Topology of polyhedral products

Topology of polyhedral products over simplicial multiwedges

Ivan Limonchenko School of Mathematical Sciences, Fudan University, Handan Road 220, Yangpu, Shanghai, 200433, P.R. China ilimonchenko@fudan.edu.cn To the memory of Professor William S. Massey (1920-2017)
Abstract.

We prove that certain conditions on multigraded Betti numbers of a simplicial complex imply existence of a higher Massey product which contains a unique element (a strictly defined product) in the cohomology ring of a moment-angle-complex . Using the simplicial multiwedge construction, we find a family of polyhedral products being smooth closed manifolds, such that for any there exists an -connected manifold with a nontrivial strictly defined -fold Massey product in . Finally, we establish a combinatorial criterion for a simple graph to produce a rationally formal moment-angle manifold over a graph-associahedron and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.

Key words and phrases:
Polyhedral product, moment-angle manifold, simplicial multiwedge, Stanley–Reisner ring, Massey product, graph-associahedron
2010 Mathematics Subject Classification:
Primary 13F55, 55S30, Secondary 52B11
The author was supported by the General Financial Grant from the China Postdoctoral Science Foundation, Grant No. 2016M601486.

1. Introduction

A Massey product is a multivalued indefinite higher operation in cohomology of a space which generalizes the notion of a cup product. Since 1957, when the triple Massey product in the cohomology ring of a topological space was introduced and applied [35] in order to prove the Jacobi identity for Whitehead products in , the higher order (ordinary) Massey operation, introduced in [36], and its generalization, matric Massey product [37], found numerous applications in geometry, topology, group theory, and other areas of research.

Among the most remarkable of these applications, we can mention the one which belongs to rational homotopy theory [45]: a nontrivial higher Massey product in (singular) cohomology ring is an obstruction to the (rational) formality of . This allows one to prove that certain complex manifolds are non-Kähler due to the classical result of [18] on formality of Kähler manifolds, as well as to construct nonformal manifolds, which arise in symplectic geometry [2], the theory of subspace arrangements [20] and toric topology [6].

Since the pioneering work [17] appeared, toric topology has become a branch of geometry and topology that provides algebraic and equivariant topology, symplectic geometry, combinatorial commutative algebra, enumerative combinatorics and polytope theory with a class of new objects, such as small covers and quasitoric manifolds [17], (generalized) moment-angle-complexes  [12] and polyhedral products  [3], on which the general theory can be worked out explicitly, and a number of remarkable properties of toric spaces with applications in other areas of research can be obtained.

CW complexes with a compact torus action over a simplicial complex on vertices and moment-angle manifolds of simple polytopes with facets (being equivariantly homeomorphic to when is a boundary of the dual simplicial polytope ) have become the key object of study in toric topology, and there have already been shown various connections of these spaces with objects arising in different areas of geometry and topology. Most of these developments have been recently summarized in the fundamental monograph [11].

In particular, it was proved in [12] that a moment-angle-complex is a deformation retract of the complement of the coordinate subspace arrangement in corresponding to , thus being homotopy equivalent to such a complement. Moreover, a moment-angle manifold of an -dimensional simple polytope is a smooth -dimensional 2-connected manifold which is embedded into with a trivial normal bundle as a nondegenerate intersection of Hermitian quadrics.

A -action on is induced from the standard (coordinatewise) action of on , and the orbit space of this action is the convex simple polytope itself. Moreover, when the quasitoric manifold over exists (in particular, when has a Delzant realization, and one of the quasitoric manifolds over is then its nonsingular projective toric variety ), is a total space of a principal -bundle over and a composition of the latter bundle with a projection of onto the orbit space of -action coincides with the projection of onto the orbit space of the big torus action mentioned above.

In a series of more recent works, starting with [26, 27, 31, 29], the topological properties of moment-angle-complexes were related to the algebraic properties of their Stanley–Reisner algebras (over a field, or a ring with unit ). One of such connections is provided by the notion of a Golod ring, which appeared firstly in homological algebra of local rings in 1960s and acquired the following interpretation in toric topology (see Theorem 2.9): a face ring is called Golod (over ) if multiplication (cup product) and all higher Massey products in are vanishing. An up-to-date comprehensive survey on homotopy theory of polyhedral products with applications to the problem of finding conditions on that imply Golodness of (over ) can be found in [28].

It turned out that most of the classes of Golod complexes that arise in the way described above produce that is homotopy equivalent to a suspension space (or even to a wedge of spheres). Moreover, there have already been established several classes of (or ) for which (resp. ) is homeomorphic to a connected sum of products of spheres, thus also being formal spaces [7, 25, 29]. On the other hand, it was first shown in [6] that there exist polyhedral products that have nontrivial higher Massey products in their cohomology, thus being nonformal. Later on, several rational models for toric spaces, among which are moment-angle-complexes, Davis-Januszkiewicz spaces and quasitoric manifolds, were constructed [42, 20, 41] and, in particular, it was proved that all quasitoric manifolds are formal [41].

The simplicial multiwedge operation (or, -construction) was introduced in the framework of toric topology in [5] and was then used to prove that an important family of generalized moment-angle-complexes are all homeomorphic to (real) moment-angle-complexes, see Theorem 2.14. Therefore, a natural problem arises: to determine a class of (generalized) moment-angle-complexes that are (rationally) formal, see Problem 3.10, and to construct a nontrivial Massey product of an arbitrary large order with a possibly small indeterminacy in (rational) cohomology of generalized moment-angle-complexes. In this paper we are going to deal with these questions.

The structure of the paper is as follows. In Section 2 we give a survey of the main definitions, constructions and results concerning simplicial complexes, -construction and polyhedral products (in particular, multiplicative structure in the cohomology ring of moment-angle-complexes) that we need in the next section in order to state and prove our main results. In Section 3 we show that certain conditions on the induced subcomplexes in a simplicial complex , more precisely, vanishing of particular multigraded Betti numbers of imply existence of a strictly defined higher Massey product in (Theorem 3.3). We then apply this theorem and the simplicial multiwedge construction in order to find a family of generalized moment-angle-complexes being smooth closed manifolds, such that for any this family contains an -connected manifold with a nontrivial strictly defined -fold Massey product in (Theorem 3.6). It turns out that the underlying polytopes are Delzant, thus a nonformal toric space from our family is a total space of a principal toric bundle over a nonsingular projective toric variety. The latter one is a formal manifold and is equivariantly symplectomorphic to a compact connected symplectic manifold with an effective Hamiltonian action of a half-dimensional compact torus [19]. We end this chapter with a combinatorial criterion for a simple graph to produce a rationally formal moment-angle manifold over a graph-associahedron (Theorem 3.15) and give all the diffeomorphism types of those of them which are formal.

We are grateful to Martin Bendersky and Taras Panov for drawing our attention to studying higher order nontrivial Massey operations in cohomology of highly connected polyhedral products. We also thank Lukas Katthän for fruitful discussions on the combinatorial commutative algebra of face rings and Massey products in their Tor-algebras.

2. Moment-angle-complexes and simplicial multiwedges

We start with the following basic definition.

Definition 2.1.

An (abstract) simplicial complex on a vertex set is a subset of , such that:

  • The empty set belongs to ;

  • If and , then .

The elements of are called its simplices, and the maximal dimension of a simplex is the dimension of and is denoted by , where . Finally, for any vertex set a subset of which equals the intersection is obviously a simplicial complex itself which is called an induced subcomplex in and denoted by .

In what follows we assume that there are no ghost vertices in , that is, for every one has . Note that a simplicial complex is a poset, the natural ordering is by inclusion. Thus, (and any of its induced subcomplexes ) is defined by its maximal (w.r.t. inclusion) simplices and is equal to the dimension of one of them. If all maximal simplices of have the same dimension, then is called a pure simplicial complex.

Example 2.2.

In what follows we denote by an -dimensional convex simple polytope with facets , i.e. faces of codimension 1. In this paper we are interested only in its face poset structure (i.e. its combinatorial equivalence class), not in its particular embedding in the ambient Euclidean space . Consider the nerve of the (closed) covering of by all the facets . The resulting simplicial complex on the vertex set will be called the nerve complex of the simple polytope and denoted by . Note that is geometrically a boundary of the combinatorially dual simplicial polytope . It is obviously a pure simplicial complex and its dimension is equal to .

One can see easily that a simplicial complex can either be defined by all of its maximal simplices, or, alternatively, by all of its minimal non-faces, that is, elements , minimal w.r.t. inclusion. By definition above, the latter is equivalent to the property that but any of its proper subsets is a simplex in . We denote the set of minimal non-faces of by . Therefore, if and only if is a boundary of a simplex on the vertex set . Using this set, we can easily define the following combinatorial operation on the sets of simplicial complexes and (combinatorial) simple polytopes, which appeared in toric topology in the work of Bahri, Bendersky, Cohen and Gitler [5].

Construction 2.3.

Let be an -tuple of positive integers. Consider the following vertex set:

To define a simplicial multiwedge, or a -construction of , which is a simplicial complex on the vertex set , we say, that the set consists of the subsets of of the type

where . Note that if , then .

Example 2.4.

Suppose is a polytopal sphere on vertices, that is, a nerve complex of a simple -dimensional polytope with facets. Due to [5, Theorem 2.4] and [16, Proposition 2.2] its simplicial multiwedge is always a polytopal sphere and thus a nerve complex of a simple polytope , and so . If we denote by then it is easy to see, that and , therefore, remains the same after performing a -construction.

Note that when one gets the so called doubling of the simple polytope ; for the geometric description of the -construction on polytopes and its applications in toric topology see also [25].

In what follows we shall denote by a field of zero characteristic, or the ring of integers. Let be a graded polynomial algebra on variables, .

Definition 2.5.

The Stanley–Reisner ring, or the face ring of (over ) is the quotient ring

where is the ideal generated by square free monomials such that . The monomial ideal is called the Stanley–Reisner ideal of . Then has a natural structure of a -algebra and a module over the polynomial algebra via the quotient projection.

Note that if and only if . By a result of Bruns and Gubeladze [9] two simplicial complexes and are combinatorially equivalent if and only if their Stanley–Reisner algebras are isomorphic. Thus, is a complete combinatorial invariant of a simplicial complex .

Definition 2.6.

A simplicial complex is called flag if it coincides with the clique complex of its 1-dimensional skeleton , that is, for any subset of vertices of , which are pairwisely connected by edges in , its induced subcomplex . A simple polytope is called flag if its nerve complex is flag.

A simplicial complex is called q-connected () if for any subset of vertices of with , its induced subcomplex . Thus, any simplicial complex is 1-connected; is -connected implies is -connected for any ; is flag if and only if is 1-connected, but not 2-connected.

Remark.

(1) is flag if and only if for any one has , i.e. is generated by monomials of degree 4. (2) is -connected if and only if the minimal with has vertices.

Suppose is a set of topological pairs. A particular case of the following construction appeared firstly in the work of Buchstaber and Panov [12] and then was studied intensively and generalized in a series of more recent works, among which are [3, 26, 31].

Definition 2.7.

([3]) A polyhedral product over a simplicial complex on the vertex set is a topological space

where for , if , and , if .

Example 2.8.

Suppose and for all . Then the following spaces are particular cases of the polyhedral product construction .

  • The moment-angle-complex ;

  • The real moment-angle-complex ;

  • The Davis–Januszkiewicz space ;

  • A complement to the coordinate subspace arrangement in

    defined by a simplicial complex ;

  • A cubical subcomplex in which is PL homeomorphic to a cone over a barycentric subdivision of .

Remark.

The following properties of the polyhedral products defined above hold.

  • is homeomorphic to ;

  • If is -connected, then is a -connected CW complex. Moreover, if then is a 2-connected smooth closed -dimensional manifold for any simple -dimensional polytope with facets;

  • There is a -equivariant deformation retraction:

    in particular, and have the same homotopy type and ;

  • One has a commutative diagram

    where is an embedding of a cubical subcomplex, induced by the inclusion of pairs: , and the maps and are projections onto the orbit spaces of -action induced by the coordinatewise action of on the unitary complex polydisk in ;

  • One has a homotopy fibration

    where is the universal -bundle and the map is induced by the inclusion of the pairs . Moreover, its homotopy fiber and is homotopy equivalent to the Borel construction . Therefore, for the equivariant cohomology of we have:

    The details can be found in [11, Chapter 4, 8].

Buchstaber and Panov [12] analysed the homotopy fibration above and proved that the corresponding Eilenberg-Moore spectral sequence over rationals, which converges to , degenerates in the term. Moreover, together with Baskakov they proved that the following result on the cohomology algebra of holds with coefficients in any ring with unit .

Theorem 2.9 ([11, Theorem 4.5.4]).

Cohomology algebra of a moment-angle-complex is given by the isomorphisms

where bigrading and differential in the cohomology of the differential (bi)graded algebra are defined by

In the third row, (we drop from the notation in what follows) denotes the reduced simplicial cohomology of . The last isomorphism is the sum of isomorphisms

To determine the product of two cohomology classes and define a natural inclusion of sets and the canonical isomorphism of cochain modules:

Then, the product of and is given by:

The additive structure of the Tor-algebra which appears in the theorem above can be computed either using the Koszul minimal free resolution for viewed as a -module, or the Taylor free resolution for viewed as a -module. In general, the latter one is not minimal, however, it is sometimes more useful in combinatorial proofs. In fact, the -module structure of is determined by all the reduced cohomology groups of all the induced subcomplexes in (including and itself). More precisely, the following result of Hochster holds.

Theorem 2.10 ([30]).

For any simplicial complex on vertices:

The ranks of the bigraded components of the Tor-algebra

are called the bigraded (algebraic) Betti numbers of or , when is fixed. Note that by Theorem 2.9 they determine the (topological) Betti numbers of .

Consider the polytopal case . Let us denote by the union of the corresponding facets of , and by – the number of connected components of . It can be seen that the following equality arises due to Theorem 2.10 when :

In Chapter 3 we shall use a refinement of Theorem 2.9. Namely, if we denote by a graded algebra with the differential as in Buchstaber-Panov theorem, then one can see that also acquires multigrading and the next statement holds.

Theorem 2.11 ([11, Construction 3.2.8, Theorem 3.2.9]).

For any simplicial complex on vertices we have:

where and , if is not a -vector. Moreover,

Now consider the real case. By [10, Theorem 8.9] one has additive isomorphisms:

The multiplicative structure was given firstly by Li Cai [15] and for generalized moment-angle-complexes by Bahri, Bendersky, Cohen and Gitler [5], which is equivalent to the result of [15] in the case of a real moment-angle-complex. Consider the differential graded algebra which is a quotient algebra of a free graded algebra on variables , where , over the Stanley-Reisner ideal of in variables and the following commutation relations:

Theorem 2.12 ([15]).

The following graded ring isomorphism holds:

where and .

Moreover, if is a simplicial multiwedge construction over then one has the next result on cohomology algebra of a generalized moment-angle-complex , see [15, Theorem 5.1] and [5, Proposition 6.2] (for even ):

Theorem 2.13.

The following isomorphism of graded algebras holds:

Note, that both the algebras and are finitely generated -modules, on the contrary to the case of the Koszul algebra which has countably many additive generators. By the Baskakov and Cai explicit constructions of multiplication in and , respectively, both and are quasi-isomorphic to the cellular cochain algebras of and , respectively.

It can be seen that in the case Li Cai’s algebra is isomorphic to the algebra , constructed by Baskakov, Buchstaber and Panov. Moreover, the real moment-angle-complex is homeomorphic to the moment-angle-complex of the original simplicial complex and, more generally, if then by [5]:

Theorem 2.14.

The generalized moment-angle-complex is homeomorphic to the real moment-angle-complex .

3. Rational formality and higher Massey products

In this section we are going to introduce our main results concerning rational formality and (ordinary) higher Massey product structure in cohomology for certain polyhedral products. For more details on rational homotopy theory we refer to the monographs [22, 23]. The applications of rational homotopy theory in toric topology can be found in [4, 20, 24].

We start with a definition of a higher Massey product in cohomology of a differential graded algebra. First, we need a notion of a defining system for an ordered set of cohomology classes. Our presentation follows [32, 10] in which all the algebraic and functorial properties of the Massey operation, necessary in what follows, can be found.

Suppose is a dga, and for . Then a defining system for is a -matrix , such that the following conditions hold:

  • , if ,

  • ,

  • for some , where and is a -matrix with ’1’ in the position and all other entries being zero.

A straightforward calculation shows that and , . Thus, is a cocycle for any defining system , and its cohomology class is defined.

Definition 3.1.

A Massey -product is said to be defined, if there exists a defining system for it. If so, this Massey product consists of all , where is a defining system. The product is called

  • trivial (or, vanishing), if for some ;

  • decomposable, if for some ;

  • strictly defined, if for some (and thus, for any) .

Remark.

Due to [32, Theorem 3], the set of cohomology classes depends on the cohomology classes rather than the particular representatives . Therefore, if for some and is defined, then , thus the -fold Massey product is trivial.

Example 3.2.

Let us give examples for the 2-, 3- and 4-fold defined Massey product.

  • Suppose .
    If is defined, then we have:

    Thus, in cohomology of a manifold our definition gives a usual cup product (up to sign).

  • Suppose .
    If is defined, then we have:

  • Suppose .
    If is defined, then we have:

Therefore, from Definition 3.1 and Example 3.2 one can see easily that for a higher Massey product to be defined it is necessary, that all the Massey products of consecutive elements in the -tuple are defined and vanish simultaneously. If all the (consecutive) subproducts vanish, but not simultaneously, then the whole product may even not exist, see [39, Example I].

Remark.

In [37] matric Massey products were defined and studied, and it was also proved, that the differentials in the Eilenberg-Moore spectral sequence of the path loop fibration for any path connected simply connected space are completely determined by higher Massey products. However, the converse statement is not true, see [39, Example II].

It follows immediately from Definition 3.1, that a strictly defined Massey product is nontrivial if and only if there exists a defining system such that and indecomposable if and only if there exists a defining system such that .

Our goal now is to determine the conditions sufficient for a higher Massey product in cohomology of a moment-angle-complex over a simplicial complex to be strictly defined.

Let us consider a set of induced subcomplexes on pairwisely disjoint subsets of vertices for and their cohomology classes , where by Theorem 2.9. If an -fold Massey product () of consecutive classes for is defined, then by Theorem 2.9 and Theorem 2.11

where and .

Remark.

If all are odd numbers, then for a defining system is an even number for all . It follows, that all the elements in have odd degrees in .

Theorem 3.3.

Suppose, in the notation above, and

  • ;

  • Any of the following two conditions holds:

    • The -fold Massey product is defined, or

    • .

Then the -fold Massey product is strictly defined.

Proof.

Let us use induction on the order of the Massey product. For the base case the condition (2b) implies that both the 2-fold products and vanish simultaneously and, by Example 3.2 (2), we get that the triple Massey product is defined. Moreover, by the condition (1) and Theorem 2.11, the indeterminacy in is trivial, thus this triple Massey product is strictly defined.

Now assume that the statement holds for Massey products of order less than . We first prove that conditions (1) and (2b) imply is defined. To make the induction step here note, that in Definition 3.1 all the Massey products of consecutive elements of orders for are defined by the inductive assumption and are not only trivial (by (2b) and the multiplicative structure in given by Theorem 2.9) but, moreover, strictly defined, and thus contain only zero elements. Therefore, a defining system exists and a cocycle from Definition 3.1 (3) represents a cohomology class in .

Now we shall prove that this Massey product contains a unique element, that is, we need to prove that for any two defining systems and . By the inductive assumption we suppose that this statement is true for defined higher Massey products of order less than . We divide the induction step into 2 parts, in both we also proceed by induction.
I. Let us determine a sequence of defining systems for such that the following conditions hold:

  • ;

  • , if ;

  • , for all .

Note that (2) implies that for all , and . We use induction on here to determine the defining systems . As by (1), we need to prove the induction step assuming that is already defined.

Consider a cochain

for . By Definition 3.1 one has: by the property (2) of above. Thus, is a cocycle and, therefore, by Theorem 2.11, for any , by the condition (1) of the statement we are to prove (as here by (3) above).
II. The construction of the defining system will be completed if we determine a sequence of defining systems () for , such that:

  • ;

  • , if , and

    when ;

  • , for all .

Note that condition (2’)(**) for implies that which is equal to by (2) above, and thus, we can define and the proof will be finished by induction.

Therefore, to complete the proof it suffices to determine a sequence of defining systems . We shall do it by induction on . The base case is done by (1’) above. Now assume, we already constructed and let us determine the defining system .

If , then we can set , see (2’)(*). Similarly, we can also set if , see (2’)(**). Now suppose and let us determine a set of cochains such that

by induction on .

By (**) which is equal to due to (*). Therefore, we can set . Suppose by inductive assumption, that the cochains are already defined for . Then (***) implies that , where the last equality holds because , since when , and when , and one gets the following equality for :