Quantitative aspects of acyclicity

Quantitative aspects of acyclicity

Abstract.

The Cheeger constant is a measure of the edge expansion of a graph, and as such plays a key role in combinatorics and theoretical computer science. In recent years there is an interest in -dimensional versions of the Cheeger constant that likewise provide quantitative measure of cohomological acyclicity of a complex in dimension . In this paper we study several aspects of the higher Cheeger constants. Our results include methods for bounding the cosystolic norm of -cochains and the -th Cheeger constants, with applications to the expansion of pseudomanifolds, Coxeter complexes and homogenous geometric lattices. We revisit a theorem of Gromov on the expansion of a product of a complex with a simplex, and provide an elementary derivation of the expansion in a hypercube. We prove a lower bound on the maximal cosystole in a complex and an upper bound on the expansion of bounded degree complexes, and give an essentially sharp estimate for the cosystolic norm of the Paley cochains. Finally, we discuss a non-abelian version of the -dimensional expansion of a simplex, with an application to a question of Babson on bounded quotients of the fundamental group of a random -complex.

Key words and phrases:
simplicial complexes, cohomology, high dimensional expansion

1. Introduction

The Cheeger constant is a parameter that quantifies the edge expansion of a graph, and as such plays a key role in combinatorics and theoretical computer science (see, e.g., [HLW06, Lu2]). The -dimensional version of the graphical Cheeger constant, called ”coboundary expansion”, came up independently in the work of Linial, Meshulam and Wallach [LiM, MW] on homological connectivity of random complexes and in Gromov’s remarkable work [Gr10] on the topological overlap property; see also the paper by Dotterer and Kahle [DK]. Roughly speaking, the -th coboundary expansion of a polyhedral complex is a measure of the minimal distance of from a complex that satisfies . Likewise, is a measure of the minimal distance of from a complex that satisfies . We proceed with the formal definitions.

1.1. Chains and cochains.

Let be a finite polyhedral complex. In this paper we shall mostly work with coefficients. Accordingly, let denote the space of -chains of over and let denote the space of -valued -cochains of . Let and denote the usual -boundary and -coboundary operators.

Let denote the usual evaluation of -cochains on -chains. Note that

(1.1)

whenever and .

For each non-negative , let denote the -th skeleton of , which is itself a finite polyhedral complex. Furthermore, let denote the set of all -cells of . The degree of a face is . In particular, if is a graph, then is the usual degree of a vertex .

When is a simplicial complex and , we write for the corresponding element in . As we are working over , the order of the ’s does not matter. We will use the convention that if for some .

For any set of cells and any cochain , we let denote the restriction cochain, i.e., the cochain that is equal to on the set  and is equal to  otherwise. Furthermore, we let denote the cochain which is equal to on the set and is otherwise.

In our convention the set is empty, and accordingly , forcing and . This is the so-called non-reduced setting. In the reduced setting we let be the set containing a single element, denoted . This is the so-called empty simplex. For convenience we also set , for . Accordingly, the reduced boundary and coboundary operators coincide with the non-reduced ones except for the following cases:

  • , for all ;

  • .

For a subcomplex let denote the space of relative -chains, with its induced -boundary map . Identifying with the subspace of consisting of all -cochains whose support is contained in , let denote the space of relative -cochains, with its induced -coboundary map . Let and denote the spaces of relative -boundaries and relative -coboundaries.

1.2. Homology expansion

Let be a finite polyhedral complex and let . Write where the ’s are in .

Definition 1.1.

The norm of is

The systolic norm of is

The chain is called a -systole if . A systolic form of is any , such that .

Definition 1.2.

The boundary expansion of a -chain is

The -th homological Cheeger constant of is

The definition of expansion can be extended to the relative case as follows. Let be a subcomplex of . A relative chain in has a unique representative . The norm of is then defined by . The notions of the relative systolic norm, relative expansion and relative homological Cheeger constants are then defined as in the absolute case.

1.3. Cohomology expansion

Let be a -cochain of .

Definition 1.3.

The norm of is

The cosystolic norm of is

A cochain is a cosystole if . A cosystolic form of is any , such that .

Definition 1.4.

The coboundary expansion of a -cochain is

The -th Cheeger constant of is

Let us now turn to the relative case. The -th coboundary expansion of -cochain is again , and the -th Cheeger constant of the pair is:

Clearly, the non-relative norm, (co)systolic norm and (co)boundary expansion are obtained by taking to be the void complex (see [Ko07]), e.g., , .

Remark 1.5.

(1) Let and . Then if and only if , and if and only if .

(2) Note that if and only if and if and only if . One can therefore view the expansion constants and as refining the notion of acyclicity, trying to catch phenomena which the regular (co)homology does not. A possible analogy could be Whitehead torsion refining the notion of homotopy equivalence.

Here we study several aspects of the higher dimensional Cheeger constants. The plan of the paper is as follows. In Section 2 we discuss some general tools that include:

  • A combinatorial lower bound on the cosystolic norm (Theorem 2.2).

  • A lower bound on Cheeger constant using chain homotopy (Theorem 2.5).

  • An Alexander type duality between the homological and cohomological Cheeger constants (Theorem 2.8).

Section 3 is concerned with cosystoles and expansion of some concrete complexes and includes:

  • A determination of the codimension one cosystoles and Cheeger constants of certain -pseudomanifolds (Theorem 3.3), and of Coxeter complexes (Theorem 3.5).

  • A lower bound on the Cheeger constants of a homogenous geometric lattices (Theorem 3.8).

In Section 4 we revisit results of Gromov on expansion of products, including

  • An elementary derivation of Gromov’s computation of the expansion of the hypercube (Theorem 4.3).

  • A detailed proof of a theorem of Gromov on the expansion of a product of a complex with a simplex (Theorem 4.4).

In Section 5 we consider some extremal problems on cosystoles and expansion. These include

  • A lower bound on the maximal cosystole in a complex (Theorem 5.1).

  • An upper bound on the expansion of bounded degree complexes (Theorem 5.3).

  • A nearly sharp estimate on the cosystolic norm of the Paley cochains (Theorem 5.5).

In Section 6 we discuss

  • The non-abelian -dimensional expansion of a simplex (Proposition 6.7).

  • An application to a problem of Babson on bounded quotients of the fundamental group of a random -complex (Theorem 6.3).

We conclude in Section 7 with some comments and open problems.

2. General Tools

2.1. Detecting large cosystolic norm using cycles

A question that frequently arises in specific examples, as well as in theoretical context, is that of determining whether given cochain is a cosystole. Using the definition directly is impractical at best, since it would involve going through all possible coboundaries and trying to see whether adding one would reduce the norm of the cochain at hand.

The new idea which we introduce here is to use cycles to detect in an indirect way that that our cocycle has a large cosystolic norm. For this, we recall that coboundaries evaluate trivially on cycles, see (1.1); therefore, the evaluation of a cochain on a cycle does not change if we add a coboundary to that cochain. In particular, if a cochain evaluates nontrivially on a cycle, its support must intersect the support of that cycle, and that will not change if we add a coboundary. The intersection cells may vary, but the fact that the intersection is non-trivial will remain.

In its most basic form, our method is based on the fact that if we have a family of cycles with pairwise disjoint supports, and a cochain which evaluates nontrivially on each of these cycles, then the cosystolic norm of is at least . Let us now formalize these observations.

Definition 2.1.

Let be a family of finite sets. A subset is a piercing set of if for all . The minimal cardinality of a piercing set of , denoted by , is called the piercing number of .

Theorem 2.2 (The cycle detection theorem).

Let be a polyhedral complex, and let be a -cochain of . Let be a family of -cycles of , such that for all . Let .

(2.1)

Proof. Let . Then for any

In particular, . It follows that is a piercing set of and therefore . Since this is true for all , we get (2.1). ∎

Corollary 2.3.

Let be a polyhedral complex, and let be a -cochain of . Let be a family of -cycles of with pairwise disjoint supports, such that for all . Then .

Example: Let and let denote the -simplex on the vertex set where the ’s are disjoint of cardinality . Consider the collection of -simplices and let . The following fact was mentioned in [MW].

Claim 2.4.

Proof: For a -tuple let . Identify each with a copy of the cyclic group and let

Let . Clearly for . Furthermore

Corollary 2.3 therefore implies that . ∎

2.2. Lower bounds for expansion via cochain homotopy

Let be an -dimensional simplicial complex and let . Let be a finite probability space. Let be a family of -chains of and let be a family of -chains of , such that for all we have

(2.2)

where denotes the -th face of . For and , define as follows. For and let

(2.3)

A natural way to think about (2.3) and (2.2) is to say that the map is a chain homotopy between the trivial map and the identity map of , and that is its dual. It follows that satisfies

(2.4)

For we set

(2.5)

For and let . Let denote the expectation of a random variable on . The following result is an elaboration of an idea from [LMM].

Theorem 2.5.

Proof. Let and . By (2.4)

and hence . Taking expectation over we obtain

∎ For let

Specializing Theorem 2.5 to the case of uniform distribution , we obtain the following

Corollary 2.6.

2.3. Alexander duality and expansion

Let denote the -simplex on an -element vertex set and let be a simplicial subcomplex of . For a subset let . The Alexander dual of is the simplicial complex given by

Note, that . We also have , , .

Let be a subcomplex of . The combinatorial version of the relative Alexander duality is the following

Theorem 2.7 (Alexander Duality).

For

In fact, there is a chain complex isomorphism , for an arbitrary abelian group . The counterpart of Alexander duality for expansion is the following

Theorem 2.8.

For

Letting be the void simplicial complex in Proposition 2.8, we obtain the following corollary.

Corollary 2.9.

Let . Then:

Proof of Theorem 2.8. Define a linear map as follows. For a generator of and let

Note that is well-defined: If and then , thus . Moreover, if then for all , i.e., . It is straightforward to check that is an isomorphism and that it commutes with the differentials, i.e.,

(2.6)

Observe that if and , then . Therefore

(2.7)

Combining (2.6) and (2.7) it follows that

(2.8)

Next note that (2.6) implies that maps injectively onto . Therefore, if then by (2.7) and (2.8):

(2.9)

Theorem 2.8 now follows by minimizing (2.9) over all . ∎

3. Cosystoles and Expansion of Pseudomanifolds and Geometric Lattices

3.1. The -th Cheeger constant of an -pseudomanifold

Let be an -dimensional simplicial complex. The flip graph of , is the graph whose vertex set is - the set of all -simplices of , and whose edge set consists of all pairs such that .

Suppose now that is a triangulation of an -pseudomanifold, i.e., is a finite pure -dimensional simplicial complex such that any is contained in exactly two -simplices of  and such that is connected. For let be the subgraph of with edge set

Let denote the family of all subgraphs of such that

for any Eulerian subgraph of . We note the following properties of the family .

Claim 3.1.

Let . Then the following hold.

  1. The graph is a forest.

  2. If vertices are in the same tree component of then we have . In particular, if is a path in , then .

Proof. To prove (i) note that if is a cycle in then

so in particular . To show (ii), let be the path in between and and let be a minimal path in . Consider the Eulerian graph where . Then

(3.1)

Hence . ∎

Using Claim 3.1 we next give a combinatorial description of the -cosystoles in .

Claim 3.2.

Let be an arbitrary -pseudomanifold and let . Then the following hold.

  1. The mapping maps injectively onto all subgraphs of the graph .

  2. We have

    (3.2)
  3. Suppose . Then is a cosystole if and only if .

Proof. Parts (i) and (ii) follow directly from the definitions. We proceed to prove part (iii).

First suppose that . Assume is an Eulerian subgraph of with edge set

Set . Clearly is a cocycle. On the other hand, we assumed that , so must also be a coboundary. We therefore have

Conversely, suppose that and let . Then is Eulerian and hence . Therefore

(3.3)

We conclude that . ∎ Claim 3.2 implies the following combinatorial characterization of the -coboundary expansion of -pseudomanifolds. See Lemmas 2.4 and 2.5 in [SKM] for a related result.

Theorem 3.3.

Let be an -pseudomanifold such that . Then

(3.4)

Proof. In view of Claim 3.2, it suffices to show that

(3.5)

To prove the lower bound let and set

Claim 3.1(i) implies that is a forest. It follows (see, e.g., Theorem 2.1.10 in [We96]) that there exist edge disjoint paths in such that . Claim 3.1(ii) implies that , hence

(3.6)

Finally, we show that equality in (3.5) is attained for some . Let such that and let be a minimal path in . Clearly and

This shows (3.4). ∎

3.2. The Expansion of Coxeter Complexes

Let be an arbitrary finite Coxeter group with the set of generators and a root system . We refer to the books by Humphrey [Hu90] and by Ronan [Ro89] for the theory of Coxeter groups and Coxeter complexes. For let be the subgroup of generated by . For let .

Definition 3.4.

The Coxeter complex is the simplicial complex on the vertex set whose maximal simplices are , for .

The simplicial complex is a triangulation of -dimensional sphere. It is well-known, see, e.g., [Ro89, Theorem 2.15], that . Therefore by Theorem 3.3 we have

Corollary 3.5.

.

Examples:
(i) Let be the symmetric group on with the set of generators where for . Then and is isomorphic to , the barycentric subdivision of the boundary of the -simplex. Hence

(3.7)

We next describe an explicit -cochain of such that . With a permutation we associate the -face