The face numbers of homology spheres
Kai Fong Ernest Chong, Tiong Seng Tay

TL;DR
This paper proves the $g$-conjecture for simplicial $bR$-homology spheres by introducing a new algebraic approach based on stresses and rigidity theory, confirming the face number characterization.
Contribution
The paper establishes the $g$-conjecture for simplicial $bR$-homology spheres using a novel algebraic framework involving stress algebras and rigidity theory.
Findings
Proves the $g$-conjecture for simplicial $bR$-homology spheres.
Introduces a new algebra structure for polytopal complexes based on stresses.
Shows the stress algebra is Gorenstein and has the weak Lefschetz property for generic realizations.
Abstract
The -theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The -conjecture asserts that the same numerical conditions given in the -theorem also characterizes the face numbers of all simplicial spheres, or even more generally, all simplicial homology spheres. In this paper, we prove the -conjecture for simplicial -homology spheres. A key idea in our proof is a new algebra structure for polytopal complexes. Given a polytopal -complex , we use ideas from rigidity theory to construct a graded Artinian -algebra of stresses on a PL realization of in , where overlapping realized -faces are allowed. In particular, we prove that if is a simplicial -homology sphere, then for generic…
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Taxonomy
TopicsCommutative Algebra and Its Applications · Advanced Combinatorial Mathematics · Topological and Geometric Data Analysis
The face numbers of homology spheres
Kai Fong Ernest Chong
Division of Mathematical Sciences
Nanyang Technological University
Singapore
and
Tiong Seng Tay
Singapore
Abstract.
The -theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The -conjecture asserts that the same numerical conditions given in the -theorem also characterizes the face numbers of all simplicial spheres, or even more generally, all simplicial homology spheres.
In this paper, we prove the -conjecture for simplicial -homology spheres. A key idea in our proof is a new algebra structure for polytopal complexes. Given a polytopal -complex , we use ideas from rigidity theory to construct a graded Artinian -algebra of stresses on a PL realization of in , where overlapping realized -faces are allowed. In particular, we prove that if is a simplicial -homology sphere, then for generic PL realizations , the stress algebra is Gorenstein and has the weak Lefschetz property.
Key words and phrases:
rigidity, stresses, -vector, homology sphere, -conjecture, weak Lefschetz property
2010 Mathematics Subject Classification:
Primary: 05E45, Secondary: 05E40, 13A02, 13E10, 13J30, 52B70, 52C25.
1. Introduction and overview
The possible sequences of numbers counting the faces (for different dimensions) of a simplicial convex polytope have been completely characterized. This is known as the -theorem, and it was proven in two parts by Billera–Lee [3] (sufficiency) and Stanley [50] (necessity). At first glance, characterizing these face numbers looks like a problem in combinatorics or polyhedral geometry. Indeed, Billera–Lee used an ingenious “shadow” construction on some suitable cyclic polytope to prove sufficiency. What was perhaps unexpected was Stanley’s proof of necessity: He applied the hard Lefschetz theorem (from algebraic geometry) to the intersection cohomology ring of the toric variety associated to a rational convex polytope. Subsequently, McMullen [32] (corrected and simplified in [29]) gave a different proof of necessity using convex geometry and an -algebra construction [31] associated to convex polytopes. Remarkably, McMullen’s proof also gives a direct combinatorial proof of the hard Lefschetz theorem for simplicial fans.
The -theorem was previously called the -conjecture; this conjecture is due to McMullen [28] (1971). In his same paper [28], McMullen also remarked on extending his conjecture to all simplicial spheres. Today, the -conjecture refers to the conjecture that the numerical conditions given in the -theorem also characterizes the face numbers of all simplicial spheres, or even more generally, all simplicial homology spheres. For decades, this -conjecture had resisted multiple attempts at a complete proof, despite much concerted effort using various methods.
In this paper, we prove the -conjecture for simplicial -homology spheres. Our proof requires a confluence of algebraic, combinatorial, geometric, number-theoretic, and topological ideas. For the rest of this section, we shall give a precise statement of the -conjecture, discuss the prior progress made towards the -conjecture, and provide an overview of our proof. For a comprehensive survey of what has been done and what strategies have been proposed (including variants and further extensions), see [53] (cf. [18]). For a rapid introduction to the subject, see Stanley’s “green book” [51].
To state the -conjecture, we first need to review some definitions. Given a simplicial -complex , its -vector is , where each equals the number of -dimensional faces of . The -vector of is , where
[TABLE]
for each . (By default, , which corresponds to the empty face .) It is an easy exercise to show that the -vector and -vector of determine each other.
Let be a field, and let be a graded -algebra generated by . (All -algebras in this paper are assumed to be unital, finitely generated, associative, and commutative.) If has Krull dimension , then a fundamental result in commutative algebra says that the Hilbert series of (in terms of ) can be written as for some unique polynomial . The vector is called the -vector of , and a sequence of integers is called an M-vector if it is the -vector of some graded -algebra generated by its degree elements. A classic theorem by Macaulay [25] gives a complete numerical characterization of all possible M-vectors (see also [50] or [51, Sec. II.2]). Thus, an assertion that some sequence of integers is an M-vector would be equivalent to a purely numerical condition.
Theorem 1.1** (-theorem).**
Let be the boundary of a simplicial convex -polytope. A sequence of integers is the -vector of if and only if the following two conditions hold. {enumerate}*
* for all .*
* is an M-vector.*
Condition 1.1 is commonly known as the Dehn–Sommerville equations [9, 12, 49]. In particular, the equation is implied by the Euler–Poincaré equation, i.e. has reduced Euler characteristic . The vector in condition 1.1 is also called the -vector of , hence the name “-theorem”. For convenience, we define and for each , so that is the -vector of . The Dehn–Sommerville equations are known to hold more generally for -homology spheres [20], so it is natural to extend the notion of -vectors to -homology spheres.
Conjecture 1.2** (-conjecture).**
Let be a simplicial (-homology) -sphere. A sequence of integers is the -vector of if and only if the following two conditions hold. {enumerate}*
* for all .*
* is an M-vector.*
Notice that Billera–Lee’s result also proves sufficiency for Conjecture 1.2. Thus, the remaining (and difficult) part is to prove that the -vector of a simplicial (-homology) sphere is an M-vector.
Mani [26] proved that simplicial -spheres with vertices are boundaries of simplicial convex -polytopes, thus the -theorem implies Conjecture 1.2 in this case. The -conjecture also holds for certain -homology -spheres with small . Swartz [53] (cf. [43]) proved that if , then is an M-vector for the following cases: (i) , (ii): , , (iii): .
If the full -conjecture is true, then the -vector of a simplicial (-homology) sphere must have non-negative entries, which we write as . In fact, before McMullen formulated his -conjecture, Walkup [57] had already proven that for all simplicial -spheres of dimension . After the -theorem was proven, there was a major breakthrough due to Kalai [16], who used rigidity theory to prove that for all simplicial -homology -spheres satisfying . (In fact, Kalai’s result holds more generally for normal -pseudomanifolds, and Nevo [42] subsequently extended Kalai’s result to -Cohen–Macaulay complexes, again using rigidity theory; cf. [23, 45].) Consequently, the -conjecture for simplicial -homology -spheres is true for . Using sheaf theory, Karu [17] proved that when is the order complex of a Gorenstein* poset. Examples of such order complexes include the barycentric subdivisions of -homology spheres. Subsequently, Kubitzke–Nevo [21] proved the -conjecture for the barycentric subdivisions of -homology spheres; see also [40].
PL spheres (also known as combinatorial spheres) are an important class of simplicial spheres. Strongly edge-decomposable (s.e.d.) spheres, introduced by Nevo [41], form a large subclass of PL spheres that include generalized Bier spheres [36] and Kalai’s squeezed spheres [34]. Babson–Nevo [1] proved that the -conjecture holds for s.e.d. spheres. As part of their proof, they showed that a generic Artinian reduction of the Stanley–Reisner ring of a s.e.d. sphere has the strong Lefschetz property in characteristic zero. (Murai [34] later proved this strong Lefschetz property in arbitrary characteristic; see also [6].) By studing how the strong Lefschetz property relates to bistellar moves, Swartz [53] also proved the -conjecture for another subclass of PL spheres obtained from the boundary of a simplex via certain bistellar moves.
The main result of this paper is a proof of the -conjecture for all -homology spheres:
Theorem 1.3**.**
If is a simplicial -homology -sphere, then the -vector of is an M-vector.
In contrast to recent results related to the -conjecture (e.g. [1, 21, 53]), we will not be using Stanley–Reisner rings. Instead, we shall look at the stresses on certain realizations, and construct what we call the “stress algebra”. One key concept we need is the notion of PL realizations of (abstract) simplicial complexes in some Euclidean space; such PL realizations are more general than geometric realizations and other embeddings that the reader might be accustomed to.
Let be integers. Let and denote the projection maps given by and respectively. Given any simplicial -complex whose set of vertices is linearly ordered, we define a PL realization of in to be a map satisfying the following conditions. {enumerate*}
, and for all .
If is a (non-empty) sequence of distinct vertices in (whose order is consistent with the given linear order on ) that form a -face of , then
[TABLE]
for all , and for all .
For convenience, we simply say that is a PL realization (of ) in . Later in Section 3, we shall define PL realizations more generally for polytopal complexes.
Given such a PL realization , there is an associated set-valued map (induced by ) such that every vertex is mapped to the singleton containing the point \big{(}\widehat{\pi}_{N}(\nu(v))\big{)}^{-1}\pi_{N}(\nu(v))\in\mathbb{R}^{N}, and more generally, every -face is mapped to the convex hull of . Notice that maps faces of to simplices of the same dimension, since for all . Note also that we allow overlaps; could possibly be non-empty, even if are faces with no common vertices. Consequently, a PL realization of in can be thought of as a realization of the vertices of as points in Euclidean -space, together with a choice of some fixed homogeneous coordinates for every realized vertex, such that the realized faces of could possibly overlap.
Let be a PL realization of a simplicial -complex in . An -stress on is a function satisfying whenever , such that the equilibrium equation holds for every -face of , i.e.
[TABLE]
where denotes the set of vertices in the link of (in ). Note that there are no equilibrium equations to check for [math]-stresses, so a [math]-stress is any scalar assignment to the empty face of . Denote the -vector space of -stresses by , and define . We will prove in Section 5 that has a graded -algebra structure, so for this reason, we shall call the stress algebra of .
In fact, this graded -algebra construction works more generally for arbitrary polytopal complexes; see Section 5 for details. In comparison, Stanley–Reisner rings are defined only for simplicial complexes, while McMullen’s polytope algebras [31, 32] (or weight algebras [29]) are defined only for convex polytopes. We also remark that when restricted to convex polytopes, the dual to the multiplication map of our stress algebra is different from McMullen’s multiplication map on weights of the corresponding dual polytopes. To prove this graded -algebra structure for polytopal complexes, we need to look at stresses in relation to Hodge duality, which we do so in Section 4.
In Section 8, we prove that if is an arbitrary field, and if is an orientable simplicial -homology -manifold (without boundary), such that the homology group is trivial whenever , then for generic PL realizations of in , the stress algebra is Gorenstein and is generated by the degree elements, i.e. the -stresses on . This genericity has a precise meaning that we cover in Section 7, and we use the term “-generic” to refer to this precise meaning. Perhaps surprisingly, our proof of the Gorenstein property for -generic PL realizations relies on a number theoretic result on algebraic number fields in a crucial manner.
In this same proof, we also require the homological condition that whenever . This is because we used a three-way interplay between liftings, reciprocals, and -stresses on from Maxwell–Cremona theory, which is no longer true without this homological condition. (For example, there is a PL realization of a -torus in for which this interplay does not hold.) The relevant results from Maxwell–Cremona theory, including important advances made by Rybnikov and coauthors [11, 48], and the connection to Poincaré duality, will be discussed in Section 6.
In Section 9, we introduce the notion of “pivot-compatibility”, which serves as a prelude to the much more technical Section 10. Suppose is a simplicial -homology -sphere, and let be a cone on . In Section 10, we look at the stresses on a -generic PL realization in , and we construct a map . We then use pivot-compatibility and a rather technical argument to prove that if is “sufficiently generic”, then is injective for all . Much of the difficulty of our proof arises from the need for “very careful bookkeeping” of various sets of parameters.
In Section 11, we use the homological interpretation of skeletal rigidity developed by Tay–Whiteley [56] as our main tool. In particular, the space of -stresses of any simplicial complex is isomorphic to the top homology group of the -skeletal chain complex introduced in [56]. Building on the results in Section 10, we extend to a chain map from the -skeletal chain complex to the -skeletal chain complex. Combined with results in [56], we can then construct the following commutative diagram
[TABLE]
where the two horizontal maps in this diagram are isomorphisms, and is a PL realization of in that is obtained from via a central projection from the conepoint of onto a generic hyperplane. Since the first vertical map is injective for all when is “sufficiently generic”, it then follows that the second vertical map is also injective for all for “sufficiently generic” PL realizations of in .
We have deliberately constructed , so that there is some such that for all , the map coincides with the multiplication map defined by . Thus, combined with the Gorenstein property of , we get the following important part of our proof.
Theorem 1.4**.**
If is a “sufficiently generic” PL realization of a simplicial -homology -sphere in , then the stress algebra is Gorenstein and has the weak Lefschetz property.
A more precise statement of this theorem is given in Theorem 11.3. At the end of Section 11, we show how Theorem 11.3 implies our main result (Theorem 1.3).
Finally, we remark that our proof of Theorem 1.3 raises several questions. What about -homology spheres for fields other than ? What about the strong Lefschetz property? What can we say about the face numbers of other homology manifolds? In Section 12, we conclude our paper by addressing these exciting questions with further remarks and open problems.
2. Basic terminology
Throughout, we use the prefix “-” on objects (e.g. -subspace, -complex, -face, etc.) to mean objects of dimension . Let denote the zero vector, and let be an arbitrary field. For any subset , let be the convex hull of , let be the affine span of , and let be the unique linear subspace of that is parallel to (and of the same dimension). For polytopes and related terminology, we follow the definitions given in [58]. In particular, polytopes are assumed to be convex, bounded, and embedded in some Euclidean space. Given a polytope , let be the dimension of , and let be the (unsigned) -volume of .
A geometric polytopal complex is a finite set of polytopes in (for some sufficiently large ) such that the intersection of any two polytopes in is always a common (possibly empty) face of both polytopes, and such that all faces of each polytope in are also polytopes in . We shall assume that the empty polytope , which has vertex set , is always an element of . An abstract polytopal complex is a set whose elements are the vertex sets of the polytopes in some geometric polytopal complex . Unless otherwise stated, a polytopal complex is assumed to be abstract. The dimension of , denoted by , is the dimension of the polytope in that corresponds to. (We define .) The dimension of is . Elements of are called faces, [math]-faces of are called vertices, and the inclusion-wise maximal faces of are called facets. If all facets of have the same dimension, then we say that is pure. Given any , we write (or equivalently, ) to mean that is a face of that contains and satisfies . A saturated flag of faces in is a sequence of faces in .
Given a polytopal -complex and any integer , let be the set of all -faces in . Let be the set of vertices in , and for any face , let be the set of vertices in . A subcomplex of is a subset of that is also a polytopal complex. The -skeleton of is the subcomplex consisting of all faces of of dimension . The open star of a face in is the set of faces , and the antistar of in is the subcomplex . The closed star of a face in , denoted by , is the (unique) minimal subcomplex of that contains . The link of a face in is the subcomplex . A vertex is called a conepoint of if . We say that is a cone if it has at least one conepoint. Given another polytopal complex , we say that is a cone on with conepoint if .
The dual graph of a pure polytopal complex is the graph whose vertices are the facets of , such that two vertices in this graph are adjacent if and only if the corresponding facets share a common codimension face in . We say is strongly connected if its dual graph is connected. A pseudomanifold is a strongly connected pure polytopal complex for which every codimension face is contained in exactly two facets. A -homology -manifold is a -pseudomanifold such that has the same homology groups (over ) as a -sphere for all non-empty faces . A -homology -sphere is a -homology -manifold with the same homology groups (over ) as a -sphere.
A polytopal complex such that for all is called simplicial or a simplicial complex. Subcomplexes of simplicial complexes are simplicial. Given another simplicial complex such that , the join of and is . In particular, a cone on with conepoint is the simplicial complex .
3. Stresses on PL realizations of polytopal complexes
In this section, we extend the definitions of PL realizations and -stresses (as given in Section 1) to allow for the consideration of polytopal complexes. We shall also discuss some basic properties of stresses that will be used throughout this paper.
Let be integers. First, we construct the map as follows. Define , and for each , define . For every non-empty ordered -set such that is an affine -subspace of , define
[TABLE]
More generally, for every non-empty finite ordered set such that is an affine -subspace of , let be any oriented triangulation of the polytope whose orientation is consistent with the given linear order on (e.g. could be a barycentric subdivision), and define , where the sum is over all ordered -simplices in (treated as ordered -sets). Note that does not depend on the choice of , and note that . In fact, for any ordered -subset of that forms an affine basis for , we can always write as a scalar multiple of , so in particular, is decomposable, i.e. the wedge product of elements of .
Definition 3.1**.**
Let be a polytopal -complex, and fix a linear order on . A PL realization of in is a map satisfying the following conditions. {enumerate*}
, and for all .
If , and if is treated as an ordered set whose order is consistent with the given linear order on , then
[TABLE]
for all , and for all .
For convenience, we simply say that is a PL realization (of ) in .
For the rest of this section, let be a polytopal -complex. A PL set-valued map on is a map satisfying \psi(F)=\operatorname{conv}\big{(}\bigcup_{v\in\mathcal{V}(F)}\psi(v)\big{)} for all , such that each is a polytope in of dimension . In particular, , and is a singleton for all . Given any PL realization of in , the associated set-valued map induced by is the map such that every vertex of is mapped to the singleton containing the point \big{(}\widehat{\pi}_{N}(\nu(v))\big{)}^{-1}\pi_{N}(\nu(v))\in\mathbb{R}^{N}, and more generally, every -face is mapped to \operatorname{conv}\big{(}\overline{\nu}(v_{0})\cup\dots\cup\overline{\nu}(v_{t})\big{)}. By default, let . (Whenever is a PL realization, we always reserve the overline in “” to mean this associated set-valued map induced by .) Since for all , it follows that maps faces of to polytopes of the same dimension, thus is a PL set-valued map. Similar to the case of simplicial complexes, and could possibly overlap for any , even if they share no common vertices.
Remark 3.2**.**
If is a PL set-valued map on , then for any function , consider the map on given by , where denotes the unique contained in the singleton . This map on extends (uniquely) to a PL realization of in , and we shall call it the PL realization induced by . More generally, we say that a PL realization of is induced by if it is the PL realization induced by for some .
Definition 3.3**.**
Let be a PL realization of a polytopal -complex in . An -stress on is a function satisfying whenever , such that the equilibrium equation holds for every , i.e.
[TABLE]
Let be the -vector space of -stresses on , and let . By default, when or .
An -stress on is called trivial if for all , and called non-trivial otherwise. A [math]-stress on is any scalar assignment to the empty face of , hence . Note that the -stresses on encode the linear relations on , or equivalently, the affine relations on . For , we have the following geometric interpretation of -stresses.
Theorem 3.4** ([54]; cf. [24]).**
Let , and let be a function satisfying whenever . Then is an -stress on if and only if for all ,
[TABLE]
where denotes the outer unit normal vector to at the codimension face .
Remark 3.5**.**
Theorem 3.4 implies that the definition of an -stress on is independent of the choice of a linear order on .
Usually, we consider PL realizations in . A simplicial -complex obviously admits PL realizations in ; simply consider any PL set-valued map that maps the vertices to points in general position; such a PL set-valued map induces a PL realization in (see Remark 3.2). For an arbitrary polytopal -complex, begin with an embedding in , then project onto a generic affine -subspace of to get a PL set-valued map that induces a PL realization in .
Suppose is a PL realization in . If has vertices, then . To see why, consider any , choose a linear basis of from among the vectors in , and notice that can be assigned arbitrarily for those vertices not equal to . Also, if is a -pseudomanifold, then if is orientable, and otherwise. Indeed, for any , if is a -face of contained in -faces and , then (3) says that for some sign completely determined by the PL realization . Since is strongly connected, must have a common value (up to sign) on all -faces of . This common value can be arbitrarily chosen when is orientable, and must be [math] when is non-orientable. For an arbitrary polytopal -complex , the vector space has dimension , where denotes the -th Betti number of ; see [54, Thm. 4.1].
A local -stress of on a face is a function such that the equilibrium equation (2) holds for every . Note that if is an -stress on , then the map on given by is a local -stress of on . In particular, a local -stress of on the empty face is precisely an -stress on . Similar to the case of the (usual) -stresses, we shall denote the -vector space of local -stresses on by , and we define .
We end this section with comments on the terminology used by other authors. Stresses first appeared in Maxwell’s study of planar frameworks in classical mechanics [27]; they coincide with our notion of -stresses on -dimensional simplicial complexes. A generalized notion of -stresses on simplicial complexes of arbitrary dimension was first introduced by Lee [22, 24], and our definition of -stresses is equivalent to what Lee calls affine -stresses in [24]. In fact, Lee defined two types of -stresses: affine and linear. Given a simplicial -complex and any integer , the space of Lee’s linear -stresses on a PL realization of in is isomorphic to our space of -stresses on a PL realization of in . Another common equivalent definition for the space of (affine) -stresses is to define it as the cokernel of some -rigidity matrix. (There are several “kinds” of rigidity matrices.) The next section gives another useful equivalent definition of stresses. For more details and other equivalent definitions of (affine) -stresses, see [48, 54, 55].
4. Stresses and Hodge duality
In this section, we give another (equivalent) definition for stresses that is related to Hodge duality. An expert familiar with rigidity theory would notice that it corresponds to the cokernel of the minimal rigidity matrix. We shall use this definition to show that stresses “remain” as stresses on links of faces in a natural way (Theorem 4.2). Later in Section 5, we use this definition in a crucial way to construct a (well-defined) multiplication map on stresses.
Let be integers. Assume that is equipped with the usual inner product , and fix the standard orthonormal basis for . Clearly, extends to an inner product on the exterior algebra . Let . Elements of are called -vectors. The Hodge star operator on is a linear operator that is completely determined by the relation for all -vectors , where . Notice that maps -vectors to -vectors for all . Hodge duality refers to the fact that for all -vectors , hence we can define an inverse by for all -vectors . The Grassmann–Cayley operator on , which we denote by , is defined by for arbitrary . An exterior algebra equipped with the Grassmann–Cayley operator is also known as the Grassmann–Cayley algebra. For such algebras, we caution the reader that some authors (e.g. in [10]) instead use and to denote wedge product and the Grassmann–Cayley operator respectively.
Given an ordered set , and any ordered subset of , let be the complement of relative to (i.e. and ), and order the elements of so that they are consistent with the fixed linear order on . Thus, by treating ordered sets as sequences, the concatenated sequence is a permutation of . We shall denote the sign of this permutation by .
Let be a PL realization of a polytopal -complex in , and fix a linear order on . Given any faces in , let denote the sign , where and are treated as ordered sets, whose orders are consistent with the given linear order on . Also, define , which is a non-zero -vector in .
Proposition 4.1** ([55]).**
Let , and let be a function satisfying whenever . Then is an -stress if and only if for every ,
[TABLE]
For the rest of this section, let be a PL realization of a simplicial -complex in . For any saturated flag of faces in , let for each . The following identities are straightforward consequences of the definition of the Grassmann–Cayley operator:
[TABLE]
Given any , write as a decomposable -vector (which is possible since is decomposable), and define . Note that for all satisfying . Given any ordered basis for , define the linear map by
[TABLE]
We say that is distinguished if for all faces in . Notice that the distinguished ordered bases for are dense among arbitrary ordered bases for .
Given a distinguished ordered basis for , let be the PL realization of in that is uniquely determined by for all . Note in particular that for all implies for all .
Theorem 4.2**.**
Let be a PL realization of a simplicial -complex in . Let , let , and define the function by . If is a distinguished ordered basis for , then .
Proof.
Consider an arbitrary . Since
[TABLE]
for all satisfying , it follows from the definition of a -stress that
[TABLE]
so the identity yields
[TABLE]
which implies
[TABLE]
Now, use (5) and apply the linear map to get
[TABLE]
Therefore is a -stress on , i.e. . ∎
5. Stress algebra
Throughout this section, let be a PL realization of a polytopal -complex in . The main aim of this section is to define multiplication on stresses of different dimensions, and prove that is an associative commutative graded -algebra.
For every , let be the set of all pairs satisfying and . Note that if is pure, then is non-empty for all . For every , let be the set of all pairs satisfying either , , or , , such that whenever .
Lemma 5.1**.**
If , then
[TABLE]
as sets of pairs in .
Proof.
The assertion is true by definition if , so assume henceforth that .
Suppose for some . If , (resp. , ), then since and , it follows that (resp. ), thus , and therefore .
Conversely, suppose instead that for some face of . This means there is some vertex that is not a vertex of . Let be an inclusion-wise maximal face of that does not contain . Note that , , and , thus by definition, , and . ∎
Given stresses and , define their multiplication by
[TABLE]
By default, if .
Theorem 5.2**.**
If and , then .
Proof.
Consider an arbitrary , and let . By Proposition 4.1,
[TABLE]
and similarly,
[TABLE]
thus
[TABLE]
Next, let \phi_{P^{\prime}}:\big{(}\bigwedge(\mathbb{R}^{d+1})\big{)}\otimes_{\mathbb{R}}\big{(}\bigwedge(\mathbb{R}^{d+1})\big{)}\to\big{(}\bigwedge(\mathbb{R}^{d+1})\big{)} be the map defined by
[TABLE]
Given any , note that
[TABLE]
so since and , it follows that
[TABLE]
for all .
Consequently, by applying to (6) and summing over all , we get
[TABLE]
It then follows from Lemma 5.1 that
[TABLE]
or equivalently,
[TABLE]
Finally, for any , the definition of implies that whenever satisfies , therefore Proposition 4.1 implies . ∎
In the rest of this paper, we shall call the stress algebra of . The following theorem justifies our terminology.
Theorem 5.3**.**
If is a PL realization of a polytopal -complex in , then is an associative commutative graded -algebra.
Proof.
Theorem 5.2 gives us for all , thus is a graded -algebra. (Recall that if or .) The commutativity of follows from the symmetry in the definition of , while the associativity of is obvious. ∎
Remark 5.4**.**
Since by definition, the proofs of the results in this section hold verbatim when -stresses are replaced by local -stresses. The multiplication of local stresses is defined in exactly the same manner and yields another local stress. In particular, is an associative commutative graded -algebra for any .
For readers familiar with McMullen’s proof of the -theorem [29], recall that the multiplication of two weights on a polytope requires the notion of tight coherent subdivisions of in its definition. McMullen proved that this multiplication map is well-defined (independent of the choice of the tight coherent subdivision) by using properties of fiber polytopes [4]. As proven by Lee [24], the -weights of coincide with the -stresses on the boundary complex of the polar dual of realized in , hence a multiplication map on stresses analogous to McMullen’s multiplication of weights would require a dual notion to tight coherent subdivisions (as well as a dual notion to fiber polytopes). It is possible to define such duals to tight coherent subdivisions for faces of ; they are certain sets of pairs of polytopes satisfying . To define multiplication on stresses in terms of , so that it agrees with McMullen’s multiplication on weights of the dual polytope, it would actually take considerable effort to show that the definition is independent of the choice of .
Instead, we have defined multiplication on stresses in terms of , which corresponds to the union of all possible (for a fixed ). Dually, we get a different multiplication map on weights of the dual polytope that is defined in terms of the union of all possible tight coherent subdivisions of (faces of) , which would simplify McMullen’s proof of the -theorem without the need for fiber polytopes.
6. Liftings, reciprocals, and -stresses on orientable homology manifolds
Liftings and reciprocals are fundamental concepts in Maxwell–Cremona theory [7, 8] that deal with the realizations of pure polytopal complexes. Intuitively, liftings (into Euclidean spaces of one dimension higher) are the inverses of vertical projections onto hyperplanes, while reciprocals are rectilinear realizations of dual graphs whose edge directions are fixed in a specific manner. In this section, we build upon the work by Rybnikov and coauthors [11, 48] on the close connections between -stresses, liftings, and reciprocals. We will relate to the liftings and reciprocals of in the case when is an orientable (polytopal) -homology manifold satisfying a certain homological condition, and in particular, we will consider the special subcase when is simplicial.
Let be a PL realization of a pure polytopal -complex in . For brevity, denote the projection maps and simply as and respectively. Given -vector spaces , let the space of -valued affine functions on be denoted by . A lifting of is a map such that for all , and for all . Note that adjacent -faces of are not required to be mapped to affine maps with distinct hyperplanes in as their images. For any fixed , let be the vector in contained in the singleton , i.e. let . Notice that the vector is invariant over all -faces of that contain , so for convenience, we denote this common vector by .
By definition, every lifting is completely determined by the map defined by . Thus, we can add two liftings of by specifying that . Similarly, scalar multiplication is defined by for all . (If is an embedding, then the -vector space of these maps is sometimes called a -spline; see [2].) Given any -face of , we shall denote by the -vector space of all liftings of that satisfy .
Theorem 6.1** ([48]).**
Let be a PL realization of an orientable (polytopal) -homology -manifold in with trivial first homology group over whenever . Then for any , there is an isomorphism between and as -vector spaces.
Remark 6.2**.**
For any -homology manifold , the condition necessarily implies that is orientable; see [14, Cor. 3.28]. This homological condition cannot be omitted from Theorem 6.1. For example, Rybnikov gave a PL realization of a -torus that has non-trivial -stresses, but does not admit any non-trivial liftings; see [48, Fig. 3].
Theorem 6.3**.**
Let be a PL realization of an orientable simplicial -homology -manifold in , such that whenever . Then , where is the number of vertices in .
Proof.
Let be an arbitrary -face of . Theorem 6.1 yields the isomorphism . Every lifting is completely determined once we know the values of for all vertices . Since is simplicial, all the vertices not in can take on arbitrary real values, therefore . ∎
Remark 6.4**.**
In general, for non-simplicial , the dimension of would depend on the combinatorial structure of . Without the explicit knowledge of this combinatorial structure, we can only conclude that , where is the maximum number of vertices in any -face of . See [48] for an algorithmic approach to finding the exact value of the dimension of the space of liftings for certain classes of non-simplicial -homology manifolds.
For the rest of this section, let be a PL realization of a (polytopal) -homology -manifold in . A reciprocal of is a map such that is parallel to for all adjacent -faces of . We say is non-degenerate if for all adjacent -faces . Typically, we consider reciprocals when is orientable. A PL orientation of is a map such that adjacent -faces of satisfy if and only if the outer unit normal vectors to and at their common codimension face have opposite directions. It can be shown that is orientable in the usual sense (e.g. as defined in [14]) if and only if admits a PL orientation [48]. In particular, every orientable has two possible PL orientations: Starting with any -face of , the value of , together with , would uniquely determine the values of for all remaining faces .
Suppose is a PL orientation of , and let be a reciprocal of . It follows from definition that for every pair of adjacent -faces of , there exists a unique scalar such that
[TABLE]
The map defined by is called the edge-length map of , and the map defined by is called the normalized edge-length map of . Notice that uniquely determines up to translation. Thus, if denotes the set of all reciprocals of satisfying for some given -face of , then every reciprocal can be identified with its edge-length map . We check that for any reciprocals , the sum of their edge-length maps is always the edge-length map of a uniquely determined reciprocal in . Thus, we can define addition on by specifying that . Scalar multiplication on is obvious, therefore is an -vector space.
Theorem 6.5** ([48]).**
Let be a PL realization of an orientable (polytopal) -homology -manifold in . Then for any , there is an isomorphism of -vector spaces between and .
Remark 6.6**.**
Note that Theorem 6.5 does not require the homological condition (for ). However, we do require that is orientable so that the addition of reciprocals is well-defined.
Given a lifting of and any , the map is by definition an affine map, so there is a unique vector , and a unique scalar , such that is the map . The following two theorems refine Theorem 6.1 and Theorem 6.5 respectively.
Theorem 6.7** ([48]).**
Let be a PL realization of an orientable (polytopal) -homology -manifold in . Then for every , there is an injective map (of -vector spaces) given by
[TABLE]
where is a -face of contained in the two (uniquely determined) -faces of , and is a PL orientation of . Furthermore, if or , then is an isomorphism.
Theorem 6.8** ([48]).**
Let be a PL realization of an orientable (polytopal) -homology -manifold in . Then for every , there is an isomorphism (of -vector spaces) given by .
Remark 6.9**.**
The proof of Theorem 6.8 (as given in [48]) requires the existence of dual polyhedral decompositions, which are used in proving Poincaré duality for -homology manifolds.
Remark 6.10**.**
Similar to local -stresses, we can define local liftings and local reciprocals by replacing every instance of with for some ; cf. Remark 5.4. The local versions of Theorem 6.7 and Theorem 6.8 still hold true. (Their proofs in [48] hold verbatim.) Thus, for any and any , we have an isomorphism of -vector spaces:
[TABLE]
Recall that a collection of vectors in is said to be in general position if every subcollection of at most vectors is affinely independent. Given any PL realization , we say that the vertices of are realized in general position if is a collection of vectors in general position. Notice that such a polytopal complex is necessarily simplicial. By combining Theorem 6.7 and Theorem 6.8 with our definition of the multiplication of stresses of different dimensions (see Theorem 5.3), we get the following useful corollary.
Corollary 6.11**.**
Let be a PL realization of an orientable simplicial -homology -manifold in such that whenever . Let and . Fix some -face of , and let be the normalized edge-length map of the reciprocal corresponding to the -stress under the isomorphism . If the vertices of are realized in general position, then is the [math]-stress given by
[TABLE]
where .
7. Generic PL realizations and multiplication by -stresses
Throughout this section, let be a PL realization of an oriented simplicial -homology -manifold in satisfying whenever , and let be the PL orientation of . The purpose of this section is twofold: First, we introduce a specific notion of genericity, which we call -genericity. Next, we prove that if is -generic, then for every non-trivial -stress , we can always find some -stress such that is non-trivial. The existence of such a -stress would serve as the base case for our induction argument in Section 8, where we prove that the stress algebra is Gorenstein for -generic PL realizations .
Intuitively, choosing a -generic PL realization amounts to choosing rational coordinates for a certain collection of points (arising from certain “basis reciprocals” and completely determined by ) such that the “normalized” distance between any pair of distinct points in this collection must be the square root of a squarefree rational number. (A rational number is called squarefree if .) We now state a number theoretic result on square roots that will be useful later.
Lemma 7.1** ([33]).**
Let be a real algebraic number field, let be square roots of elements in , and suppose that the product of the elements in every non-empty subcollection of is not contained in . If is a polynomial in variables with coefficients in , such that is linear in each variable , then if and only if all the coefficients of are [math].
Fix a linear order on the vertices of . For each set of vertices such that , let be the -by- matrix whose rows are (as row vectors), and let be the matrix obtained from as follows: For every , divide all the entries in the -th row of by the scalar . Notice that would have a column of ones as its right-most column. For convenience, we write simply as for each face of . Recall that the Gram matrix of a collection of vectors in is the matrix whose -th entry equals the inner product . Given any matrix , let be the Gram matrix of the collection of the rows of (treated as vectors).
Following the notation in Section 6, we recall that any lifting of is completely determined by the collection of real scalars . For each , let be the lifting of given by , and for all remaining . Given any -face of , it follows from definition that if is not a vertex of , and if is the -th vertex of induced by the linear order , where is the standard basis column vector whose -th entry equals . In this latter case, Cramer’s rule says that the -th entry of equals , where is the matrix obtained from replacing the -th column of by . Consequently, if for all (in which case we say that is rational), then for all and all .
Consider an arbitrary . Since is a -homology manifold, there are two uniquely determined -faces of that contain . For each , define the non-negative real scalar
[TABLE]
where denotes the usual Euclidean norm, and define the real scalar
[TABLE]
Notice that Theorem 6.7 and Theorem 6.8 together imply that is the absolute value of .
Definition 7.2**.**
A PL realization of in is called -generic if is rational, the vertices of are realized in general position, the collection of real scalars
[TABLE]
is linearly independent over (i.e. the scalars in are distinct, non-zero, and do not satisfy any non-trivial linear equations with coefficients in ), and the product of the elements in every non-empty subcollection of is irrational.
Geometrically, the last two conditions in this definition mean that the normalized edge-length maps of the reciprocals () map each -face of to either [math] or an irrational number (cf. Theorem 6.8), such that the irrational “normalized edge-lengths” (over all these reciprocals) do not satisfy any non-trivial monomial or linear equations over .
Notice that every PL realization of is completely determined by and hence can be identified with a vector in . Under this identification, the following proposition justifies our terminology “-generic”.
Proposition 7.3**.**
The set of all -generic PL realizations of is a dense subset of all (not necessarily rational) PL realizations of (with respect to the usual Euclidean metric).
Proof.
First of all, rational PL realizations and PL realizations with vertices realized in general position, are each dense among arbitrary PL realizations. Hence, if denotes the set of rational PL realizations of with vertices realized in general position, then it suffices to show that the set of -generic PL realizations of is a dense subset of .
Suppose . Let , let be the two uniquely determined -faces of that contain , and let . Note that , while for each , the scalar equals the -volume of the orthogonal projection of onto the coordinate hyperplane containing all points whose -th coordinate is zero. (Similar statements hold when is replaced by .) By definition, . Note also that
[TABLE]
Consequently, is a rational function with rational coefficients in terms of the coordinates of the vectors in as its variables. We check that this rational function is not the square of any rational function (with rational coefficients) over the same variables.
Now, the set of squarefree rational numbers is dense in , thus if some scalar in is rational, then we can always perturb (as a vector in ) so that the scalar becomes the square root of a squarefree rational number, i.e. the scalar becomes irrational. Consequently, there is a dense subset of such that for every PL realization in , the scalars in are distinct and irrational.
Finally, a routine argument shows that there is a dense subset of such that for any , the product of scalars in every non-empty subset of is irrational. Therefore, it follows from Lemma 7.1 that is linearly independent over . By definition, this dense subset is identically the set of -generic PL realizations of . ∎
Before we prove the main result (Theorem 7.7) of this section, we need to introduce more notation and definitions. Given a real algebraic number field , and any set of real numbers, let denote the smallest subfield of that contains and . Given any matrix , let be its transpose, and let be its row space. For any pure simplicial -complex with linearly ordered vertices and linearly ordered -faces, the vertex-ridge incidence matrix of is an real matrix whose -entry equals if the -th vertex is contained in the -th -face, and equals [math] otherwise.
Lemma 7.4** ([5, Cor. 3]).**
If is a pure strongly connected simplicial complex, then its vertex-ridge incidence matrix has full rank.
For the rest of this section, fix a linear order on the -faces of . Let be an matrix whose -th entry equals . Let be an real matrix whose -th entry equals if , and equals [math] otherwise. Also, let be the vertex-ridge incidence matrix of . Note that have constant column sums respectively, so , and by Lemma 7.4, both and have full rank. An elementary fact from linear algebra says that if are matrices (note that ), then is invertible if and only if has full rank and , thus is invertible if and only if is invertible.
Lemma 7.5**.**
Let be positive integers. For each , let be a set of distinct square roots of squarefree positive rational numbers, and assume that the product of the elements in every non-empty subcollection of is irrational. Suppose is an matrix that satisfies the following conditions. {enumerate}*
Every row or column of has at least one non-zero entry.
*For each , the entries in the -th row of are elements of the number field , and the non-zero entries among them are linearly independent over .
Then must be an invertible matrix.*
Proof.
Let be the set of all non-zero entries in the first row of , and define . We shall first show that is linearly independent over . Suppose not, then there exist , not all zero, such that . Let be a vector of variables, and let be a vector of doubly-indexed variables, where the indices satisfy and . For every , since (resp. ), there exists a polynomial (resp. ) in terms of variables (resp. ) with coefficients in , that is linear in each variable in (resp. ), such that is the value of evaluated at for (resp. is the value of evaluated at for , ).
Since are not all zero, at least one of is not the zero polynomial. By assumption, are non-zero, so none of is the zero polynomial. Choose some such that whenever . Define the polynomial , and note that evaluated at and gives a non-trivial -linear combination of . This -linear combination must be non-zero by condition 7.5, thus is not the zero polynomial. Now, is the value of evaluated at for and for , , hence Lemma 7.1 forces the contradiction that . Consequently, is linearly independent over as claimed.
Next, we prove the lemma by induction on . Using Laplace’s expansion along the first row, we have , where each is the -cofactor of . This implies that is a -linear combination of the elements in . Now, by condition 7.5, we know that there is a permutation on such that for all , thus by induction hypothesis. Therefore, it follows from the linear independence of over that . ∎
Proposition 7.6**.**
If is -generic, then is invertible.
Proof.
We shall prove the equivalent statement that is invertible. Let be the collection of non-zero scalars as defined in (9). For each , let be the reciprocal of given by , and recall that denotes the normalized edge-length map of . By definition, the -th entry of equals . Thus, every non-zero entry of is an element of up to some sign. Since is -generic, the absolute values of these non-zero entries are distinct square roots of squarefree positive rational numbers, and the product of any non-empty subset of these absolute values is irrational.
For each , let be the set of the absolute values of all non-zero entries in the -th row of , and note that the entries in the -th row of are elements of the number field . By definition, the -th entry of equals , where the sum is over all . This sum is in particular a non-trivial -linear combination of the elements in (where the coefficients are in ), so it must be non-zero by the -genericity of .
Next, we prove that the non-zero entries in each row of are linearly independent over . For each , let be the row vector consisting of all non-zero entries in the -th row of , and let be the vertex-ridge incidence matrix of . Note that the entries in correspond to the -faces in . By definition, the non-zero entries in the -th row of , when written as a row vector, equals the product . Since is a pure strongly connected simplicial complex, Lemma 7.4 says that has full rank. Now, since the -genericity of implies that the entries in are linearly independent over , it then follows that the entries in are linearly independent over . Therefore, we can apply Lemma 7.5 and conclude that is invertible. ∎
Theorem 7.7**.**
If is a -generic PL realization of an oriented simplicial -homology -manifold in , such that whenever , then for every non-trivial , there exists some such that is non-trivial.
Proof.
By relabeling the vertices of if necessary, assume that the last vertices form a -face of . Choose arbitrary and , and let be the isomorphism of -vector spaces given in Theorem 6.7. By Theorem 6.3, is a basis for , thus we can write for some uniquely determined . For convenience, let denote the row vector , and let denote the column vector . Notice that Corollary 6.11 yields , where is the submatrix of corresponding to the first rows.
By the definition of a -stress, is a -linear combination of for all . (The coefficients are rational since is rational.) Explicitly, if we define the matrix , and let denote the -th entry of the rational matrix , then b(v_{n-d-1+i})=-\big{(}\sum_{j=1}^{n-d-1}w_{i,j}b(v_{j})\big{)} for all .
Let (resp. ) be the matrix obtained by applying the following sequence of column operations to (resp. ): For each , , subtract times the -th column from the -th column. Let (resp. ) be the principal submatrix of (resp. ) containing the first rows and columns of (resp. ). Similar to how we showed that is invertible if and only if is invertible, we can also show that is invertible if and only if is invertible.
In the proof of Proposition 7.6, we showed that satisfies the conditions of Lemma 7.5. Since the non-zero entries in each row of are linearly independent over , and since every is rational, we infer that also satisfies the conditions of Lemma 7.5. In particular, the diagonal entries of are non-zero. Consequently, also satisfies the conditions of Lemma 7.5, which implies is invertible.
Observe that , where is a column vector. By assumption, the vertices of are realized in general position, so is a linear basis of , and , whereby each uniquely determines . Therefore, if is non-trivial, then , and it follows from the invertibility of that we can always choose some (which corresponds uniquely to some -stress ) such that . ∎
8. Gorenstein property of the stress algebra
In this section, we shall establish a symmetry in the -vector of the stress algebra. Specifically, if is a -generic PL realization of an orientable simplicial -homology -manifold in , such that whenever , then we show that is an Artinian Gorenstein -algebra generated by the degree elements, so in particular, the -vector of satisfies for all .
Theorem 8.1**.**
Let be a -generic PL realization of an orientable simplicial -homology -manifold in , such that whenever . Then for every and every non-trivial , there exists some such that is non-trivial.
Proof.
The case is trivial (since ), while the case is proven in Theorem 7.7, so assume that . Let be non-trivial, and suppose for some . Let be any -face of contained in , and define the vertex . Note that is a -homology -sphere. Choose a distinguished ordered basis for such that the PL realization is -generic; the existence of such a is implied by Proposition 7.3. Define the map by for all . Theorem 4.2 implies that is a -stress. Since , it follows from Theorem 7.7 that there exists some -stress such that is non-trivial, i.e. .
Define by for all . Using Theorem 4.2, we check that is a local -stress of on , and we get
[TABLE]
Fix some . By Theorem 6.7 and Remark 6.10, there is an isomorphism
[TABLE]
of -vector spaces that extends to the isomorphism given by (7).
Let . Following the notation in Section 6, note that is completely determined by the values of the scalars for all . Now, choose an arbitrary assignment of real scalars to the remaining vertices in , so that we get a lifting satisfying for all . Finally, define . Note that for all , therefore (10) implies that . ∎
For each , let denote the -vector subspace of spanned by the -fold products of elements in . Also, define .
Corollary 8.2**.**
Let be a -generic PL realization of an orientable simplicial -homology -manifold in , such that whenever . Then for every and every non-trivial , there exists some such that is non-trivial.
Theorem 8.3**.**
If is a -generic PL realization of an orientable simplicial -homology -manifold in , such that whenever , then and for all . In particular, the stress algebra is generated as an -algebra by .
Proof.
For any subspaces , , we say that separates if for every non-trivial , there exists some such that is non-trivial. Note that if separates , then . Given any , Corollary 8.2 says in particular that separates , and that separates . Now, since , a dimension count yields
[TABLE]
Thus, all dimensions must be equal, and we get . ∎
Given an Artinian graded -algebra that is generated (as an -algebra) by , the socle of (as an -module) is . We say that is Gorenstein if . Note that , and that for all , thus we have the following immediate consequence of Theorem 8.1 and Theorem 8.3.
Corollary 8.4**.**
If is a -generic PL realization of an orientable simplicial -homology -manifold in , such that whenever , then is Gorenstein.
9. Pivot-compatibility and the rigidity matrix
In this section, we shall introduce the notion of “pivot-compatibility” for simplicial complexes whose vertices are realized in general position. We begin with a brief overview of what it entails, and why it is important. Recall that the space of -stresses on a PL realization of any simplicial -complex in is isomorphic to the cokernel of some -rigidity matrix. The rows of this rigidity matrix correspond bijectively to the -faces of , so if we reduce the transpose of this matrix to reduced row-echelon form, then the -faces that correspond to the pivot columns of the resulting matrix in reduced row-echelon form would depend on some choice of a linear order on . Roughly speaking, we want to choose a suitable linear order on so that the non-pivot columns correspond to a subset that satisfies a certain “nice” property. We say that is “pivot-compatible” if such a suitable linear order on exists. The existence of a pivot-compatible will later be crucial in Section 10, where we use it to prove a technical result that our proof of the -conjecture will subsequently rely on.
For the rest of this section, let , let be an arbitrary simplicial -complex, and let be a PL realization in , such that the vertices of are realized in general position. Let be the -vector of , and let .
Let be a set of variables indexed by the -faces of , and consider the following system of vector equations
[TABLE]
where ranges over all -faces of . Since every is in , these vector equations are equivalent to a system of linear equations in the variables . Given some choice of linear orders on and respectively, let be the coefficient matrix of this linear system (that is consistent with the given linear orders). Notice that is the transpose of the truncated face-ring rigidity matrix of as defined in [54, Sec. 6]. In particular, [54, Prop. 6.1] says that the nullspace of is isomorphic to as -vector spaces.
Let be the reduction of to reduced row-echelon form. By assumption, the nullspace of has dimension . A linear order on is a bijective map . Given a linear order on such that for all , we say that is pivotal if the first variables of the above linear system correspond to pivot columns of , or equivalently, if has the matrix structure
[TABLE]
where denotes the -by- identity matrix, denotes a (possibly empty) zero matrix of an appropriate size, and is a -by- submatrix of . For convenience, let denote the set consisting of the first -faces of (relative to ), and let denote the set consisting of the last -faces of (relative to ). For any pivotal linear order on , we call the elements of (resp. ) the pivot -faces (resp. non-pivot -faces) of (relative to ). For each (resp. ), let (resp. ) denote the -th pivot (resp. non-pivot) -face of relative to .
Let be a pivotal linear order on , and let . By definition, knowing the values of for all non-pivot -faces would uniquely determine the values of for all pivot -faces . Furthermore, the values of for non-pivot -faces can be arbitrarily chosen. Since the vertices of are realized in general position, it follows from the vector equation (11) that for each , knowing the values of on vertices in would uniquely determine the values of for all remaining vertices in .
Given any , the pivotal weight of relative to , which we denote by , is defined recursively to be the smallest integer such that the value of can be uniquely determined once we know the values of for and for all satisfying . Equivalently, is the smallest integer such that knowing the values of on the first non-pivot -faces (relative to ) would uniquely determine the value of . Note in particular that for all .
Also, given any , the pivotal subweight of relative to , denoted by , is the smallest integer such that knowing the values of on the first non-pivot -faces (relative to ) would uniquely determine the value of for every containing . The following lemma is a direct consequence of the definitions of pivotal weights and pivotal subweights.
Lemma 9.1**.**
For any pivotal linear order on , the following statements hold. {enumerate}*
If , then .
If , then .
Given any permutations and on and respectively, let denote the bijective map \big{(}[\widehat{\sigma},\sigma]\cdot\phi\big{)}:\mathcal{F}_{k}(\Delta)\to\{1,\dots,f_{k}\} given by
[TABLE]
Notice that is always a pivotal linear order on for all possible permutations and . (A much more general theory on permutations of rows and columns of matrices in relation to pivot columns is known as “pivoting” in numerical linear algebra.) Given any linear order on , we say that is a pivotal reordering of if for some permutations and on and respectively.
For any satisfying , define the bijective map so that
[TABLE]
By the definition of a pivotal weight, the -th column of is not contained in the subspace spanned by the first columns of , thus must also be a pivotal linear order on .
Definition 9.2**.**
Given any set , we say that is pivot-compatible if for some pivotal linear order on , and for every , there exists some -face contained in that is not contained in any other -face in , i.e. if for some , then .
Definition 9.3**.**
A subset of vertices is called -autonomous if for every -face of that does not contain any of the vertices in , there exists some such that is a vertex of . By default, we define to be -autonomous if and only if is non-empty.
Theorem 9.4**.**
Let be a -autonomous set of vertices. Let be all the distinct -faces of that do not contain any of the vertices in , and for each , let be any vertex in such that . (Such a vertex exists since is -autonomous.) Then the set is pivot-compatible.
Proof.
First of all, for each , note that is the only -face in that contains . Let be a pivotal linear order on such that the value is minimized over all possible pivotal linear orders on . Suppose , and let . Without loss of generality, assume that . Since for all pivotal reorderings of , we can assume that . By Lemma 9.19.1, , thus there exist at least -faces of distinct from that contain and have pivotal weight relative to .
Let be one such -face. Since and , we get , so we can obtain a new pivotal linear order from . Now, is the only -face in that contains , thus , and we have , which contradicts the minimality of . Therefore, , i.e. , and so is pivot-compatible. ∎
10. Stresses on cones of homology spheres
Our strategy for proving the -conjecture for -homology spheres is to look at the stresses on cones of -homology spheres instead, which (fortunately for us) is easier to work with. In this section, we shall construct a map on the -stresses on the cone of an -homology -sphere, and prove that if our PL realization is “sufficiently generic”, then this map is injective for all (see Theorem 10.5). Our proof on the injectivity of this map is rather technical and will later (in Section 11) play a crucial role in establishing that the stress algebra of an -homology sphere has the weak Lefschetz property.
Homology spheres and cones on them are examples of Cohen–Macaulay complexes. Recall that a simplicial -complex is called -Cohen–Macaulay if for all and all . (Here, denotes the -th reduced homology group of with coefficients in .)
Theorem 10.1** ([24] [56]).**
Let be a PL realization of a simplicial -complex in , such that the vertices of are realized in general position. If is -Cohen–Macaulay, and if is the -vector of , then h_{i}=\dim_{\mathbb{R}}\big{(}\Psi_{d+1-i}(\Delta,\nu)\big{)} for all .
For the rest of this section, fix some integers , let be a simplicial -homology -sphere, and fix some -face of . Also, let be a cone on with conepoint , and note that . For convenience, let (resp. ) denote the set of all -generic PL realizations of in (resp. in ).
Consider arbitrary PL realizations and , and let be a pivotal linear order on that corresponds to . Recall from Section 9 that is the transpose of the truncated face-ring rigidity matrix of , the null-space of is isomorphic to as -vector spaces, and is the reduction of to reduced row-echelon form, i.e. has the matrix structure
[TABLE]
Since is -Cohen–Macaulay, it follows from Theorem 10.1 that the nullspace of has dimension , hence is a -by- submatrix of . Note that has rational entries, since is rational. Conversely, every -by- matrix with rational entries equals for some rational PL realization on . Thus by Proposition 7.3, the set is dense among arbitrary -by- matrices with rational entries.
Recall that and are the sets of pivot -faces and non-pivot -faces of respectively (relative to ), and that denotes the -th non-pivot -face of relative to . For each such that , let be the -th entry of . Let be the -vector space isomorphic to that contains all -valued maps on , and let be the dense subset
[TABLE]
Since is a fixed -face of , each choice of would uniquely determine a PL orientation on that satisfies . For each and , define the scalar
[TABLE]
where we note that was previously already defined in (8).
Next, for every pair , and every , define
[TABLE]
if , and define
[TABLE]
if . Notice that if , then . Our motivation for defining will later become apparent in the proof of Theorem 11.3.
Given any , , , any pivotal linear order on that corresponds to , and any non-empty subset , let denote the -by- matrix whose entries are indexed by pairs , such that the -th entry of equals . Notice that every entry of is a -linear combination of the irrational scalars in . Given any non-empty subset , let denote the -by- submatrix of induced by the rows indexed by .
We shall prove a technical lemma that says has full rank when and is “sufficiently generic” (Lemma 10.4), where “sufficiently generic” has a precise meaning that we shall determine. To state this technical lemma, we need the following definitions. (By default, we define the empty product to be equal to .)
Definition 10.2**.**
Let , and define .
Given any real number , we say that the -support of , denoted by , is the set of (possibly empty) subsets of , such that can be written as
[TABLE]
for some collection of non-zero rational scalars . (By Lemma 7.1, the scalars are uniquely determined once and are fixed.) For each , we say that is the rational coefficient of in relative to , while for each subset of not in , we say that [math] is the rational coefficient of in relative to . Observe that if and only if is a non-zero rational number, while if and only if .
Let be a set of doubly indexed variables, and let be the ring of polynomials on the variables in with coefficients in . For any polynomial , we write to denote the evaluation of on the values for all in . Given non-empty subsets , satisfying , any map , and any subset , let denote the uniquely determined polynomial in , such that for every and , the evaluation equals the rational coefficient of in relative to .
Lemma 10.3**.**
Let be an integer, let , , , and let be a pivotal linear order on that corresponds to . Also, let , let , and let be a sequence of (not necessarily distinct) pivot -faces in . If such that the coefficient of the monomial in the polynomial is non-zero, then there exists some permutation on such that and for all .
Proof.
This follows from the Leibniz formula for determinants applied to , as well as from the definitions of and . ∎
Lemma 10.4**.**
Let , let , let , and let . Also, let be a pivotal linear order on such that . (The existence of is implied by Theorem 9.4, since is -autonomous.) Suppose that , and that satisfies for all . Then the following statements hold. {enumerate}*
There exists a dense subset of , such that for every satisfying , the -support of contains .
The coefficient of the monomial in the polynomial equals the non-zero rational scalar , where , and is the number of permutations on satisfying for each . (In particular, is a non-zero rational scalar, and is a positive integer.)
Proof.
For any , note that is by definition the rational coefficient of in relative to . If statement 10.4 holds, then the coefficient of in the polynomial is non-zero, so in particular, is not the zero polynomial. This implies that the subset , defined by the condition that the polynomial equation holds for all satisfying , is a Zariski dense subset of . Consequently, statement 10.4 implies statement 10.4.
Next, we shall prove statement 10.4 by induction on . The base case trivially follows from the definitions of and , so assume that . Let , and for each , define . By the cofactor expansion along the -th column,
[TABLE]
thus the rational coefficient of in relative to equals
[TABLE]
For each , let , and let
[TABLE]
Next, let be all the indices in that satisfy all of the following conditions. {enumerate*}
.
There exists a bijective map such that for all .
In particular, is among these indices , so . By Lemma 10.3, the coefficient of in the polynomial q_{\{F_{m}\},\{H_{m}\}}^{\{(G_{j},v_{j})\};{\bf w}}\big{(}X\big{)} is non-zero only if condition 10 holds, while the coefficient of in the polynomial q_{\mathcal{F}\backslash\{F_{m}\},\mathcal{H}^{\prime}}^{\mathcal{A}_{j};{\bf w}}\big{(}X\big{)} is non-zero only if condition 10 holds, thus it follows from (13) that the coefficient of in is identical to the coefficient of in the following polynomial:
[TABLE]
For every , notice that
[TABLE]
is the rational coefficient of in relative to , thus the definition of implies that the coefficient of in the linear polynomial q_{\{F_{m}\},\{H_{m}\}}^{\{(G_{j_{t}},v_{j_{t}})\};{\bf w}}\big{(}X\big{)} is the non-zero rational scalar .
Let , be subsets of , and note that . Also, let be the map such that for all . Treat as a polynomial in the variables in with coefficients in . (Here, denotes the polynomial ring on the set of variables with coefficients in .) Then from (14), the coefficient (contained in ) of in is the polynomial
[TABLE]
of degree .
Suppose . By induction hypothesis, the coefficient of in q_{\mathcal{F}\backslash\{F_{m}\},\mathcal{H}^{\prime}}^{\mathcal{A}_{j_{t}};-{\bf 1}}\big{(}X\big{)} is a positive integer that equals the number of bijective maps satisfying for all . This coefficient is positive, since by assumption, there is a bijective map such that for all . Thus, it follows from (15) that the coefficient of the monomial in (as a polynomial in variables in with rational coefficients) equals .
For every , note that . Observe that if we treat as a polynomial in the variables in with coefficients in , then any variable with a non-zero coefficient in must necessarily satisfy . Consequently, the coefficient of the monomial in (as a polynomial in variables with rational coefficients) equals [math]. Finally, from (12), we conclude that the coefficient of in equals . ∎
Theorem 10.5**.**
Let and . There exists a dense subset of such that for every PL realization in satisfying , the map defined by
[TABLE]
is injective for all .
Proof.
For each , note that is -autonomous, thus by Theorem 9.4, the set is pivot-compatible. In particular, there exists a pivotal linear order on such that . Observe that an equivalent reformulation of (1) yields
[TABLE]
Note also that is -Cohen–Macaulay, which implies for all (see [51, Thm. II.3.3]). If , then , thus it follows from (16), Theorem 8.3 and Theorem 10.1 that
[TABLE]
Consider an arbitrary integer satisfying . Let be an enumeration of all -faces in , and let be the ordered set consisting of the first -faces in . Then by Lemma 10.410.4, there exists a dense subset of such that for every satisfying , and every
[TABLE]
satisfying for all , the -support of contains . Since is -generic (cf. Lemma 7.1), we would then get that , hence would be a matrix of rank (i.e. of full rank). Now, define
[TABLE]
The intersection of finitely many dense subsets is dense, thus is a dense subset of . Henceforth, we shall fix some .
Let , and let . Given any , it follows from definition that
[TABLE]
By the definition of the pivotal linear order on , the value of on every pivot -face is completely determined once the values of on all non-pivot -faces are known. In particular,
[TABLE]
for every . Thus, (17) and (18) together imply that
[TABLE]
for all .
Since if and only if , it then follows from the definition of that the matrix has rank (i.e. full rank). Consequently, it follows from (19) that \big{(}\varphi_{\nu,{\bf w}}^{r}(x)\big{)}(F)=0 for all if and only if for all . Therefore, is injective. ∎
11. The weak Lefschetz property for homology spheres
An Artinian graded -algebra is said to have the weak Lefschetz property if there exists some such that the multiplication map (given by ) is either injective or surjective for all . This element is called a weak Lefschetz element of . In this section, we shall prove that if is a “sufficiently generic” PL realization of an -homology -sphere in , then the stress algebra has the weak Lefschetz property, where “sufficiently generic” here has a precise meaning; see Theorem 11.3. We end this section by completing the proof of Theorem 1.3 (i.e. the -conjecture for -homology spheres) using this weak Lefschetz property.
A key tool used in this section is the homological interpretation of skeletal rigidity developed by Tay–Whiteley [56], which requires some preparation. We shall build on the notation introduced in Section 4 and Section 10. Let be a PL realization of a simplicial -complex in . For each and , note that induces an equivalence relation on given by . Let be the quotient space . Given any chain complex , let denote the -th homology group of .
Let . The -skeletal chain complex of , which we denote by , is
[TABLE]
where the boundary map is defined by
[TABLE]
for all and all . From [56, Thm. 4.1(i)], we have
[TABLE]
as -vector spaces. For convenience, let
[TABLE]
for each .
Let be the cone on with conepoint , and let be a PL realization in . Fix a codimension subspace of that does not contain , and let be the central projection from to . Identify with , and let denote the PL realization of in determined by for all . Next, extend linearly so that it becomes the map .
Let be a sequence of homomorphisms determined by
[TABLE]
on elementary chains . As proven in [56, Thm. 8.1], is a (well-defined) surjective chain map. This chain map descends to a map on homology
[TABLE]
and it was further proven in [56, Thm. 8.2(iv)] that
[TABLE]
is an isomorphism.
Suppose is an arbitrary PL realization of a simplicial -complex in . Given , and any , let be a sequence of homomorphisms determined by
[TABLE]
on elementary chains .
Lemma 11.1**.**
{enumerate*}
* is a chain map.*
The chain maps and commute, i.e. if is a cone on with conepoint as above, then the following diagram
[TABLE]
commutes for all .
Proof.
It is straightforward to check that both and equal
[TABLE]
for all elementary chains , thus , i.e. is a chain map.
Also, we check that both and equal
[TABLE]
for all elementary chains , therefore and commute. ∎
Lemma 11.2**.**
Let be a simplicial -homology -sphere, and let be a cone on with conepoint . Let and . Then there exists a dense subset of such that for every PL realization in satisfying , the chain map induces an injective map
[TABLE]
for all .
Proof.
This is a straightforward translation of Theorem 10.5. In particular, we will have to use the isomorphism given in (20). ∎
Theorem 11.3**.**
Let be a simplicial -homology -sphere. Then there exists a dense subset of such that for every PL realization in satisfying , the stress algebra has the weak Lefschetz property. In particular, for every , there exists a weak Lefschetz element such that the multiplication map is injective for and surjective for .
Proof.
Let be a cone on with conepoint , and let be a PL realization in . Recall that is a surjective chain map, so we have the following short exact sequence of chain complexes
[TABLE]
where is the chain map induced by the natural inclusions . Thus by the snake lemma, we have the following induced long exact sequence:
[TABLE]
where denotes the connecting homomorphism.
Let and . Lemma 11.1 says that the chain maps and commute, thus for all , the following diagram commutes.
[TABLE]
From (21), the two middle horizontal maps of this commutative diagram are isomorphisms. Thus it follows from (20) that for all , the following diagram commutes.
[TABLE]
Now, by Lemma 11.2, there exists a dense subset of such that for every PL realization in satisfying , the second vertical map in the commutative diagram (22) is injective for all . Define
[TABLE]
and note that is a dense subset of . Henceforth, fix some such that .
Let . By the four lemma, the third vertical map in the commutative diagram (22) is injective. By definition,
[TABLE]
for all and all .
Recall that in the definition of , we have implicitly fixed some -face of . For each , let be the lifting of defined by , and for all remaining vertices . (Recall that denotes the projection map .) Let be the isomorphism given by Theorem 6.7, and note that Theorem 6.3 says is a basis for . Now, define
[TABLE]
and observe that by definition. Since the vertices of are realized in general position, it thus follows from the definition of the multiplication of stresses that
[TABLE]
for all , which implies that the multiplication map is injective for all . It then follows that the dual map is surjective for all .
Finally, since is Gorenstein (by Corollary 8.4), it follows from [51, Thm. I.12.5] and [51, Thm. I.12.10] that , hence is surjective for all , and therefore has the weak Lefschetz property. ∎
We now complete the proof of the -conjecture for -homology spheres.
Proof of Theorem 1.3.
By Theorem 11.3, we can choose some -generic PL realization of in , such that the stress algebra has a weak Lefschetz element for which the multiplication map is injective for , and surjective for . Let be the graded quotient ring . For each , consider the following short exact sequence:
[TABLE]
By Theorem 10.1 and Theorem 8.3, the -vector of is the -vector of . Consequently, by the additivity of dimensions (of -vector spaces) on exact sequences, we get (where ), thus the -vector of the truncated ring is the -vector of , and therefore the -vector of is an M-vector.∎
12. Further Remarks
12.1. Homology spheres over other fields.
Our proof of Theorem 1.3 uses the assumption that is a homology sphere over the reals in several instances. To show both the Gorenstein property and the weak Lefschetz property in Theorem 1.4, we used Mordell’s result on algebraic number fields (Lemma 7.1), as well as the fact that is a field extension of containing infinitely many quadratic fields. (Although we work with -homology spheres, our proof actually works for homology spheres over any subfield of that contains the square roots of all squarefree positive rational numbers.) To show that -generic PL realizations are dense among arbitrary PL realizations (Proposition 7.3), we used the fact that is an infinite metric space.
To relate rigidity theory to -vector theory, we used Theorem 10.1, which relates the -vector of an -Cohen–Macaulay simplicial -complex to the -vector of the stress algebra on a generic PL realization of in . Tay–Whiteley [56] gave a homological proof of Theorem 10.1, so an analogous statement holds for generic PL realizations of -Cohen–Macaulay simplicial -complexes in . (The definitions of PL realizations and stresses extend in the obvious way.) However, we do require PL realizations in to apply Rybnikov’s results in Section 6. Specifically, we used the fact that is an inner product space over an ordered field, so that the notion of outer unit normal vectors makes sense. Thus, without an analog of Maxwell–Cremona theory (and in particular, an analogous three-way interplay between liftings, reciprocals, and -stresses) over non-ordered fields, we do not know how to extend our proof of Theorem 1.3 to homology spheres over fields of non-zero characteristic.
12.2. Lefschetz properties.
Given an Artinian Gorenstein graded -algebra that is generated (as a -algebra) by , and whose socle is contained in , we say that has the strong Lefschetz property if there exists some such that for all . It is easy to see that the strong Lefschetz property implies the weak Lefschetz property. A less obvious fact is that this implication is strict: There are Artinian Gorenstein graded algebras that have the weak Lefschetz property but not the strong Lefschetz property [13] (cf. [1]).
Problem 12.1**.**
Let be a -generic PL realization of a simplicial -homology -sphere in . Is it possible for to have the weak Lefschetz property, but not the strong Lefschetz property?
The strong Lefschetz property for generic Artinian reductions of Stanley–Reisner rings is preserved under joins, connected sums, stellar subdivisions, and certain bistellar moves [1, 53]. Does an analogous statement hold for stress algebras? More fundamentally, how are Stanley–Reisner rings related to stress algebras?
Problem 12.2**.**
Given a PL realization of an -Cohen–Macaulay simplicial -complex in , is it possible to find an Artinian reduction (in terms of ) of the Stanley–Reisner ring of (over ) that is isomorphic to as graded -algebras?
The generalized lower bound conjecture (GLBC), posed by McMullen–Walkup [30] (cf. [12, Sec. 10]), characterizes “stacked” simplicial convex polytopes in terms of their -vectors. A -homology -ball is called -stacked if every face of dimension intersects the boundary non-trivially. A -homology -sphere is called -stacked if it is the boundary of an -stacked -homology -ball.
Conjecture 12.3** (McMullen–Walkup (1971)).**
Let be the boundary of a simplicial convex -polytope. For every , we have , with equality holding if and only if is -stacked.
The -theorem implies that , while McMullen–Walkup [30] had already proven (when they proposed the conjecture) that if is -stacked, then . Recently, Murai–Nevo [37] (cf. [38]) proved the remaining (difficult) part of the conjecture. In fact, they proved the following more general result.
Theorem 12.4** ([37, Thm. 1.3]).**
Let be a simplicial -homology -sphere. If has characteristic [math], and if there exists an Artinian reduction of the Stanley–Reisner ring of (over ) that has the weak Lefschetz property, then for every , the equality implies is -stacked.
Is there an analog of Theorem 12.4 in terms of stress algebras? If so, then Theorem 1.4 would yield a proof of an extension of the GLBC to -homology spheres.
More recently, Klee–Novik [19] introduced the notion of “-vector” for balanced simplicial complexes, and they proposed a balanced analog of the GLBC in terms of -vectors. (A simplicial -complex is called balanced if its -skeleton, treated as a graph, admits a -coloring.) Soon after, Kubitzke–Murai [15] proved the first part of the balanced GLBC, i.e. if is the boundary of a balanced simplicial convex polytope, then the -vector of has non-negative entries. In their proof, they showed a certain “weaker Lefschetz property” for a particular Artinian reduction of the Stanley–Reisner ring of . Can we use stress algebras to show the remaining part of the balanced GLBC, as well as prove an extension of the balanced GLBC for balanced -homology spheres?
12.3. Extending the -conjecture to homology manifolds.
Kalai’s manifold -conjecture [44] (see also [18, 53]) is a far-reaching generalization of the -conjecture to orientable -homology manifolds without boundary. To state this conjecture, we need the notion of -vectors introduced by Kalai.
Given a simplicial -complex , define for each integer . Let , and for every , define
[TABLE]
Next, define for each , and define . We say that is the -vector of .
Conjecture 12.5** (Kalai’s manifold -conjecture).**
Let be an orientable simplicial -homology -manifold without boundary. A sequence of integers is the -vector of if and only if the following two conditions hold. {enumerate}*
* for all .*
* is an M-vector.*
Condition 12.5 was proven combinatorially by Novik [44], and subsequently proven algebraically by Novik–Swartz [46]. See [39, 47] for a general algebraic treatment of the face numbers of -Buchsbaum complexes. (-homology manifolds are examples of -Buchsbaum complexes.) Murai [35] showed that condition 12.5 is true when is the barycentric subdivision of an orientable simplicial -homology -manifold. Swartz [46, 53] (cf. [52]) proved that condition 12.5 holds if the generic Artinian reductions of the Stanley–Reisner rings of the links (in ) of at least vertices have the weak Lefschetz property. We believe that stress algebra analogs of the results by Swartz and Novik–Swartz hold.
To tackle Kalai’s manifold -conjecture, we pose the following two conjectures, which if true would prove Conjecture 12.5 for a large class of orientable -homology manifolds (without boundary).
Conjecture 12.6**.**
Let be an orientable simplicial -homology -manifold (without boundary), such that whenever . There exists a dense subset of all -generic PL realizations of in , such that for every , the stess algebra has the weak Lefschetz property.
Conjecture 12.7**.**
Let be a PL realization of an orientable simplicial -homology -manifold (without boundary) in . If the vertices are realized in general position, then for all .
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