n$ be all the facets containing p. By the property of $\lambda _W$,

$$\begin{aligned} e_1=\lambda _W(F_1),\cdots , e_n=\lambda _W(F_n) \end{aligned}$$

form a linear basis of $\mathbb Z_2^n$ over $\mathbb Z_2$. Then, since the $\mathbb Z_2^n$-action on W is locally standard, it is easy to see that only when $g=e_1+\cdots +e_n$ could the fixed point set $W^{\tau _g}$ be discrete. So we must have $\tau =\tau _{e_1+\cdots +e_n}$, and in particular

$$\begin{aligned} W^{\tau }=W^{\tau _{e_1+\cdots +e_n}}=W^{\mathbb Z_2^n}. \end{aligned}$$

Hence

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(W^{\mathbb Z_2^n};\mathbb Z_2) = \dim _{\mathbb Z_2} H^*(W^{\tau };\mathbb Z_2) =\dim _{\mathbb Z_2} H^*(W;\mathbb Z_2) \end{aligned}$$

where the second “$=$” is due to the assumption that $\tau$ is an $\textrm{m}$-involution. So by Theorem 1.1, W is equivariantly formal. Then, Q is mod 2 face-acyclic by Theorem 1.3. In particular, every face of Q has a vertex and the 1-skeleton of Q is connected (by Lemma 2.2).

It remains to prove that the image of $\lambda _W: \mathcal {F}(Q)\rightarrow \mathbb Z_2^n$ is exactly $\{e_1,\cdots , e_n\}$. Indeed, take an edge e of Q whose vertices are p and $p'$. So the n facets of Q that contain $p'$ are $F_1,\cdots , F_{i-1}, F'_i, F_{i+1},\cdots , F_n$ for some $1\le i \le n$. Then, since $\tau _{e_1+\cdots +e_n}$ is an $\textrm{m}$-involution, we must have

$$\begin{aligned} \lambda _W(F_1)+\cdots + \lambda _W(F_{i-1}) +\lambda _W( F'_i)+ \lambda _W(F_{i+1}) + \cdots +\lambda _W(F_n) = e_1+\cdots +e_n. \end{aligned}$$

This implies $\lambda _W( F'_i)=e_i$. Then, since the 1-skeleton of Q is connected and every facet F of Q contains a vertex, we can iterate the above argument to prove that every $\lambda _W(F)$ must take value in $\{e_1,\cdots , e_n\}$.

Next, we prove the “only if” part. Suppose Q is mod 2 face-acyclic and the values of the characteristic function $\lambda _W$ of Q consist exactly of a linear basis $e_1,\cdots , e_n$ of $\mathbb Z^n_2$. By Theorem 1.3 (i), W is equivariantly formal. So we have

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(W^{\mathbb Z_2^n};\mathbb Z_2)=\dim _{\mathbb Z_2} H^*(W;\mathbb Z_2) \ \ \text {(by Theorem~1.1)}. \end{aligned}$$

On the other hand, our assumption on $\lambda _W$ implies that the regular involution $\tau =\tau _{e_1+\cdots +e_n}$ satisfies $W^{\tau }=W^{\mathbb Z^n_2}$ which is a discrete set. Then, we have

$$\begin{aligned} \dim _{\mathbb Z_2} H^*(W^{\tau };\mathbb Z_2) = \dim _{\mathbb Z_2} H^*(W^{\mathbb Z_2^n};\mathbb Z_2)=\dim _{\mathbb Z_2} H^*(W;\mathbb Z_2). \end{aligned}$$

So $\tau$ is a regular $\textrm{m}$-involution on W by definition. The theorem is proved.$\square$

Remark 1.4

If we do not assume a 2-torus manifold W to be locally standard, even if W admits a regular $\textrm{m}$-involution, W may not be equivariantly formal or locally standard. For example: let

$$\begin{aligned} S^2=\{(x_1,x_2,x_3)\in \mathbb R^3\,|\, x_1^2+x^2_2 + x^2_3 =1 \}. \end{aligned}$$

Define two involutions $\sigma$ and $\sigma '$ on $S^2$ by

$$\sigma (x_1,x_2,x_3)=(-x_1,-x_2,x_3), \quad \sigma '(x_1,x_2,x_3) = (x_1,x_2,-x_3). $$

It is easy to see that $\sigma$ is an $\textrm{m}$-involution on $S^2$ with two isolated fixed points (0, 0, 1) and $(0,0,-1)$. But since the $\mathbb Z^2_2$-action on $S^2$ determined by $\sigma$ and $\sigma '$ has no global fixed point, it is not equivariantly formal. We can also directly check that this $\mathbb Z^2_2$-action on $S^2$ is not locally standard.

Finally, we propose some questions on weakly equivariantly formal 2-torus manifolds:

Question-3: Does there exist a weakly equivariantly formal 2-torus manifold which is not equivariantly formal?

Question-4: If a 2-torus manifold is weakly equivariantly formal, are there any restrictions on the topology and combinatorial structure of its orbit space?

Question-5: Whether or not a 2-torus manifold being weakly equivariantly formal is determined only by the topology and combinatorial structure of its orbit space?