A GENERALIZATION OF MOMENT-ANGLE MANIFOLDS WITH NON-CONTRACTIBLE ORBIT SPACES

. We generalize the notion of moment-angle manifold over a simple convex polytope to an arbitrary nice manifold with corners. When Q is a nice PL-manifold with corners, we obtain a formula to compute the homology groups of such manifolds via the strata of Q , which generalizes the Hochster’s formula for computing the homology groups of moment-angle manifolds.


Introduction
The construction of a moment-angle manifold over a simple polytope is first introduced in Davis-Januszkiewicz [17].Suppose P is a simple (convex) polytope with m facets (codimension-one faces).A convex polytope in a Euclidean space is called simple if every codimension-k face is the intersection of exactly k facets of the polytope.The moment-angle manifold Z P over P is a closed connected manifold with an effective action by the compact torus T m = (S 1 ) m whose orbit space is P .It is shown in [17] that many important topological invariants of Z P can be computed easily from the combinatorial structure of P .These manifolds play an important role in the research of toric topology.The reader is referred to Buchstaber-Panov [9,10] for more discussions on the topological and geometrical aspects of moment-angle manifolds.
The notion of moment-angle manifold over a simple convex polytope has been generalized in many different ways.For example, Davis and Januszkiewicz [17] define a class of topological spaces now called moment-angle complexes (named by Buchstaber and Panov in [8]) where the simple polytope is replaced by a simple polyhedral complex.Later, Lü and Panov [26] defined the notion of momentangle complex of a simplicial poset.In addition, Ayzenberg and Buchstaber [1] defined the notion of moment-angle spaces over arbitrary convex polytopes (not necessarily simple).Note that in all these generalizations, the orbit spaces of the canonical torus actions are all contractible.Yet an even wider class of spaces called generalized moment-angle complexes or polyhedral products over simplicial complexes were introduced by Bahri, Bendersky, Cohen and Gitler in [3], which has become the major subject in the homotopy theoretic study of toric topology.
In this paper, we generalize the construction of moment-angle manifolds by replacing the simple polytope P by a nice manifold with corners Q which is not necessarily contractible.Such a generalization has been considered by Poddar and Sarkar [29] for polytopes with simple holes.
A motive for the study of this generalized construction is to compute the equivariant cohomology ring of locally standard torus actions.Recall that an action of a compact torus T n on a smooth compact manifold M of dimension 2n is called locally standard if it is locally modeled on the standard representation of T n on C n .Then the orbit space Q = M/T n is a manifold with corners.Conversely, every manifold with a locally standard T n -action and with Q as the orbit space is equivariantly homeomorphic to the quotient construction Y / ∼, where Y is a principal T n -bundle over Q and ∼ is an equivalence relation determined by the characteristic function on Q (see [35]).Generally speaking, it is difficult to compute the equivariant cohomology ring of M from the corresponding principal bundle Y and the characteristic function on Q.But we will see in Corollary 5.5 that when Y is the trivial T n -bundle over Q, the equivariant cohomology ring of M can be computed from the strata of Q directly.Examples of such kind include many toric origami manifolds (see [12,22,2]) with coorientable folding hypersurface where the faces of the orbit spaces may be non-acyclic.
Recall that an n-dimensional manifold with corners Q is a Hausdorff space with a maximal atlas of local charts onto open subsets of R n ≥0 such that the transitional functions are homeomorphisms which preserve the codimension of each point.Here the codimension c(x) of a point ≥0 is the number of x i which are 0.So we have a well defined map c : Q → Z ≥0 where c(q) is the codimension of a point q ∈ Q.In particular, the interior Q • of Q consists of points of codimension 0, i.e.Q • = c −1 (0).
Suppose Q is an n-dimensional manifold with corners with ∂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.Note that a face of codimension zero in Q is just a connected component of Q.
A manifold with corners Q is said to be nice if either its boundary ∂Q is empty or ∂Q is non-empty and any codimension-k face of Q is a component of the intersection of k different facets in Q.
Let Q be a nice n-manifold with corners.Let F (Q) = {F 1 , • • • , F m } be the set of facets of Q.For any subset J ⊆ [m] = {1, • • • , m}, let It is clear that Let λ : F (Q) → Z m be a map such that {λ(F 1 ), • • • , λ(F m )} is a unimodular basis of Z m ⊂ R m .Since S 1 = {z ∈ C | z = 1}, we can identify the m-torus (S 1 ) m = R m /Z m .The moment-angle manifold over Q is defined by: (1) where (x, g) ∼ (x ′ , g ′ ) if and only if x = x ′ and g −1 g ′ ∈ T λ x where T λ x is the subtorus of (S 1 ) m determined by the linear subspace of R m spanned by the set {λ(F j ) | x ∈ F j }.There is a canonical action of (S 1 ) m on Z Q defined by: (2) g ′ • [(x, g)] = [(x, g ′ g)], x ∈ Q, g, g ′ ∈ (S 1 ) m .
Since the manifold with corners Q is nice and λ is unimodular, it is easy to see from the above definition that Z Q is a manifold.
Convention: In the rest of this paper, we assume that any nice manifold with corners Q can be equipped with a CW-complex structure such that every face of Q is a subcomplex.In addition, we assume that Q has only finitely many faces.Note that a compact smooth nice manifold with corners always satisfies these two conditions since it is triangulable (see Johnson [25]).But in general we do not require Q to be compact or smooth.We do not assume Q to be connected either.
Similarly to the stable decomposition of (generalized) moment-angle complexes obtained in [3], we have the following stable decomposition of Z Q .Theorem 1.1.Let Q be a nice manifold with corners with facets There is a homotopy equivalence where denotes the wedge sum and Σ denotes the reduced suspension.
It is indicated in [10, Exercise 3.2.14]that Theorem 1.4 holds for any simple polytope.Moreover, we can generalize Theorem 1.4 to describe the cohomology ring of the polyhedral product of any (D, S) = { D n j +1 , S n j , a j } m j=1 over Q (see Theorem 4.8).In particular, we have the following result for RZ Q .
Theorem 1.5 (Corollary 4.10).Let Q be a nice manifold with corners with facets Moreover, the integral cohomology ring of RZ Q is isomorphic as a graded ring to the ring (R * Q , ∪) where ∪ is the relative cup product We can describe the equivariant cohomology ring of Z Q with respect to the canonical action of (S 1 ) m as follows.
Let k denote a commutative ring with a unit.For any Definition 1.6 (Topological Face Ring).Let Q be a nice manifold with corners with m facets F 1 , • • • , F m .For any coefficients ring k, the topological face ring of Q over k is defined to be (7) k Here if In addition, we can consider k Q as a graded ring if we choose a degree for every indeterminate Then the equivariant cohomology ring of Z Q (or RZ Q ) with Z-coefficients (or Z 2 -coefficients) with respect to the canonical (S 1 ) m -action (or (Z 2 ) m )-action) is isomorphic as a graded ring to the topological face ring Z Q (or Moreover, the natural H * (BT m )-module structure on the integral equivariant cohomology ring H * T m (Z Q ) is described in (52) where T m = (S 1 ) m .Remark 1.8.A calculation of the equivariant cohomology group of Z Q with Z-coefficients was announced earlier by T. Januszkiewicz in a talk [24] in 2020.The formula given in Januszkiewicz's talk is equivalent to our Z Q .But the ring structure of the equivariant cohomology of Z Q was not described in [24].
For a nice manifold with corners Q, there are two other notions which reflect the stratification of Q.One is the face poset of Q which is the set of all faces of Q ordered by inclusion, denoted by S Q (note that each connected component of Q is also a face).The other one is the nerve simplicial complex of the covering of ∂Q by its facets, denoted by K Q .The face ring (or Stanley-Reisner ring) of a simplicial complex is an important tool to study combinatorial objects in algebraic combinatorics and combinatorial commutative algebra (see [28] and [30]).
When Q is a simple polytope, all faces of Q, including Q itself, and all their intersections are acyclic.Then it is easy to see that the topological face ring of Q is isomorphic to the face ring of K Q (see Example 5.2).But in general, the topological face ring of Q encodes more topological information of Q than the face ring of K Q .
There is another way to think of the topological face ring k Q .Let where product * on R * ∩Q,k is defined by: for any with respect to their 2 [m] -gradings.By definition, the Segre product of two rings R and S graded by a common semigroup A (using the notation in [21]) is: So R ⊗ S is a subring of the tensor product of R and S (as graded rings).The Segre product of two graded rings (or modules) is studied in algebraic geometry and commutative algebra (see [13] and [21,19] for example).
Here we can think of 2 [m] as a semigroup where the product of two subsets of [m] is just their union.Then by this notation, we can write From this form, we see that k Q is essentially determined by R * ∩Q,k .The paper is organized as follows.In Section 2, we first construct an embedding of Q into Q × [0, 1] m which is analogous to the embedding of a simple polytope into a cube.This induces an embedding of Z Q into Q × (D 2 ) m from which we can do the stable decomposition of Z Q and give a proof of Theorem 1.1.Our argument proceeds along the same line as the argument given in [3,Sec. 6] but with some extra ingredients.In fact, we will not do the stable decomposition of Z Q directly, but the stable decomposition of the disjoint union of Z Q with a point.In Section 3, we obtain a description of the product structure of the cohomology of Z Q using the stable decomposition of Z Q and the partial diagonal map introduced in [4].From this we give a proof of Theorem 1.4.In Section 4, we define the notion of polyhedral product of a sequence of based CW-complexes over a nice manifold with corners Q and obtain some results parallel to Z Q for these spaces.In particular, we obtain a description of the integral cohomology ring of real moment-angle manifold RZ Q (see Corollary 4.10).In Section 5, we compute the equivariant cohomology ring of Z Q and prove Theorem 1.7.In Section 6, we discuss more generalizations of the construction of Z Q and extend our main theorems to some wider settings.

Stable Decomposition of Z Q
Let Q be a nice manifold with corners with m facets.To obtain the stable decomposition of Z Q , we first construct a special embedding of Q into Q×[0, 1] m , called the rim-cubicalization of Q.This construction can be thought of as a generalization of the embedding of a simple polytope with m facets into [0, 1] m defined in [9,Ch. 4].

Rim-cubicalization of
be all the facets of Q.For a face f of Q, let I f be the following subset of [m] called the strata index of f .
Then we define a subset f of Q × [0, 1] m associated to f as follows.We write and define In particular, Let S Q be the face poset of Q and define It is easy to see that Q is a nice manifold with corners whose facets are can think of Q as inductively gluing the product of all codimension-k strata of Q with a k-cube to ∂Q (see Figure 1).
Lemma 2.1.Q is homeomorphic to Q as a manifold with corners.
Proof.For any face f of Q and 0 ≤ t ≤ 1, let Then Q(t) determines an isotopy (see Figure 1) from There is a strong deformation retraction from C n k (−1) to C n k (0) defined by where It is easy to see that for any t ∈ [0, 1], the image of H( , t) is So H actually defines an isotopy from C n k (−1) to C n k (0) (see Figure 2).
In the following, we consider [0, 1] as a subset of D 2 and the cube [0, 1] m as a subset of (D 2 ) m ⊂ C m .For any j ∈ [m], let S 1 (j) and D 2 (j) denote the corresponding spaces indexed by j.
There is a canonical action of (S 1 ) m on Q × (D 2 ) m defined by where x ∈ Q, g j ∈ S 1 (j) and z j ∈ D 2 (j) for 1 ≤ j ≤ m.The orbit space of this action can be identified with Q × [0, 1] m .We denote the quotient map by For any face f of Q, we define (10) (D 2 , S 1 ) There is a canonical action of (S 1 ) m on (D 2 , S 1 ) Q induced by the canonical action of (S 1 ) ) m which can be written explicitly as Notice that the facets of Q are the intersections of We can easily check that the restriction of Moreover, for any face f of Q, we have So we have a homeomorphism Clearly, the above homeomorphism is equivariant with respect to the canonical actions of (S 1 ) m on Z Q and (D 2 , S 1 ) Q .So the lemma is proved.
By Lemma 2.3, studying the stable decomposition of Z Q is equivalent to studying that for (D 2 , S 1 ) Q .To do the stable decomposition as in [3], we want to first think of (D 2 , S 1 ) Q as the colimit of a diagram of CW-complexes over a finite poset (partially ordered set).The following are some basic definitions (see [38]).
• Let CW be the category of CW-complexes and continuous maps.
• Let CW * be the category of based CW-complexes and based continuous maps.• A diagram D of CW-complexes or based CW-complexes over a finite poset P is a functor D : P → CW or CW * such that for every p ≤ p ′ in P, there is a map where ∼ denotes the equivalence relation generated by requiring that for each x ∈ D(p ′ ), x ∼ d pp ′ (x) for every p < p ′ .To think of (D 2 , S 1 ) Q as a colimit of CW-complexes, we need to introduce a finer decomposition of (D 2 , S 1 ) Q as follows.By the notations in Section 2.1, for any face f of Q and any subset Corresponding to this decomposition, we define a poset associated to Q by ( 15) It follows from the definition (13) that: Note that P Q is a finite poset since by our convention Q only has finitely many faces.
Definition 2.4.Let D : P Q → CW be a diagram of CW-complexes where Clearly, (D 2 , S 1 ) Q is the colimit of the diagram D. So we have ( 16) Remark 2.5.Here we do not write (D 2 , S 1 ) Q as the colimit of a diagram of based CW-complexes.This is because in general it is not possible to choose a basepoint in each (D 2 , S 1 ) (f,L) to adapt to the colimit construction of a diagram in CW * .

Stable decomposition of Z Q .
First of all, let us recall a well-known theorem (see [23,34]) which allows us to decompose the Cartesian product of a collection of based CW-complexes into a wedge of spaces after doing a suspension. Let where I runs over all the non-empty subsets of [m].Furthermore, the map h commutes with colimits.
In our proof later, we need a slightly generalized version of Theorem 2.6.Before that, let us first prove three simple lemmas.
Proof.The deformation retraction from X to x 0 naturally induces a deformation retraction from X ∧Y = X ×Y /({x 0 }×Y )∪(X ×{y 0 }) to its canonical basepoint Then there is a homotopy equivalence Proof.This follows easily from the definition of reduced suspension.
Proof.(a) By the definition of smash product, we have a homeomorphism (b) It follows directly from the definition of smash product.
We can generalize Theorem 2.6 to the following form.
Theorem 2.10.Let (X i , x i ) for 1 ≤ i ≤ m be based CW-complexes.Assume that for some 1 ≤ n ≤ m, where Y i is a connected CW-complex and There is a based, natural homotopy equivalence which commutes with colimits: We choose a basepoint for each Y i , 1 ≤ i ≤ n.So by Lemma 2.8, we have are all connected based CW-complexes, we can apply Theorem 2.6 to each Y I × X [n+1,m] and obtain On the other hand, for any Then by iteratively using Lemma 2.9(a), we obtain • If J = ∅ and I = ∅, by iteratively using Lemma 2.9(b), we can deduce that X I is the disjoint union of Y I and a point represented by x I .So by Lemma 2.8, Σ( By comparing the above expression with (17), we prove the theorem.
Remark 2.11.By Theorem 2.10, it is not hard to see that all the main theorems in [3] also hold for based CW-complex pairs {(X i , A i , a i )} m i=1 where each of X i and A i is either connected or is a disjoint union of a connected CW-complex with its basepoint.In particular, [3, Corollary 2.24] also holds for (D 1 , S 0 ).Remark 2.12.It is possible to extend Theorem 2.10 further to deal with spaces each of which is a disjoint union of a connected CW-complex with finitely many points.But since Theorem 2.10 is already enough for our discussion in this paper, we leave the more generalized statement to the reader.Definition 2.13.For any based CW-complexes (X, x 0 ) and (Y, y 0 ), let If each of X and Y is either connected or is a disjoint union of a connected CW-complex with its basepoint, there is a homotopy equivalence by Theorem 2.10 We can further generalize Theorem 2.6 to the following form.We will use the following convention in the rest of the paper: Theorem 2.14.Let (X i , x i ), 1 ≤ i ≤ m and (B, b 0 ) be a collection of based CW-complexes where each of X i and B is either connected or is a disjoint union of a connected CW-complex with its basepoint.Then there is a based, natural homotopy equivalence which commutes with colimits: Proof.By definition, we have Then by (18), we have To apply the above stable decomposition lemmas to (D 2 , S 1 ) Q , we need to choose a basepoint for each (D 2 , S 1 ) (f,L) in the first place.But by Remark 2.5, there is no good way to choose a basepoint inside each (D 2 , S 1 ) (f,L) to adapt to the colimit construction of (D 2 , S 1 ) Q .So in the following, we add an auxiliary point to all (D 2 , S 1 ) (f,L) as their common basepoint.
• Let 1 (j) be the basepoint of S 1 (j) and D 2 (j) for every j ∈ [m].
). Next, we analyze the reduced suspension Σ colim(D + ) from the colimit point of view.Since all the (D 2 , S 1 ) (f,L) + share the same basepoint q 0 , we have Lemma 2.15.For any (f, L) ∈ P Q , there is a natural homeomorphism which commutes with taking the colimit: Proof.By our definitions, The above homeomorphism " ∼ =" is induced by the global homeomorphism with q 0 .The lemma follows.
Since we assume that each face f of Q is a CW-complex in our convention, we can deduce from Theorem 2.14 and Lemma 2.15 that ≃ According to (21), we define a family of diagrams of based CW-complexes where Since here the reduced suspension commutes with colimits up to homotopy equivalence (see [3,Theorem 4.3]), we obtain a homotopy equivalence (23) Σ colim(D Σ colim( D J + ) .
The following theorem from [3] will be useful in our proof of Theorem 1.1.It is a modification of the "Homotopy Lemma" given in [31,38,33].
Theorem 2.16 (Corollary 4.5 in [3]).Let D and E be two diagrams over a finite poset P with values in CW * for which the maps colim q>p D(q) ֒→ D(p), and colim q>p E(q) ֒→ E(p) are all closed cofibrations.If f is a map of diagrams over P such that for every p ∈ P, f p : D(p) → E(p) is a homotopy equivalence, then f induces a homotopy equivalence f : colim(D(P )) → colim(E(P )).Now we are ready to give a proof of Theorem 1.1.
Proof of Theorem 1.1.By ( 20) and ( 23), we obtain a homotopy equivalence Notice that when J ∩ (I f \L) = ∅, D J + ((f, L)) is contractible by Lemma 2.7.So for any J ⊆ [m], we define another diagram of based CW-complexes is either the natural inclusion or the constant map c [ q J 0 ] (mapping all points to [ q J 0 ]).The basepoint of + be a map of diagrams over P Q defined by: Then by Theorem 2.16, there exists a homotopy equivalence: Note that we always have j) .To understand colim E J + , we need to figure out in (25) what are those faces f of Q with some L I f such that J ∩ (I f \L) = ∅.
• There exists L I f with J ∩ (I f \L) = ∅ if and only if J ∩ I f = ∅, which is equivalent to f ⊆ F J .Conversely, we have This implies ( 26) The above discussion implies: (28) By the definition of E J + , if we have a face f of Q and two subsets L, So in this case, we have Then in colim E J + , the image of any of such Combining all the above arguments, we obtain homotopy equivalences: On the other hand, we have Then the theorem follows.

Cohomology ring structure of Z Q
The cohomology ring of the moment-angle complex over a simplicial complex K was computed in Franz [18] and Baskakov-Buchstaber-Panov [7].The cohomology rings of a much wider class of spaces called generalized moment-angle complexes or polyhedral products were computed in Bahri-Bendersky-Cohen-Gitler [4] via partial diagonal maps and in Bahri-Bendersky-Cohen-Gitler [5] by a spectral sequence under certain freeness conditions (coefficients in a field for example).The study in this direction is further extended in [6].A computation using different methods was carried out in Wang-Zheng [32] and Zheng [37].
It was shown in Bahri-Bendersky-Cohen-Gitler [4] that the product structure on the cohomology of a polyhedral product over a simplicial complex can be formulated in terms of the stable decomposition and partial diagonal maps of the polyhedral product.For a nice manifold with corners Q, since we also have the stable decomposition of Z Q , we should be able to describe the cohomology ring of Z Q in a similar way.
Let us first recall the definition of partial diagonal in product spaces from [4].Let X 1 , • • • , X m be a collection of based CW-complexes.Using the notations in Section 2.4, for any I ⊆ [m], there are natural projections X [m] → X I obtained as the composition where Π I : X [m] → X I is the natural projection and ρ I is the quotient map in the definition of the smash product X I .In addition, let Note that the smash products W J,J ′ I and X J ∧ X J ′ have the same factors, but in a different order arising from the natural shuffles.Let (29) be the natural homeomorphism given by a shuffle.Define the partial diagonal be the composition of Θ J,J ′ I and ψ J,J ′

I
. There is a commutative diagram is the reduced diagonal map of X [m] .Let k denote a commutative ring with a unit.For any J ⊆ [m], there is a homomorphism of rings given by the reduced cross product × (see [20, p. 223]): In particular, this ring homomorphism becomes a ring isomorphism if all (possibly except one) H * (X j ; k) are free k-modules (see [20,Theorem 3.21]).Lemma 3.1.For any φ j ∈ H * (X j ; k), j ∈ J and any φ Proof.The above formula follows easily from the definition of ∆ J,J ′ I .Note that the shuffle Θ J,J ′ I (see (29)) sorts all the cohomology classes {φ j } j∈J and {φ ′ j } j∈J ′ in order without introducing any ± sign.This is because for any space X and Y , So when Θ J,J ′ I transposes the space factors, the cohomology classes in the reduced cross product are transposed accordingly.
The following lemma will be useful for our proof of Theorem 1.4 later.Lemma 3.2.Let X be a CW-complex and A, B be two subcomplexes of X.The relative cup product which can be factored as where ∆ X : X → X × X is the diagonal map and φ × φ ′ is the reduced cross product of φ and φ ′ .
Proof.This can be verified directly from the following diagram when A, B are nonempty.
where the lower × −→ is the reduced cross product on H * (X/A) ⊗ H * (X/B).
Another useful fact is when X i is the suspension of some space, the reduced diagonal ∆ i : X i → X i ∧ X i is null-homotopic (see [4]).So we have the following lemma.
Lemma 3.3.If for some j ∈ J ∩ J ′ , X j is a suspension space, then the partial diagonal ∆ J,J ′ I : X I → X J ∧ X J ′ is null-homotopic, where I = J ∪ J ′ .Now we are ready to give a proof of Theorem 1.4.Our argument is parallel to the argument used in the proof of [4,Theorem 1.4].
Proof of Theorem 1.4.
For brevity, we will use the following notation in the proof.
Considering the partial diagonals (30) for , we obtain a map for any J, J ′ ⊆ [m] and a commutative diagram: is the reduced diagonal map of Q + × (D 2 ) [m] .By restricting the above diagram to colim(D + ), we obtain a commutative diagram for ∀J, J ′ ⊆ [m]: (32) colim(D + ) is the reduced cross product of u and v.This defines a ring structure on J⊆[m] H * colim D J + .The commutativity of diagram (32) implies , where ∪ is the cup product for colim(D + ).
By (23), the direct sum of Π * J induces an additive isomorphism ( 34) Then since Π * J : J is a ring isomorphism.Then by the proof of Theorem 1.1, this induces a ring isomorphism ( 35) Finally, let us show how to define a ring isomorphism from (R * Q , ⋒) to the cohomology ring ).Then for any subset For each J ⊆ [m], there is a canonical linear isomorphism (see [20, p. 223]): By Lemma 3.2, there is natural ring structure on R * Q , denoted by ⋒, that is induced from the product ⋒ on R * Q (see (57)).We have a commutative diagram is null-homotopic.So by (33), ⊛ is trivial in this case which corresponds to the definition of ⋒ on R * Q .
• When J ∩ J ′ = ∅, suppose in (35), we have elements Then Lemma 3.1 and Lemma 3.2 imply that So we have a commutative diagram below, which implies that the product ⋒ on R * Q corresponds to the product ⊛ in (35) in this case. (37) Combining the above arguments, we obtain isomorphisms of rings: It follows that there is a ring isomorphism from R * Q , ⋒ to H * (Z Q ) up to a sign.
Note that the above ring isomorphism is not degree-preserving.But by the diagram in (37), we can make this ring isomorphism degree-preserving by shifting the degrees of all the elements in H * (Q, F J ) up by |J| for every J ⊆ [m].The theorem is proved.

Polyhedral product over a nice manifold with corners
Let Q be a nice manifold with corners whose facets are where X j and A j are CW-complexes with a basepoint a j ∈ A j ⊆ X j .
For any face f of Q, define If (X, A) = {(X j , A j , a j ) = (X, A, a 0 )} m j=1 , we also denote (X, A) Q by (X, A) Q .We call (X, A) Q the polyhedral product of (X, A) over Q.Note that in general, the homeomorphism type of (X, A) Q depends on the ordering of the facets of Q and the ordering of the X j 's.We consider (X, A) Q as an analogue of polyhedral products over a simplicial complex (see [8]).
In the rest of this section, we assume that each of X j and A j in (X, A) is either connected or is a disjoint union of a connected CW-complex with its basepoint.Then we can study the stable decomposition and cohomology ring of (X, A) Q in the same way as we do for Z Q .
where q (X,A) 0 is the basepoint defined by q (X,A) 0 with basepoint q (X,A) 0 .
Let D (X,A)+ : P Q → CW * be the diagram of based CW-complexes where .
By Theorem 2.10, we can prove the following lemma parallel to Lemma 2.15.
Lemma 4.1.For any (f, L) ∈ P Q , there is a natural homeomorphism which commutes with taking the colimit: Remark 4.6.If any combination of Q/F J and A j 's satisfies the strong smash form of the Künneth formula as defined in [3, p. 1647] over a coefficient ring k, i.e. the natural map we can write the cohomology ring structure of (X, A) Q with k-coefficients more explicitly via the formula in Lemma 3.1.
In the following, we demonstrate the product ⊛ for (D, S) Q where Here D n+1 is the unit ball in R n+1 and S n = ∂D n+1 .
In particular, if (D, S) = D n j +1 , S n j , a j = D n+1 , S n , a 0 m j=1 , we also write We define a graded ring structure ⋒ (D,S) on R * Q according to (D, S) as follows.
and there exists n j ≥ 1 for some j ∈ J ∩ J ′ , then • If J ∩ J ′ = ∅ and there exists n j ≥ 1 for some j ∈ J ∩ J ′ , then We have the following theorem which generalizes Theorem 1.1 and Theorem 1.4.Theorem 4.8.Let Q be a nice manifold with corners with facets Then for any (D, S) = D n j +1 , S n j , a j m j=1 , (a) There is a homotopy equivalence (b) There is a ring isomorphism (up to a sign) from (R * Q , ⋒ (D,S) ) to the integral cohomology ring of (D, S) Q .Moreover, we can make this ring isomorphism degree-preserving by shifting the degrees of the elements in Proof.For brevity, we use the following notation in our proof.
Statement (a) follows from Theorem 4.4 and the simple fact that: For statement (b), note that by Theorem 4.5 we have a ring isomorphism (43) For any 1 ≤ j ≤ m, let ι n j denote a generator of H n j (S n j ).Let S n j ) be a generator.
(i) Assume J ∩J ′ = ∅ and there exists n j ≥ 1 for some j ∈ J ∩J ′ .Then since S n j is a suspension space, the map ∆ J,J ′ J∪J ′ ,Q + in (41) is null-homotopic.This implies that the product ⊛ in (43) is trivial which corresponds to the definition of ⋒ (D,S) on R * Q in this case.(ii) Assume J ∩ J ′ = ∅ but n j = 0 for all j ∈ J ∩ J ′ .Let So the condition on J and J ′ is equivalent to J ∩ J ′ ⊆ J 0 which implies (44) Since X ∧ S 0 ∼ = X for any based space X, we have for any J ⊆ [m]: By Lemma 3.1 and Lemma 3.2, we can derive an explicit formula for the product ⊛ in (43) as follows.For any elements .
So we have a commutative diagram parallel to diagram (37) below This implies that the product ⋒ (D,S) on R * Q corresponds to the product ⊛ in (43) in this case.(iii) When J ∩ J ′ = ∅, the proof of the correspondence between the product ⋒ (D,S) on R * Q and the product ⊛ in (43) is the same as case (ii).The above discussion implies that there is an isomorphism of rings: ) is isomorphic (up to a sign) to the integral cohomology ring H * (D, S) Q .Moreover, according to the above diagram, we can make the ring isomorphism between (R * Q , ⋒ (D,S) ) and H * (D, S) Q degreepreserving by shifting the degrees of all the elements in H * (Q, F J ) up by N J\J 0 for every J ⊆ [m].The theorem is proved.Remark 4.9.S 0 is not a suspension of any space and the reduced diagonal map ∆ S 0 = id S 0 : S 0 → S 0 ∧ S 0 ∼ = S 0 is not null-homotopic.This is the essential reason why for a general (D, S), the cohomology ring of (D, S) Q is more subtle than that of Z Q .
A very special case of Theorem 4.8 is (D 1 , S 0 ) Q = RZ Q where the product ⋒ (D 1 ,S 0 ) on R * Q is exactly the relative cup product for all J, J ′ ⊆ [m].
When J = ∅, we have So by Theorem 4.2, we have homotopy equivalences: By Definition 2.13, we have Then since Σ(Q ∪ q 0 ) ≃ S 1 ∨ Σ(Q), the theorem is proved.
The cohomology ring structure of (X, A) Q can be computed by Theorem 4.3.In particular, if any combination of F ∩J and X j 's satisfies the strong smash form of the Künneth formula over a coefficient ring k, we can give an explicit description of the cohomology ring of (X, A) Q with k-coefficients.Indeed, by Theorem 4.3 and Theorem 4.12 we obtain an isomorphism of rings (49) where the product ⊛ on the left-hand side is defined by (42) via the partial diagonal maps.We will do some computation of this kind in the next section to describe the equivariant cohomology ring of the moment-angle manifold Z Q .
5. Equivariant cohomology ring of Z Q and RZ Q Let Q be a nice manifold with corners whose facets are Since there is a canonical action of (S 1 ) m on Z Q (see ( 2)), it is a natural problem to compute the equivariant cohomology ring of Z Q with respect to this action.
For a simple polytope P , it is shown in Davis-Januszkiewicz [17] that the equivariant cohomology of Z P with integral coefficients is isomorphic to the face ring (or Stanley-Reisner ring) Z[P ] of P defined by where I P is the ideal generated by all square-free monomials We can also think of Z[P ] as the face ring of ∂P * where P * is the dual simplicial polytope of P (see [9,Ch. 3]).
For brevity, let T m = (S 1 ) m .By definition, the equivariant cohomology of Z Q , denoted by where (e, x) ∼ (eg, g −1 x) for any e ∈ ET m , x ∈ Z Q and g ∈ T m .Here we let Associated to the Borel construction, there is a canonical fiber bundle where BT m = (BS 1 ) m = (S ∞ /S 1 ) m = (CP ∞ ) m is the classifying space of T m .By Lemma 2.3, Z Q is equivariantly homeomorphic to (D 2 , S 1 ) Q .So computing the equivariant cohomology of Z Q is equivalent to computing that for (D 2 , S 1 ) Q .
By the colimit construction of (D 2 , S 1 ) Q in ( 16) and our notation for polyhedral products (38), the Borel construction Then by the homotopy equivalence of the pairs we can derive from Theorem 2.16 that there is a homotopy equivalence We call (CP ∞ , * ) Q the Davis-Januszkiewicz space of Q, denoted by DJ (Q).So the equivariant cohomology ring of Z Q is isomorphic to the ordinary cohomology ring of DJ (Q).
Similarly, we can prove that the Borel construction of RZ Q with respect to the canonical By the proof of Theorem 4.12 and the fact that H * (CP ∞ ) is torsion free, we can deduce from (48) that From the canonical fiber bundle associated to the Borel construction in (51), we have a natural H * (BT m )-module structure on H * T m (Z Q ).By the identification we can write the H * (BT m )-module structure on H * T m (Z Q ) as: for each 1 ≤ i ≤ m, (52) F ∩J is either empty or a face of P and hence acyclic.So we can write the topological face ring of P as where for any f According to the linear basis of the face ring Z[P ] in (50), we can easily check that Z P is isomorphic to Z[P ].
Theorem 5.3.Let Q be a nice manifold with corners with m facets.If a subtorus H ⊆ T m = (S 1 ) m acts freely on Z Q through the canonical action, the equivariant cohomology ring with Z-coefficients of the quotient space Z Q /H with respect to the induced action of T m /H is isomorphic to the topological face ring Z Q of Q.
So the equivariant cohomology ring of Z Q /H is isomorphic to the equivariant cohomology ring of Z Q .Then the theorem follows from Theorem 1.7.
In Theorem 5.3, the group homomorphism which, along with the maps in (53), induce the diagram We can describe the natural H * B T m /H -module structure of the integral equivariant cohomology ring of Z Q /H as follows.The inclusion H ֒→ T m induces a monomorphism ϕ H : Z m−k → Z m whose image is a direct summand in Z m .This determines an integer m × (m − k) matrix S = (s ij ) if we choose a basis for each of Z m−k and Z m .Then since the image of ϕ H is a direct summand in Z m , there is an integer k it follows from the diagram (54) that the natural H * B T m /H -module structure of the integral equivariant cohomology ring of Z Q /H is determined by the formula in (52) along with the map H * B T m /H → H * (BT m ) given by: The above formula is parallel to the formula given in [9, Theorem 7.37] (where Q is a simple polytope).
Remark 5.4.If a subtorus H ⊆ T m of dimension m − dim(Q) acts freely on Z Q through the canonical action, the quotient space Z Q /H with the induced action of T m /H ∼ = T dim(Q) can be considered as a generalization of quasitoric manifold over a simple polytope defined by Davis and Januszkiewicz [17].
The following is an application of Theorem 5.3 to locally standard torus actions on closed manifolds.Recall that an action of T n on a closed 2n-manifold M 2n is called locally standard (see [17, § 1]) if every point in M 2n has a T n -invariant neighborhood that is weakly equivariantly diffeomorphic an open subset of C n invariant under the standard T n -action: Corollary 5.5.Let M 2n be a closed smooth 2n-manifold with a smooth locally standard T n -action and the free part of the action is a trivial T n -bundle.Then the integral equivariant cohomology ring H * T n (M 2n ) of M 2n is isomorphic to the topological face ring Z M 2n /T n .
Proof.The orbit space Q = M 2n /T n is a smooth nice manifold with corners since the T n -action is locally standard and smooth.Then Q is triangulable (by [25]) and hence all our theorems can be applied to Q.In addition, using the characteristic function argument in Davis-Januszkiewicz [17] (also see [27, § 4.2] or [35]), we can prove that M 2n is a free quotient space of Z Q by a canonical action of some torus.Then this corollary follows from Theorem 5.3.
Remark 5.6.The equivariant cohomology ring H * T n (M 2n ) in the above corollary was also computed by Ayzenberg-Masuda-Park-Zeng [2, Proposition 5.2] under an extra assumption that all the proper faces of M 2n /T n are acyclic.We leave it as an exercise for the reader to check that the formula for

Generalizations
Let Q be a nice manifold with corners with facets Observe that neither in the construction of Z Q nor in the proof of Theorem 1.1 and Theorem 1.4 do we really use the connectedness of each facet F j .So we have the following generalization of Z Q . Let Moreover, we require J to satisfy the following condition in our discussion: From Q and the partition J , we can construct the following manifold.
Let {e 1 , • • • , e k } be a unimodular basis of Z k .Let µ : F (Q) → Z k be the map which sends all the facets in F J i to e i for every where (x, g) ∼ (x ′ , g ′ ) if and only if x = x ′ and g −1 g ′ ∈ T µ x where T µ x is the subtorus of (S 1 ) k = R k /Z k determined by the linear subspace of R k spanned by the set {µ(F j ) | x ∈ F j }.There is a canonical action of (S 1 ) k on Z Q,J defined by: (56) Note that here {F J i } play the role of facets {F j } in the definition of Z Q .But F J i may not be connected.Using the term defined in Davis [14], the decomposition of ∂Q into {F J i } is called a panel structure on Q and each F J i is called a panel.Remark 6.1.For a general partition J of [m], it is possible that F j ∩F j ′ = ∅ for some j, j ′ ∈ J i .Although the definition of Z Q,J still makes sense in the general setting, the orbit space of the (S 1 ) k -action on Z Q,J may not be Q (as a manifold with corners).It would be Q with some corners smoothed.But for a general partition of [m], one can always reduce to the case where the condition (55) is satisfied by smoothing the corners of the orbit space.
Proof.We can generalize the rim-cubicalization of Q in Section 2.1 as follows.For any face f of Q, let By the same argument as in the proof of Lemma 2.1, we can show that Q J with faces f J is homeomorphic to Q as a manifold with corners.The partition J of the facets of Q naturally induces a partition of the corresponding facets of Q J , also denoted by J .So we have Z Q J ,J ∼ = Z Q,J .For any face f of Q, let There is a canonical (S 1 ) k -action on (D 2 , S 1 ) Q J induced from the canonical (S 1 ) k -action on Q × (D 2 ) k .And parallel to Lemma 2.3, we can prove that there is an equivariant homeomorphism from (D 2 , S 1 ) Q J to Z Q J ,J ∼ = Z Q,J .
For any subset L ⊆ I J f , let We can easily translate the proof of Theorem 1.1 to obtain the desired stable decomposition of Z Q,J ∼ = (D 2 , S 1 ) Q J by the following correspondence of symbols.The proof of Theorem 1.1 The proof of Theorem 6.2 (f,L) J Remark 6.3.Theorem 6.2 is an analogue of [36,Theorem 1.3].
To describe the cohomology ring of Z Q,J , let There is a graded ring structure ⋒ J on R * Q,J defined as follows.
To describe the equivariant cohomology ring of Z Q,J , let where the product on k J Q is defined in the same way as k Q in Definition 1.6.
The following theorem generalizes Theorem 1.4 and Theorem 1.7.The proof is omitted since it is completely parallel to the proof of these two theorems.• There is a ring isomorphism (up to a sign) from (R * Q,J , ⋒ J ) to the integral cohomology ring of Z Q,J .Moreover, we can make this ring isomorphism degree-preserving by shifting the degrees of all the elements in H * (Q, F ω ) up by |ω| for every ω ⊆ [k].
• There is a graded ring isomorphism from the equivariant cohomology ring of Z Q,J with integral coefficients to Z J Q by choosing deg(x i ) = 2 for all 1 ≤ i ≤ k.
By combining the constructions in Theorem 4.4 and Theorem 6.2, we have the following definitions which provide the most general setting for our study.
Let J = {J 1 , • • • , J k } be a partition of [m] = {1, • • • , m} and let where X i and A i are CW-complexes with a basepoint a i ∈ A i ⊆ X i .
For any face f of Q, let The following theorem generalizes Theorem 4.4 and Theorem 4.5.
Theorem 6.5.Let Q be a nice manifold with corners with facets F 1 , • • • , F m .Let (X, A) = {(X i , A i , a i )} k i=1 where each X i is contractible and each A i is either connected or is a disjoint union of a connected CW-complex with its basepoint.Then for any partition J = {J 1 , • • • , J k } of [m], there is a homotopy equivalence In addition, there is a ring isomorphism where ⊛ is defined in the same way as in (42).
In particular, for (D, S) = D n i +1 , S n i , a i k i=1 , we can describe the integral cohomology ring of (D, S) Q J explicitly as follows.Define a graded ring structure ⋒ (D,S) J on R * Q,J according to (D, S) by: • If ω ∩ ω ′ = ∅ or ω ∩ ω ′ = ∅ but n i = 0 for all i ∈ ω ∩ ω ′ , then • If ω ∩ ω ′ = ∅ and there exists n i ≥ 1 for some i ∈ ω ∩ ω ′ , then Σ 1+ i∈ω n i Q/F ω .

Theorem 6 . 6 .
Let Q be a nice manifold with corners with facetsF 1 , • • • , F m .For any partition J = {J 1 , • • • , J k } of [m] and (D, S) = D n i +1 , S n i , a i k i=1 , there is a homotopy equivalence Σ (D, S) Q J ≃ ω⊆[k] manifold with corners.The lemma is proved.