aa r X i v : . [ m a t h . C O ] S e p g -vectors of manifolds with boundary Isabella Novik ∗ Department of MathematicsUniversity of WashingtonSeattle, WA 98195-4350, USA [email protected]
Ed Swartz
Department of MathematicsCornell UniversityIthaca, NY 14853-4201, USA [email protected]
September 17, 2019
Abstract
We extend several g -type theorems for connected, orientable homology manifolds withoutboundary to manifolds with boundary. As applications of these results we obtain K¨uhnel-type bounds on the Betti numbers as well as on certain weighted sums of Betti numbers ofmanifolds with boundary. Our main tool is the completion ˆ∆ of a manifold with boundary∆; it is obtained from ∆ by coning off the boundary of ∆ with a single new vertex. Weshow that despite the fact that ˆ∆ has a singular vertex, its Stanley–Reisner ring shares a fewproperties with the Stanley–Reisner rings of homology spheres. We close with a discussion ofa connection between three lower bound theorems for manifolds, PL-handle decompositions,and surgery. This paper is devoted to the study of face numbers of manifolds with boundary. While [23]established the best to-date lower bounds on the g -numbers of manifolds with boundary, ouremphasis here is on Macaulay-type inequalities involving the g -numbers.The quest for characterizing possible f -vectors of various classes of simplicial complexes or atleast establishing significant necessary conditions started about fifty years ago with McMullen’s g -conjecture [18] that posited a complete characterization of f -vectors of simplicial polytopes. Inten years, this conjecture became a theorem [8, 33]. This gave rise to algebraic and combinatorialversions of the g -conjecture for simplicial spheres. Very recently Adiprasito [1] announced a proofof the most optimistic algebraic version of this conjecture. In the late 1990s, Kalai proposed afar reaching generalization of the sphere g -conjecture to simplicial manifolds without boundary.The authors proved that the (weaker) algebraic version of the g -conjecture for spheres implies allthe enumerative consequences of Kalai’s manifold g -conjecture, see [27]. Furthermore, Murai andNevo [21] establsihed a ˜ g -variation of this result. In this paper we extend both of these statementsto manifolds with boundary.The main idea (that goes back to Kalai [13, Section 10]) is as follows: given a simplicial complex∆ whose geometric realization is a connected, orientable, homology manifold with boundary, we ∗ Research is partially supported by NSF grant DMS-1664865 and by Robert R. & Elaine F. Phelps Professorshipin Mathematics v . We then show that, despite the fact that ˆ∆ hasa singular vertex, a certain quotient of a generic Artinian reduction of the Stanley–Reisner ringof ˆ∆ enjoys several properties that Artinian reductions of the Stanley–Reisner rings of simplicialspheres have. This result together with the computation of the Hilbert function of this quotientallows us to extend virtually all known results on face numbers of orientable manifolds withoutboundary to the class of orientable manifolds with boundary.The main results and the structure of the paper are as follows. In Section 2 we discuss basicsof simplicial complexes and Stanley–Reisner rings. In particular, we review Gr¨abe’s theorem onlocal cohomology [12] and introduce our main player — the completion ˆ∆ of a manifold withboundary ∆. Section 3 is devoted to the Gorensteiness and the weak Lefschetz property of acertain quotient of the Stanley–Reisner ring of ˆ∆, see Theorem 3.1 and Corollary 3.5. Section4 computes the Hilbert function of this quotient, Theorem 4.1. This result is used in Section5 to establish two versions of g -theorems for manifolds with boundary, Theorems 5.1 and 5.3.In Section 6 we use these g -results to derive K¨uhnel-type bounds on the Betti numbers andcertain weighted sums of Betti numbers of manifolds with boundary. Finally, in Section 7, weexamine the combinatorial and topological consequences of some of the known inequalities for f -vectors of homology manifolds with boundary when they are sharp. More specifically, we discussa connection between three lower bound theorems for manifolds, PL-handle decompositions, andsurgery, see Theorems 7.2, 7.11, and 7.16. In this section we review the necessary background material on simplicial complexes and theirStanley–Reisner rings with a special emphasis on homology manifolds with and without boundaryas well as on singular simplicial complexes that have only one singular vertex. We refer the readerto [34, Chapter 2] and the papers [28, 29] for more details on the subject. A simplicial complex ∆ on the vertex set V is a collection of subsets of V that is closed underinclusion and contains all singletons { v } for v ∈ V . The elements of ∆ are called faces . Themaximal faces (with respect to inclusion) are called facets . The dimension of a face F ∈ ∆ isdim F := | F | − dimension of ∆ is the maximal dimension of its faces. A complex is pure if all of its facets have the same dimension. A complex ∆ is j -neighborly if every j -elementsubset of V is a face of ∆.Let ∆ be a simplicial complex and let F be a face of ∆. The star and the link of F in ∆ arethe following subcomplexes of ∆:st F = st ∆ F := { G ∈ ∆ | G ∪ F ∈ ∆ } , lk F = lk ∆ F := { G ∈ st ∆ F | G ∩ F = ∅} . In particular, the link of the empty face is the complex ∆ itself. We refer to the links of non-empty faces as proper links . The contrastar of F in ∆ (also known as the deletion of F from ∆)is cost F = cost ∆ F := { G ∈ ∆ | G F } . If v is a vertex, it is customary to write v ∈ ∆, st v ,lk v , and cost v instead of { v } ∈ ∆, st { v } , lk { v } , and cost { v } . (In fact, we will sometimes write2 − v instead of cost { v } .) Also, if v / ∈ V is a new vertex, then the cone over ∆ with apex v is v ∗ ∆ := ∆ ∪ { v ∪ F | F ∈ ∆ } .Throughout the paper, we fix an infinite field k . We denote by ˜ H ∗ (∆; k ) the reduced simplicialhomology of ∆ with coefficients in k and by ˜ β i (∆) := dim k ˜ H i (∆; k ) the i -th reduced Betti number .One of the central objects of this paper is a k -homology manifold . A pure ( d − k -homology manifold without boundary (or a closed k -homology manifold )if the homology (computed over k ) of every proper link of ∆, lk ∆ F , coincides with the homologyof a ( d − − | F | )-dimensional sphere. In this case, we write ∂ ∆ = ∅ . Similarly, a pure ( d − k -homology manifold with boundary if every proper linkof ∆, lk ∆ F , has the homology of a ( d − − | F | )-dimensional sphere or a ball (over k ), and the boundary complex of ∆, i.e., ∂ ∆ := n F ∈ ∆ | ˜ H ∗ (lk ∆ F ; k ) = 0 o ∪ {∅} , is a ( d − k -homology manifold without boundary. The faces of ∂ ∆ are called boundary faces . The non-boundary faces of ∆ are called interior faces . When the field plays norole we simply call ∆ a homology manifold (with or without boundary).The prototypical example of a homology manifold (with or without boundary) is a triangu-lation of a topological manifold (with or without boundary). A connected k -homology manifold∆ is orientable if the top homology of the pair (∆ , ∂ ∆) is 1-dimensional. In this case, (∆ , ∂ ∆)satisfies the usual Poincar´e–Lefschetz duality associated with orientable compact manifolds. Notethat an arbitrary triangulation of any topological manifold (orientable or not, with or withoutboundary) is an orientable Z / Z -homology manifold.A k -homology ( d − d − k -homology manifold without boundarythat has the same homology as the ( d − k -homology ( d − d − k -homology manifold with boundary whose homology is trivial and whoseboundary complex is a k -homology ( d − k -homology( d − k -homology ( d − k -homologymanifold without boundary is a k -homology sphere, while a proper link of a k -homology manifoldwith boundary is a k -homology sphere or ball.Let ∆ be a k -homology manifold with or without boundary and let v V be a new vertex.A key to most of our proofs is the completion of ∆, ˆ∆, defined as follows:ˆ∆ := ∆ ∪ ( v ∗ ∂ ∆) . Note that we define v ∗ ∅ = ∅ ; hence if ∆ is a homology manifold without boundary, then ˆ∆ = ∆.A pure simplicial complex Γ is a complex with at most one singularity if all of the vertex linksof Γ but possibly one are k -homology balls or spheres. This exceptional vertex is called a singular vertex; the other vertices are called non-singular . For instance, if ∆ is a k -homology manifoldwith boundary, ˆ∆ is a completion of ∆, and v = v , then both ˆ∆ and cost ˆ∆ v are complexes with(at most) one singular vertex, namely, v .When only topological properties of a space are relevant we may use capital roman letters.For instance, “If X is a d -dimensional ball, then its boundary Y is a ( d − .2 Face numbers and the Stanley–Reisner rings Let ∆ be a ( d − V . Denote by f i (∆) thenumber of i -dimensional faces of ∆. The f -vector of ∆ is f (∆) = ( f − (∆) , f (∆) , . . . , f d − (∆))and the h -vector of ∆ is h (∆) = ( h (∆) , h (∆) , . . . , h d (∆)), where h i (∆) := i X j =0 ( − i − j (cid:18) d − jd − i (cid:19) f j − (∆) . Let A = k [ x v | v ∈ V ] be a polynomial ring, and let m = ( x v | v ∈ V ) be the graded maximalideal of A . For F ⊆ V , write x F := Q v ∈ F x v . The Stanley–Reisner ideal I ∆ of ∆ is the ideal of A defined by I ∆ = ( x F | F ⊆ V, F / ∈ ∆) . The
Stanley–Reisner ring k [∆] of ∆ (over k ) is the quotient ring k [∆] = A/I ∆ . In particular, k [∆] is a graded ring; it is also a graded A -module. If dim ∆ = d −
1, then the Krull dimensionof k [∆] is d and the Hilbert series of k [∆] is given by ([34, Chapter II.1])Hilb( k [∆]; λ ) = P di =0 h i (∆) λ i (1 − λ ) d . A linear system of parameters (or l.s.o.p for short) for k [∆] is a sequence Θ = θ , . . . , θ d of d = dim ∆ + 1 linear forms in m such that k (∆ , Θ) := k [∆] / Θ k [∆]has Krull dimension zero (i.e., it is a finite-dimensional k -space). Since k is infinite, by theNoether normalization lemma, an l.s.o.p. always exists: a generic choice of θ , . . . , θ d does the job.The ring k (∆ , Θ) is called an
Artinian reduction of k [∆].We need a few more definitions. If M is a finitely-generated graded A -module, we let M j denote the j -th homogeneous component of M . For τ ∈ A , define (0 : M τ ) := { ν ∈ M | τ ν = 0 } .The socle of M is the following graded submodule of M :Soc M = \ v ∈ V (0 : M x v ) = { ν ∈ M | m ν = 0 } . In particular, for any choice of integers i < i < · · · < i ℓ , L ℓj =1 (Soc M ) i j is a submodule of M .For a standard graded k -algebra M = A/I of Krull dimension zero, this allows us to define the interior socle of M :Soc ◦ M := d − M i =0 (Soc M ) i , where d := max { j | M j = 0 } . We say that
A/I is a level algebra if Soc ◦ ( A/I ) = 0, and that
A/I is Gorenstein if it has a1-dimensional socle. Equivalently,
A/I is Gorenstein if it is level and dim k ( A/I ) d = 1.We are interested in the Hilbert functions of k (∆ , Θ) and its quotient k (∆ , Θ) := k (∆ , Θ) / Soc ◦ k (∆ , Θ) . efinition 2.1. Let ∆ be a ( d − θ , . . . , θ d be a generic l.s.o.p. for k [∆]. The h ′ - and the h ′′ -numbers of ∆ are defined by h ′ j (∆) := dim k k (∆ , Θ) j and h ′′ j (∆) := dim k k (∆ , Θ) j (for j ≥ k is suppressed from our notation, the h ′ - and h ′′ -numbers do depend on k . For any( d − h ′ j (∆) = 0 for all j > d (see [34, Proposition III.2.4(b)]),while h ′ d (∆) = h ′′ d (∆) = ˜ β d − (∆) (see [37, Theorem 4.1] and [3, Lemma 2.2(3)]). The followingtheorem collects several other known results on h ′ - and h ′′ -numbers. Theorem 2.2.
Let ∆ be a ( d − -dimensional simplicial complex.1. If ∆ is a k -homology sphere or a ball, then h ′ i (∆) = h i (∆) for all ≤ i ≤ d .2. If ∆ is a k -homology manifold with or without boundary, then h ′ i (∆) = h i (∆) − (cid:18) di (cid:19) i − X j =1 ( − i − j ˜ β j − (∆) ∀ ≤ i ≤ d.
3. If ∆ is a connected, orientable k -homology manifold without boundary, then h ′′ i (∆) = h ′ i (∆) − (cid:18) di (cid:19) ˜ β i − (∆) = h i (∆) − (cid:18) di (cid:19) i X j =1 ( − i − j ˜ β j − (∆) ∀ ≤ i ≤ d − .
4. If ∆ is a complex with (at most) one singular vertex u , then for all ≤ i ≤ d , h ′ i (∆) = h i (∆) − i − X j =1 ( − i − j (cid:18)(cid:18) d − i − (cid:19) ˜ β j − (∆) + (cid:18) d − i (cid:19) ˜ β j − (cost ∆ u ) (cid:19) . Part 1 of this theorem is due to Stanley [32], part 2 is due to Schenzel [31], part 3 is [27, Theorem1.3], and part 4 is a special case of [29, Theorem 4.7]. When ∆ is a k -homology manifold withboundary, part 4 allows us to compute the h ′ -numbers of ˆ∆. One of the goals of this paper is tounderstand the h ′′ -numbers of ˆ∆ where we in addition assume that ∆ is connected and orientable.This requires some results on the local cohomology of k [ ˆ∆] that we review in the next subsection. Let M be an arbitrary finitely-generated graded A -module. We denote by H i m ( M ) the i -th localcohomology of M with respect to m .For a simplicial complex ∆, Gr¨abe [12] gave a description of H i m ( k [∆]) and its A -modulestructure in terms of simplicial cohomology of the links of ∆ and the maps between them. When∆ is a complex with one singular vertex u , this description takes the following simple form. For F ∈ ∆, consider the i -th simplicial cohomology of the pair (∆ , cost ∆ F ) with coefficients in k : H iF := ˜ H i (∆ , cost ∆ F ; k ) ∼ = ˜ H i −| F | (lk ∆ F ; k ) . In particular, H i ∅ = ˜ H i (∆ , ∅ ; k ) = ˜ H i (∆; k ). If G ⊆ F ∈ ∆, we let ι ∗ : H iF (∆) → H iG (∆) be themap induced by inclusion ι : cost ∆ G → cost ∆ F .5 heorem 2.3. [Gr¨abe] Let ∆ be a ( d − -dimensional simplicial complex with one singular vertex u , and let − ≤ i < d − . Then H i +1 m ( k [∆]) − j = if j < ,H i ∅ (∆) if j = 0 ,H i { u } (∆) if j > . For every vertex w = u and any integer j , the map · x w : H i +1 m ( k [∆]) − ( j +1) → H i +1 m ( k [∆]) − j isthe zero map; on the other hand, · x u = -map if j < ,ι ∗ : H i { u } (∆) → H i ∅ (∆) if j = 0 , identity map: H i { u } (∆) → H i { u } (∆) if j > . The description of H d m ( k [∆]) is quite a bit more involved. To this end, for a monomial ρ ∈ A ,define the support of ρ by s ( ρ ) := { v ∈ V | x v divides ρ } . Let M (∆) be the set of all monomialsin A whose support is in ∆, and let M j (∆) := { ρ ∈ M (∆) | deg( ρ ) = j } . Theorem 2.4. [Gr¨abe]
Let ∆ be any ( d − -dimensional simplicial complex. Then for j ∈ Z , H d m ( k [∆]) − j = M ρ ∈M j (∆) H ρ , where H ρ = H d − s ( ρ ) (∆) , and the A -structure on the ρ -th component of the right-hand side is given by · x ℓ = -map if ℓ / ∈ s ( ρ ) ,ι ∗ : H d − s ( ρ ) (∆) → H d − s ( ρ/x ℓ ) (∆) if ℓ ∈ s ( ρ ) , but ℓ / ∈ s ( ρ/x ℓ ) , identity map: H d − s ( ρ ) (∆) → H d − s ( ρ/x ℓ ) (∆) if ℓ ∈ s ( ρ ) , and ℓ ∈ s ( ρ/x ℓ ) . If ∆ is a k -homology sphere and Θ is an arbitrary l.s.o.p. for k [∆], then k (∆ , Θ) is Gorenstein(see [34, Theorem II.5.1]). This result was extended in [27, Theorem 1.4] to connected, orientable k -homology manifolds without boundary: if ∆ is such a complex and Θ is an l.s.o.p. for k [∆],then k (∆ , Θ) is Gorenstein. Here we further extend this result to manifolds with boundary.Throughout this section, we let ∆ be a k -homology manifold with boundary. We assume that∆ is ( d − V , and so ˆ∆ has vertex set V := V ∪ { v } . The mainresult of this section is: Theorem 3.1.
Let ∆ be a connected, orientable k -homology manifold with boundary and let Θ be a generic l.s.o.p. for k [ ˆ∆] . Then k ( ˆ∆ , Θ) is Gorenstein. The proof relies on a few lemmas. For these lemmas we fix a vertex v of ∆. (Hence v is anon-singular vertex of ˆ∆.) Lemma 3.2.
Let ∆ be a ( d − -dimensional, k -homology manifold with boundary, and let v bea vertex of ∆ . Then for all j ≤ d − , ˜ β j ( ˆ∆) = ˜ β j (cost ˆ∆ v ) and ˜ β j (∆) = ˜ β j (cost ∆ v ) . roof: The proof is a simple application of the Mayer-Vietoris argument. Indeed, since v = v ,the link ˆ L := lk ˆ∆ v is a k -homology ( d − L := lk ∆ v is a k -homology( d − d − β j ( ˆ L ) = ˜ β j ( L ) = 0 for all j ≤ d −
3. Since, thestars st ∆ v and st ˆ∆ v are acyclic, considering the following portion˜ H j ( L ) → ˜ H j (cost ∆ v ) ⊕ ˜ H j (st ∆ v ) → ˜ H j (∆) → ˜ H j − ( L )of the Mayer-Vietoris sequence for ∆ and its analog for ˆ∆ yields the result. (cid:3) The following lemma is a generalization of [35, Proposition 4.24]. We set A := k [ x u | u ∈ V ]and A v := k [ x u | u ∈ V \ { v } ]. Observe that k [ ˆ∆] and k [cost ˆ∆ v ] have natural A -modulestructures (where multiplication by x v on k [cost ˆ∆ v ] is the zero map), while k [lk ˆ∆ v ] has a natural A v -module structure (if u = v is not in the link of v , then multiplication by x u is the zeromap). Let Θ = θ , . . . , θ d ∈ A be a generic l.s.o.p. for k [ ˆ∆], and hence also for k [cost ˆ∆ v ]. SinceΘ is generic, θ has non-vanishing coefficients. So by scaling the variables if necessary, we canwork in an isomorphic setting and assume w.l.o.g. that all coefficients of θ are equal to 1. Let θ ′ := θ − x v , and for j >
1, let θ ′ j = θ j − c j θ where c j is the coefficient of x v in θ j . Then θ ′ , θ ′ , . . . , θ ′ d can be viewed as elements of A v , with Θ v = { θ ′ , . . . , θ ′ d } ⊂ A v forming an l.s.o.p. for k [lk ˆ∆ v ]. Furthermore, the ring k (lk ˆ∆ v, Θ v ) inherits an A v -module structure, and defining x v · y := − θ ′ · y for y ∈ k (lk ˆ∆ v, Θ v )extends it to an A -module structure. Lemma 3.3.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary, and let v be a vertex of ∆ . Then the map φ v : k (lk ˆ∆ v, Θ v ) → ( x v ) k ( ˆ∆ , Θ) given by z → x v · z, is well-defined and is a graded isomorphism of A -modules (of degree ).Proof: The proof of [35, Proposition 4.24] shows that φ v is a well-defined and surjective homo-morphism of A -modules. Thus to complete the proof, it suffices to check that for 1 ≤ i ≤ d , thedimensions of k -spaces (cid:0) k (lk ˆ∆ v, Θ v ) (cid:1) i − and (cid:0) ( x v ) k ( ˆ∆ , Θ) (cid:1) i agree. Since v = v , the link lk ˆ∆ v is a k -homology sphere, and sodim k (cid:0) k (lk ˆ∆ v, Θ v ) (cid:1) i − = h i − (lk ˆ∆ v ) for all i ≤ d. (3.1)To compute dim k (cid:0) ( x v ) k ( ˆ∆ , Θ) (cid:1) i for i ≤ d , consider the following exact sequence, induced bythe natural surjection k [ ˆ∆] → k [cost ˆ∆ v ],0 → ( x v ) k ( ˆ∆ , Θ) → k ( ˆ∆ , Θ) → k (cost ˆ∆ v, Θ) → . (3.2)If i = d , then, since ∆ is connected and orientable, dim k (cid:0) k ( ˆ∆ , Θ) (cid:1) d = ˜ β d − ( ˆ∆) = 1 whiledim k k (cost ˆ∆ v, Θ) d = ˜ β d − (cost ˆ∆ v ) = 0. Hence, in this case eq. (3.2) implies thatdim k (cid:0) ( x v ) k ( ˆ∆ , Θ) (cid:1) d = dim k (cid:0) k ( ˆ∆ , Θ) (cid:1) d = 1 = ˜ β d − (lk ˆ∆ v ) = dim k (cid:0) k (lk ˆ∆ v, Θ v ) (cid:1) d − ,
7s desired.Thus for the rest of the proof we assume that 1 ≤ i ≤ d −
1. Since both ˆ∆ and cost ˆ∆ v arecomplexes with at most one singular vertex, namely v , and since cost ˆ∆ v = ∆, we infer fromTheorem 2.2(4) thatdim k (cid:0) k ( ˆ∆ , Θ) (cid:1) i − h i ( ˆ∆) = − i − X j =1 ( − i − j (cid:18)(cid:18) d − i − (cid:19) ˜ β j − ( ˆ∆) + (cid:18) d − i (cid:19) ˜ β j − (∆) (cid:19) , (3.3)and that a similar expression holds for dim k (cid:0) k (cost ˆ∆ v, Θ) (cid:1) i − h i (cost ˆ∆ v ): to obtain it, simplyreplace all occurrences of ˆ∆ on the right-hand side of (3.3) with cost ˆ∆ v and those of ∆ withcost ∆ v . Since according to Lemma 3.2, for i ≤ d −
1, these replacements do not affect the valueof the right-hand side of (3.3), we conclude that for all 1 ≤ i ≤ d − k (cid:0) ( x v ) k ( ˆ∆ , Θ) (cid:1) i by (3.2) = dim k (cid:0) k ( ˆ∆ , Θ) (cid:1) i − dim k (cid:0) k (cost ˆ∆ v, Θ) (cid:1) i = h i ( ˆ∆) − h i (cost ˆ∆ v )= h i − (lk ˆ∆ v ) by (3.1) = dim k (cid:0) k (lk ˆ∆ v, Θ v ) (cid:1) i − , where the penultimate step uses [2, Lemma 4.1]. The result follows. (cid:3) We are now in a position to prove Theorem 3.1. Our proof follows the same outline as theproof of [27, Theorem 1.4] with an additional twist at the end.
Proof of Theorem 3.1:
Let ∆ be a ( d − k -homologymanifold with boundary, and let Θ be a generic l.s.o.p. for k [ ˆ∆]. (As before, we assume w.l.o.g. thatall coefficients of θ are equal to 1.) Then dim k k ( ˆ∆ , Θ) d = ˜ β d − ( ˆ∆) = 1. Hence, we only needto verify that the socle of k ( ˆ∆ , Θ) = k ( ˆ∆ , Θ) / Soc ◦ k ( ˆ∆ , Θ) vanishes in all degrees j = d . Since (cid:0) Soc ◦ k ( ˆ∆ , Θ) (cid:1) d = 0 and (cid:0) Soc ◦ k ( ˆ∆ , Θ) (cid:1) d − = (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) d − , this does hold for j = d − j ≤ d −
2, and let y ∈ k ( ˆ∆ , Θ) j be such that x v · y ∈ (Soc k ( ˆ∆ , Θ)) j +1 for every vertex v of ˆ∆. We must show that y ∈ Soc k ( ˆ∆ , Θ). Assume first that v = v . Then the isomorphism ofLemma 3.3 implies that y v := φ − v ( x v · y ) ∈ ( k (lk ˆ∆ v, Θ v )) j is in the socle of k (lk ˆ∆ v, Θ v ). Sincelk ˆ∆ v is a k -homology ( d − k (lk ˆ∆ v, Θ v ) is Gorenstein, and hence its socle vanishes inall degrees ≤ d −
2. Therefore, y v = 0. We conclude that x v · y = φ v ( y v ) = 0 in k ( ˆ∆ , Θ) for all v = v . (3.4)Finally, to show that x v · y = 0 in k ( ˆ∆ , Θ), recall that θ = x v + P v = v x v , and so θ · y = x v · y + X v = v x v · y. (3.5)The left-hand side of (3.5) is zero in k ( ˆ∆ , Θ) = k [ ˆ∆] / Θ k [ ˆ∆]. Furthermore, by (3.4), all summandson the right-hand side of (3.5), except possibly x v · y , are zeros in k ( ˆ∆ , Θ). Thus x v · y must bezero in k ( ˆ∆ , Θ). The result follows. (cid:3)
We now turn to some consequences of Theorem 3.1. As the Hilbert function of a Gorensteingraded k -algebra of Krull dimension zero is always symmetric, one immediate corollary is Corollary 3.4.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary. Then h ′′ i ( ˆ∆) = h ′′ d − i ( ˆ∆) for all ≤ i ≤ d . k -homology ( m − m − weak Lefschetzproperty over k (the WLP, for short) if for a generic lsop Θ for k [Γ] and an additional generic linearform ω , the map · ω : k (Γ , Θ) ⌊ m ⌋ → k (Γ , Θ) ⌊ m ⌋ +1 is surjective. It was proved in [27, Theorem 3.2]that if Λ is a ( d − k -homology manifold without boundary,and if all but at most d vertex links of Λ have the WLP over k , then for generic Θ and ω , themap · ω : k (Λ , Θ) i → k (Λ , Θ) i +1 is an injection for i < ⌊ d ⌋ and is a surjection for i ≥ ⌈ d ⌉ . Theproof relied on [35, Theorem 4.26] and on the Gorenstein property of k (Λ , Θ) established in [27,Theorem 1.4]). Noting that [35, Theorem 4.26] continues to hold for ˆ∆ and using Theorem 3.1instead of [27, Theorem 1.4], but leaving the rest of the proof of [27, Theorem 3.2] intact, yieldsthe following generalization:
Corollary 3.5.
Let ∆ be a ( d − -dimensional connected, orientable k -homology manifold withboundary.1. If d ≥ , then the map · ω : k ( ˆ∆ , Θ) i → k ( ˆ∆ , Θ) i +1 is an injection for i ≤ and is asurjection for i ≥ d − .2. If for all vertices v of ∆ , the link lk ˆ∆ v has the WLP over k , then the map · ω : k ( ˆ∆ , Θ) i → k ( ˆ∆ , Θ) i +1 is an injection for all i < ⌊ d ⌋ and is a surjection for all i ≥ ⌈ d ⌉ . Remark 3.6.
By a result of Stanley [33], the boundary complexes of all simplicial polytopeshave the WLP over Q . Furthermore, it follows from [19, Cor. 3.5] and [39] that all triangulationsof 2-dimensional spheres have the WLP over any infinite field. (This result is the reason noassumption on vertex links is needed in part 1 of the corollary.) A very recent preprint byAdiprasito [1] announces a spectacular generalization of these theorems: for an arbitrary infinitefield k , every k -homology sphere has the weak Lefschetz (and even strong Lefschetz) propertyover k , and so the hypothesis of the WLP assumption in the statement of Corollary 3.5 as wellas in the rest of the paper might be unnecessary.To apply results of this section to the study of face numbers of homology manifolds withboundary, we first need to work out the h ′′ -numbers of ˆ∆, that is, the Hilbert function of k ( ˆ∆ , Θ).This is done in the next section. h ′′ -numbers of ˆ∆ In this section we prove the following extension of Theorem 2.2(3) to manifolds with boundary.
Theorem 4.1.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary and let Θ be a generic l.s.o.p. for k [ ˆ∆] . Then for all i < d ,1. dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) i = (cid:0) d − i − (cid:1) ˜ β i − ( ˆ∆) + (cid:0) d − i (cid:1) ˜ β i − (∆) , and2. h ′′ i ( ˆ∆) = h i ( ˆ∆) − P ij =1 ( − i − j (cid:16)(cid:0) d − i − (cid:1) ˜ β j − ( ˆ∆) + (cid:0) d − i (cid:1) ˜ β j − (∆) (cid:17) . Remark 4.2.
It is instructive to rewrite both formulas of the theorem purely in terms of ∆.Indeed, by connectivity, ˜ β ( ˆ∆) = ˜ β (∆) = 0, while˜ H j − ( ˆ∆; k ) ∼ = ˜ H j − ( ˆ∆ , st ˆ∆ v ; k ) ∼ = ˜ H j − (∆ , ∂ ∆; k ) ∼ = ˜ H d − j (∆; k ) ∀ j < d, h i ( ˆ∆) = h i (∆) + h i − (cid:0) lk ˆ∆ v (cid:1) = h i (∆) + h i − ( ∂ ∆)(see [2, Lemma 4.1]). Thus, for i < d , Theorem 4.1 can be rewritten as1. dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) i = (cid:0) d − i (cid:1) ˜ β i − (∆) + (cid:0) d − i − (cid:1) ˜ β d − i (∆);2. h ′′ i ( ˆ∆) = h i (∆) + h i − ( ∂ ∆) − P ij =2 ( − i − j (cid:16)(cid:0) d − i (cid:1) ˜ β j − (∆) + (cid:0) d − i − (cid:1) ˜ β d − j (∆) (cid:17) . Note that if ∆ is a connected, orientable k -homology manifold without boundary, then (1) ˆ∆ = ∆,(2) ˜ β d − j (∆) = ˜ β j − (∆) for all 1 < j < d (by Poincar´e duality), and (3) h i − ( ∂ ∆) = 0 for all i (since ∂ ∆ = ∅ ). In this case, the above formula for h ′′ i ( ˆ∆) reduces to Theorem 2.2(3).To prove Theorem 4.1, several lemmas are in order. As in the previous section, we continueto assume that Θ is a generic l.s.o.p. for k [ ˆ∆] and that all coefficients of θ are equal to 1. Lemma 4.3.
Let ∆ be a ( d − -dimensional k -homology manifold with boundary. Then (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) i ∼ = d − M j =0 (cid:18) d − j (cid:19)(cid:0) H j m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1)(cid:1) i − j M ( SB ) i − ( d − ∀ i ∈ Z , where SB is a graded submodule of Soc H d − m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1) . Furthermore, for j ≤ d − , dim k (cid:0) H j m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1)(cid:1) ℓ = ˜ β j ( ˆ∆) if ℓ = 1˜ β j − (∆) if ℓ = 00 otherwise . Proof:
Since ˆ∆ has at most one singularity, Lemma 4.3(2) of [29] implies that k [ ˆ∆] /θ k [ ˆ∆] is aBuchsbaum A -module of Krull dimension d −
1. The first part of the statement then follows from[28, Theorem 2.2], while the second part follows from [29, Lemma 4.3(1) and Theorem 4.7]. (cid:3)
We now turn our attention to the submodule SB of Soc H d − m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1) . Proposition 4.4.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifoldwith boundary. Then, for all ℓ ≤ − , (cid:0) Soc H d − m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1)(cid:1) ℓ = 0 , and hence ( SB ) ℓ = 0 .Proof: Since depth k [ ˆ∆] ≥ θ is a non-zero divisor on k [ ˆ∆]; in other words, the sequence0 → k [ ˆ∆]( − · θ −→ k [ ˆ∆] −→ k [ ˆ∆] /θ k [ ˆ∆] → A -module M , M ( −
1) denotes M with grading defined by M ( − ℓ = M ℓ − .)The above sequence induces a long exact sequence in local cohomology. In particular, the part H d − m (cid:0) k [ ˆ∆] (cid:1) ( − · θ −→ H d − m (cid:0) k [ ˆ∆] (cid:1) −→ H d − m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1) −→ H d m (cid:0) k [ ˆ∆] (cid:1) ( − · θ −→ H d m (cid:0) k [ ˆ∆] (cid:1) is exact. Thus, H d − m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1) , considered as a vector space, is isomorphic to the direct sumof C := Coker (cid:2) H d − m (cid:0) k [ ˆ∆] (cid:1) ( − · θ −→ H d − m (cid:0) k [ ˆ∆] (cid:1)(cid:3) and K := Ker (cid:2) H d m (cid:0) k [ ˆ∆] (cid:1) ( − · θ −→ H d m (cid:0) k [ ˆ∆] (cid:1)(cid:3) .10uthermore, on the K -part of H d − m (cid:0) k [ ˆ∆] /θ k [ ˆ∆] (cid:1) , the A -module structure is induced by the A -module structure on H d m (cid:0) k [ ˆ∆] (cid:1) .Since ˆ∆ has (at most) one singular vertex, namely v , Theorem 2.3 implies that for ℓ ≤ − · θ : (cid:0) H d − m (cid:0) k [ ˆ∆] (cid:1)(cid:1) ℓ − → (cid:0) H d − m (cid:0) k [ ˆ∆] (cid:1)(cid:1) ℓ is the identity map. Hence its cokernel, C ℓ ,vanishes for all ℓ ≤ −
1. Therefore, it only remains to show that the socle (Soc K ) ℓ , vanishes forall ℓ ≤ −
1. Indeed, by definition of socles,(Soc K ) ℓ = (cid:16) Soc Ker (cid:2) · θ : H d m (cid:0) k [ ˆ∆] (cid:1) ( − −→ H d m (cid:0) k [ ˆ∆] (cid:1)(cid:3)(cid:17) ℓ = (cid:16) Soc H d m (cid:0) k [ ˆ∆] (cid:1) ( − (cid:17) ℓ = (cid:16) Soc H d m (cid:0) k [ ˆ∆] (cid:1)(cid:17) ℓ − . The following lemma verifies that the latter term vanishes, and thus completes the proof. (cid:3)
Lemma 4.5.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary. Then, for all ℓ ≥ , (cid:16) Soc H d m (cid:0) k [ ˆ∆] (cid:1)(cid:17) − ℓ = 0 . Proof:
Recall that by Theorem 2.4, H d m ( k [ ˆ∆]) − ℓ = M ρ ∈M ℓ ( ˆ∆) H ρ , where H ρ = H d − s ( ρ ) ( ˆ∆) . (4.1)Fix ℓ ≥
2, and let ρ ∈ M ℓ ( ˆ∆). Then either ρ is divisible by x v for some vertex v of ˆ∆ (possibly v )or ρ is a squarefree monomial whose support has size at least two: s ( ρ ) ⊇ { v, w } . In the formercase, by Theorem 2.4, the multiplication map · x v : H ρ → H ρ/x v is the identity map, and so nonon-zero element of H ρ is in the socle. In the latter case, at least one of v, w is not v . Assumewithout loss of generality that w = v , and consider the map · x v : H ρ → H ρ/x v , which by Theorem2.4 is simply ι ∗ : H d − s ( ρ ) ( ˆ∆) → H d − s ( ρ/x v ) ( ˆ∆). We will show that this map is an isomorphism, andhence that no non-zero element of H ρ is in the socle in this case as well.Our argument is similar to the one used in the proof of [27, Theorem 2.1]. Denote by k ˆ∆ k thegeometric realization of ˆ∆, and by b ( ρ ) and b ( ρ/x v ) the barycenters of realizations of faces s ( ρ )and s ( ρ/x v ), respectively. Consider the following commutative diagram, where the maps f ∗ and j ∗ are induced by inclusion:˜ H d − (cid:0) k ˆ∆ k (cid:1) ( j ∗ ) − −−−−→ ˜ H d − (cid:0) k ˆ∆ k , k ˆ∆ k − b ( ρ/x v ) (cid:1) f ∗ −−−−→ ˜ H d − (cid:0) ˆ∆ , cost ˆ∆ s ( ρ/x v ) (cid:1)(cid:13)(cid:13)(cid:13) ι ∗ x ˜ H d − (cid:0) k ˆ∆ k (cid:1) ( j ∗ ) − −−−−→ ˜ H d − (cid:0) k ˆ∆ k , k ˆ∆ k − b ( ρ ) (cid:1) f ∗ −−−−→ ˜ H d − (cid:0) ˆ∆ , cost ˆ∆ s ( ρ ) (cid:1) . The two maps f ∗ are isomorphisms by the usual deformation retractions. Since w = v and w ∈ s ( ρ/x v ) ⊂ s ( ρ ), the links lk ˆ∆ s ( ρ ) and lk ˆ∆ s ( ρ/x v ) are k -homology spheres, so the four k -spaces on the right and in the middle of the diagram are 1-dimensional. Furthermore, since ∆ isconnected and orientable, the k -spaces on the left of the diagram are 1-dimensional and the two j ∗ -maps are isomorphisms, so that ( j ∗ ) − -maps are well-defined and are isomorphisms as well.This implies that ι ∗ is an isomorphism and completes the proof. (cid:3) We are now ready to prove Theorem 4.1. 11 roof of Theorem 4.1:
We prove both parts simulataneously. If d = 2, then ˆ∆ is a circle, inwhich case the statement is known. So assume d ≥
3. Lemma 4.3 and Proposition 4.4 imply thatthe formula for the dimension of the socle holds for all i ≤ d −
2. Together with Theorem 2.2(4)and Definition 2.1, this also implies that the formula for h ′′ i ( ˆ∆) holds for all i ≤ d −
2. Thus, itonly remains to show that the theorem holds for i = d −
1. Since by Corollary 3.4, the h ′′ -numbersof ˆ∆ are symmetric, to complete the proof of both parts, it suffices to check that the proposedexpression for h ′′ d − ( ˆ∆) is equal to h ′′ ( ˆ∆) = h ( ˆ∆).Let ˜ χ denote the reduced Euler characteristic. Note that since ˜ β d − ( ˆ∆) = 1 and ˜ β d − (∆) = 0,the proposed expression for h ′′ d − ( ˆ∆), h d − ( ˆ∆) − P d − j =1 ( − d − j − h ( d −
1) ˜ β j − ( ˆ∆) + ˜ β j − (∆) i , canbe rewritten as h d − ( ˆ∆) − ( d − (cid:16) − d ˜ χ ( ˆ∆) (cid:17) − ( − d ˜ χ (∆) . Thus to complete the proof, we only need to verify that h d − ( ˆ∆) = h ( ˆ∆) + ( d − (cid:16) − d ˜ χ ( ˆ∆) (cid:17) + ( − d ˜ χ (∆) . To do so, observe that for all i , f i ( ˆ∆) = f i (∆) + f i (st ˆ∆ v ) − f i ( ∂ ∆) . This, together with the fact that vertex stars are contractible, implies that˜ χ ( ∂ ∆) = ˜ χ (∆) + ˜ χ (st ˆ∆ v ) − ˜ χ ( ˆ∆) = ˜ χ (∆) − ˜ χ ( ˆ∆) . (4.2)Finally, according to [29, Theorem 3.1], h d − ( ˆ∆) = h ( ˆ∆) + d (cid:16) − d ˜ χ ( ˆ∆) (cid:17) − (cid:16) − d − ˜ χ ( ∂ ∆) (cid:17) by (4.2) = h ( ˆ∆) + ( d −
1) + d ( − d ˜ χ ( ˆ∆) + ( − d (cid:16) ˜ χ (∆) − ˜ χ ( ˆ∆) (cid:17) = h ( ˆ∆) + ( d − (cid:16) − d ˜ χ ( ˆ∆) (cid:17) + ( − d ˜ χ (∆) . The result follows. (cid:3) g -theorems for manifolds with boundary Algebraic results obtained in the two previous sections along with Macaulay’s characterization ofHilbert functions of homogeneous quotients of polynomial rings allow us to easily derive severalnew enumerative results on face numbers of k -homology manifolds with boundary. This section isdevoted to results that generalize and are similar in spirit to the g -theorem for simplicial polytopes.We follow the custom and define g i := h i − h i − , g ′ i := h ′ i − h ′ i − , and g ′′ i := h ′′ i − h ′′ i − .We start by recalling that given positive integers a and i , there is a unique way to write a = (cid:18) a i i (cid:19) + (cid:18) a i − i − (cid:19) + · · · + (cid:18) a j j (cid:19) , where a i > a i − > · · · > a j ≥ j ≥ . a h i i := (cid:18) a i + 1 i + 1 (cid:19) + (cid:18) a i − + 1 i (cid:19) + · · · + (cid:18) a j + 1 j + 1 (cid:19) and 0 h i i := 0 . Macaulay’s theorem [34, Theorem II.2.2] asserts that a (possibly infinite) sequence ( b , b , . . . ) ofintegers is the Hilbert function of a homogeneous quotient of a polynomial ring if and only if b = 1 and 0 ≤ b ℓ +1 ≤ b h ℓ i ℓ for all ℓ ≥
1. A sequence that satisfies these conditions is called an M -vector .Our first g -type result is an extension of [27, Theorem 3.2] to manifolds with boundary: Theorem 5.1.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary, and let h ′′ i ( ˆ∆) = h i (∆) + h i − ( ∂ ∆) − P ij =2 ( − i − j (cid:16)(cid:0) d − i (cid:1) ˜ β j − (∆) + (cid:0) d − i − (cid:1) ˜ β d − j (∆) (cid:17) for i < d and h ′′ d ( ˆ∆) = 1 . Then1. h ′′ i ( ˆ∆) = h ′′ d − i ( ˆ∆) for all ≤ i ≤ d .2. If d ≥ , then (cid:0) , g ′′ ( ˆ∆) , g ′′ ( ˆ∆) (cid:1) is an M -vector.3. If for all vertices v of ∆ , lk ˆ∆ v has the WLP over k , then (cid:0) , g ′′ ( ˆ∆) , g ′′ ( ˆ∆) , · · · , g ′′⌊ d ⌋ ( ˆ∆) (cid:1) isan M -vector.Proof: The expressions for h ′′ i ( ˆ∆) are from Remark 4.2(2). Part 1 is the content of Corollary 3.4.Furthermore, it follows from Corollary 3.5 and Theorem 4.1/Remark 4.2 that under our assump-tions, for generic Θ and ω , and for i ≤ i ≤ ⌊ d ⌋ in part 3,dim k (cid:16) k ( ˆ∆ , Θ) /ω k ( ˆ∆ , Θ) (cid:17) i = g ′′ i ( ˆ∆ , Θ) . Together with Macaulay’s theorem, this completes the proof. (cid:3)
Remark 5.2.
Applying the same reasoning to k ( ˆ∆ , Θ) / L ℓj =0 (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) j instead of k ( ˆ∆ , Θ),part 3 of Theorem 5.1 can be strengthened to the statement that (cid:18) , g ′′ ( ˆ∆) , · · · , g ′′ ℓ ( ˆ∆) , g ′′ ℓ +1 ( ˆ∆) + (cid:18) d − ℓ + 1 (cid:19) ˜ β ℓ (∆) + (cid:18) d − ℓ (cid:19) ˜ β d − ℓ − (∆) (cid:19) is an M -vector for every ℓ < ⌊ d ⌋ (cf. discussion at the bottom of page 995 in [27]).Our second g -type result is an extension of [21, Theorem 5.4(i)] to manifolds with boundary.To this end, in the spirit of [21, Section 5], for a ( d − k -homology manifold with boundary, ∆, and for r ≤ ⌊ d/ ⌋ , define˜ g r ( ˆ∆) := g ′′ r ( ˆ∆) − (cid:18)(cid:18) d − r − (cid:19) ˜ β r − (∆) + (cid:18) d − r − (cid:19) ˜ β d − r (∆) (cid:19) (5.1)= g r (∆) + g r − ( ∂ ∆) − r X j =2 ( − r − j (cid:18)(cid:18) dr (cid:19) ˜ β j − (∆) + (cid:18) dr − (cid:19) ˜ β d − j (∆) (cid:19) , (5.2)where the last equality follows from Remark 4.2(2).13 heorem 5.3. Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary.1. If d ≥ , then (cid:0) , ˜ g ( ˆ∆) , ˜ g ( ˆ∆) (cid:1) is an M -vector.2. If for all vertices v of ∆ , lk ˆ∆ v has the WLP over k , then (cid:0) , ˜ g ( ˆ∆) , ˜ g ( ˆ∆) , · · · , ˜ g ⌊ d ⌋ ( ˆ∆) (cid:1) isan M -vector.Proof: Observe that by definition of ˜ g r ( ˆ∆),˜ g r ( ˆ∆) = h ′′ r ( ˆ∆) − h ′′ r − ( ˆ∆) − (cid:18)(cid:18) d − r − (cid:19) ˜ β r − (∆) + (cid:18) d − r − (cid:19) ˜ β d − r (∆) (cid:19) = h ′′ d − r ( ˆ∆) − h ′′ d − r +1 ( ˆ∆) − (cid:18)(cid:18) d − d − r + 1 (cid:19) ˜ β d − r (∆) + (cid:18) d − d − r (cid:19) ˜ β d − ( d − r +1) (∆) (cid:19) = h ′′ d − r ( ˆ∆) − h ′ d − r +1 ( ˆ∆) , (5.3)where the middle step is by Corollary 3.4 and the last step is by Remark 4.2(1). The rest of theproof follows the proof of [21, Theorem 5.4(i)]: the only change is that we rely on Theorem 3.1that asserts Gorensteinness of k ( ˆ∆ , Θ) instead of [27, Theorem 1.4] that asserts Gorensteinnessof the analogous ring associated with a manifold without boundary. (cid:3)
Remark 5.4.
Assume that for all vertices v of ∆, lk ˆ∆ v has the WLP over k and that for all boundary vertices v of ∆, lk ∂ ∆ v has the WLP over k ; assume also that r ≤ ⌊ ( d − / ⌋ . Underthese assumptions the non-negativity part of Theorem 5.3(2) is not new: the fact that ˜ g r ( ˆ∆) ≥ , ∂ ∆). For a detailed treatment of the case d ≥ r = 2 see theproof of Proposition 7.15. The results of previous sections can also be used to extend known K¨uhnel-type bounds on theBetti numbers (and their sums) of manifolds without boundary to the case of manifolds withboundary. Deriving such bounds is the goal of this section.Specifically, Theorem 5.3 in [20] asserts that if ∆ is a ( d − k -homology manifold without boundary that has n vertices, then (cid:0) d +12 (cid:1) ˜ β (∆) ≤ (cid:0) n − d (cid:1) as long as d ≥
4. Furthermore, Theorem 5.1 in [20] asserts that if, in addition, all vertex links of ∆ havethe WLP over k , then (cid:0) d +1 r +1 (cid:1) ˜ β r (∆) ≤ (cid:0) n − d − rr +1 (cid:1) for all r ≤ ⌊ d ⌋ −
1. (The conjecture that for r ≤ ⌊ d/ ⌋ − d − n vertices, theinequality (cid:0) d +1 r +1 (cid:1) ˜ β r (∆) ≤ (cid:0) n − d − rr +1 (cid:1) holds is due to K¨uhnel [17, Conjecture 18].) In the specialcase of orientable k -homology manifolds without boundary the same results were proved in [28,Theorem 5.2] and [26, Theorem 4.3], respectively. An easy adaptation of proofs from [28, 26]combined with our results from the previous sections leads to the following extension. We do notknow if this extension also holds in the non-orientable case. Theorem 6.1.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary, and assume that f (∆) = n . . If d ≥ , then (cid:0) d (cid:1) ˜ β (∆) + (cid:0) d (cid:1) ˜ β d − (∆) ≤ (cid:0) n − d +12 (cid:1) . If equality holds, then ∆ is -neighborlyand has no interior vertices.2. If for all vertices v of ∆ , lk ˆ∆ v has the WLP over k , then (cid:18) dr + 1 (cid:19) ˜ β r (∆) + (cid:18) dr (cid:19) ˜ β d − r − (∆) ≤ (cid:18) n − d + rr + 1 (cid:19) for all r ≤ ⌊ d ⌋ − . If equality holds, then ∆ is ( r + 1) -neighborly and has no interior faces of dimension ≤ r − .Proof: Since the proof is very similar to that of [26, Theorem 4.3], we omit some of the details.Fix an integer r : r = 1 for part 1 and any r ≤ ⌊ d ⌋ − g r +1 ( ˆ∆) is nonnegative. Hence0 ≤ h ′′ r +1 ( ˆ∆) − h ′′ r ( ˆ∆) − (cid:18)(cid:18) d − r (cid:19) ˜ β r (∆) + (cid:18) d − r − (cid:19) ˜ β d − r − (∆) (cid:19) by Remark 4.2(1) = h ′ r +1 ( ˆ∆) − h ′′ r ( ˆ∆) − (cid:18)(cid:18) dr + 1 (cid:19) ˜ β r (∆) + (cid:18) dr (cid:19) ˜ β d − r − (∆) (cid:19) . We conclude that (cid:0) dr +1 (cid:1) ˜ β r (∆) + (cid:0) dr (cid:1) ˜ β d − r − (∆) ≤ h ′ r +1 ( ˆ∆) − h ′′ r ( ˆ∆). Thus, to complete the proof,it suffices to show that h ′ r +1 ( ˆ∆) − h ′′ r ( ˆ∆) ≤ (cid:0) n − d + rr +1 (cid:1) and that if equality holds then ˆ∆ is ( r + 1)-neighborly. (The latter condition implies that ∆ is ( r + 1)-neighborly and that all faces of ∆ ofcardinality ≤ r are in the link of v , and hence that they are boundary faces.)Indeed, since f (∆) = n , h ′ ( ˆ∆) = n − d +1. Macaulay’s theorem applied to k ( ˆ∆ , Θ), then showsthat h ′ r +1 ( ˆ∆) = (cid:0) x +1 r +1 (cid:1) for some real number x ≤ n − d + r . Another application of Macaulay’stheorem, this time to k ( ˆ∆ , Θ) / (Soc k ( ˆ∆ , Θ)) r , yields that h ′ r +1 ( ˆ∆) ≤ ( h ′′ r ( ˆ∆)) h r +1 i , and hencethat h ′′ r ( ˆ∆) ≥ (cid:0) xr (cid:1) . Therefore, h ′ r +1 ( ˆ∆) − h ′′ r ( ˆ∆) ≤ (cid:0) xr +1 (cid:1) ≤ (cid:0) n − d + rr +1 (cid:1) , as desired. Furthermore,if h ′ r +1 ( ˆ∆) − h ′′ r ( ˆ∆) = (cid:0) n − d + rr +1 (cid:1) , then dim k k (cid:0) ˆ∆ , Θ (cid:1) r +1 = h ′ r +1 ( ˆ∆) = (cid:0) n − d + r +1 r +1 (cid:1) , which, in turn,implies that ˆ∆ is ( r + 1)-neighborly. (cid:3) Corollary 6.2.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifold withboundary, and assume that f (∆) = n .1. If d ≥ , then ˜ β (∆) ≤ (cid:0) n − d +12 (cid:1) / (cid:0) d (cid:1) . In particular, if ˜ β (∆) = 0 , then n ≥ d − .2. If for all vertices v of ∆ , lk ˆ∆ v has the WLP over k , then ˜ β r (∆) ≤ (cid:0) n − d + rr +1 (cid:1) / (cid:0) dr +1 (cid:1) for all r ≤ ⌊ d ⌋ − . Consequently, if ˜ β r (∆) = 0 , then n ≥ d − r . Similarly, if both ˜ β r (∆) and ˜ β d − r − are non-vansishing, then n ≥ d − r + 1 . The bounds on the number of vertices in the above corollary are similar in spirit to thebounds established by Brehm and K¨uhnel [10, Theorem B] on the number of vertices that an( r − r -connected closed PL manifold must have. Example 6.3.
K¨uhnel [14] (see also [16]) constructed for every d ≥
3, a ( d − d , with exactly 2 d − d = 3, this gives a unique 5-vertex triangulation of the M¨obius band.) His constructionthus provides a family of connected, orientable over Z / Z manifolds with boundary that havenon-vanishing ˜ β and achieve equalities in both statements of Corollary 6.2(1).15e now turn to K¨uhnel-type bounds on certain weighted sums of Betti numbers. It wasconjectured by K¨uhnel [15, Conjecture B] and proved in [28, Theorem 4.4] (see also [25, Theo-rem 7.6]) that if Λ is a 2 k -dimensional, orientable k -homology manifold without boundary, then( − k (cid:0) ˜ χ (Λ) − (cid:1) ≤ (cid:0) f (Λ) − k − k +1 (cid:1) / (cid:0) k +1 k +1 (cid:1) . In fact, the proof showed that the same upper boundapplies to ˜ β k (Λ) + ˜ β k − (Λ) + 2 P k − i =0 ˜ β i (Λ). The methods of [25, 28] combined with our resultsfrom Sections 3 and 4 lead to the following extension of this result to manifolds with boundary. Theorem 6.4.
Let ∆ be a connected, orientable, k -homology manifold with boundary. If ∆ is k -dimensional and has n vertices, then ˜ β k (∆) + k X i =2 (cid:0) n − k − k +1 (cid:1)(cid:0) k +1 k +1 (cid:1)(cid:0) n − k − ii (cid:1) · (cid:18)(cid:18) ki (cid:19) ˜ β i − (∆) + (cid:18) ki − (cid:19) ˜ β k +1 − i (∆) (cid:19) ≤ (cid:0) n − k − k +1 (cid:1)(cid:0) k +1 k +1 (cid:1) . (6.1) Equality holds if and only if ∆ is ( k +1) -neighborly and has no interior faces of dimension ≤ k − . Examples that achieve equality include ( k + 1)-neighborly triangulations of closed manifoldsof dimension 2 k with one vertex removed. Before proving Theorem 6.4 we discuss some of itsconsequences. Corollary 6.5.
Let ∆ be a connected, orientable, k -homology manifold with boundary. Assume ∆ is k -dimensional and has n vertices. Then1. ˜ β k (∆) ≤ ( n − k − k +1 )( k +1 k +1 ) . In particular, if ˜ β k (∆) = 0 , then n ≥ k + 2 .2. If n ≥ k + 2 , then ˜ β k (∆) + P k − i =2 ˜ β i − (∆) ≤ ( n − k − k +1 )( k +1 k +1 ) .3. If n ≥ k + 2 , then P k +1 i =2 ˜ β i − (∆) ≤ ( n − k − k +1 )( k +1 k +1 ) . To derive parts 2 and 3 of Corollary 6.5 from Theorem 6.4, use routine computations withbinomial coefficients to show that if n ≥ k + 2, then the coefficient of ˜ β i − in (6.1) is at least 1for all i ≤ k + 1, while if n ≥ k + 2, then such a coefficient is ≥ i ≤ k + 1 except possiblyfor i = k . The proof of Theorem 6.4 is very similar to that of [28, Theorem 4.4], and so we onlysketch the main details. Proof of Theorem 6.4 (Sketch): Let N p := (cid:0) f ( ˆ∆) − (2 k +1)+ p − p (cid:1) = (cid:0) n − k − pp (cid:1) . In particular, N k +1 − N k = (cid:0) n − k − k +1 (cid:1) .Applying Macaulay’s theorem to k ( ˆ∆ , Θ) / (Soc k ( ˆ∆ , Θ)) i , yields that h ′ i +1 ( ˆ∆) ≤ ( h ′′ i ( ˆ∆)) h i +1 i ≤ N i +1 N i h ′′ i ( ˆ∆) = N i +1 N i (cid:16) h ′ i ( ˆ∆) − dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) i (cid:17) for all i ≤ k. Iterating this process (see the proof of [28, Theorem 4.4] for more details), we obtain that h ′ k +1 ( ˆ∆) − h ′ k ( ˆ∆) ≤ (6.2)( N k +1 − N k ) − " N k +1 N k dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) k + N k +1 − N k N k k − X i =2 N k N i dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) i .
16n the other hand, since h ′ i ( ˆ∆) = h ′′ i ( ˆ∆) + dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) i and since h ′′ k +1 ( ˆ∆) = h ′′ k ( ˆ∆) byCorollary 3.4, it follows that h ′ k +1 ( ˆ∆) − h ′ k ( ˆ∆) = dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) k +1 − dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) k . (6.3)Combining equations (6.2) and (6.3), we conclude thatdim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) k +1 + N k +1 − N k N k k X i =2 N k N i dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) i ≤ N k +1 − N k . Substituting expressions for the dimensions of graded components of the socle from Remark 4.2(1)and, in particular, noting that dim k (cid:0) Soc k ( ˆ∆ , Θ) (cid:1) k +1 = (cid:0) k +1 k (cid:1) ˜ β k (∆), yields the inequality. Thetreatment of equality case is almost identical to that in [28, Theorem 4.4] and is omitted. (cid:3) In this section we examine the combinatorial and topological consequences of some of the knowninequalities for f -vectors of homology manifolds with boundary when they are sharp. This in-cludes a discussion of a connection between three lower bound theorems for manifolds, PL-handledecompositions, and surgery. Along the way we propose several problems.The right-hand side of Theorem 2.2(3) makes sense for any simplicial complex ∆ . So we define¯ h ′′ i (∆) := h i (∆) − (cid:18) di (cid:19) i X j =1 ( − i − j ˜ β j − (∆) ∀ ≤ i ≤ d − . It turns out that for homology manifolds with boundary, or more generally Buchsbaum com-plexes, ¯ h ′′ i ≥ h ′′ -numbers of Buchsbaum complexes have an algebraicinterpretation, see [24, Theorem 1.2]. Murai and Nevo determined the combinatorial implicationsof ¯ h ′′ i = 0 . To state this we recall that a homology manifold with boundary is i -stacked if it con-tains no interior faces of codimension i + 1 or more. A homology manifold without boundary is i -stacked if it is the boundary of an i -stacked homology manifold with boundary. As is customary,for both homology manifolds with or without boundary we will generally shorten 1-stacked tojust stacked. Theorem 7.1. [21, Theorem 3.1]
Let ∆ be a ( d − -dimensional homology manifold with bound-ary, ≤ i ≤ d − , and d ≥ . Then ¯ h ′′ i (∆) = 0 if and only if ∆ is ( i − -stacked. Murai and Nevo further noted that with the same hypotheses, ¯ h ′′ i = 0 also implied that ˜ β j = 0for all j ≥ i [21, Corollary 3.2]. When ∆ is a PL-manifold with boundary the above combinatorialrestriction has an even stronger topological implication in terms of the PL-handle decompositionof k ∆ k . In order to describe this we review handle decompositions of PL-manifolds. We refer thereader to Rourke and Sanderson [30] for definitions and results concerning PL-manifolds.Let B be a ( d − B = B s × B t , where B s and B t are PL-balls of dimensions s and t respectively. Hence, ∂B = ( ∂B s × B t ) [ ∂B s × ∂B t ( B s × ∂B t ) . X be a ( d − X ′ is obtainedfrom X by adding a PL-handle of index s if X ′ is the union of X and B and, in addition, theintersection of X and B is contained in the boundary of X and equals ∂B s × B t . For instance,adding a disjoint ball to X is adding a PL-handle of index 0 . A PL-handle decomposition of X isa sequence of ( d − X ⊆ X ⊆ · · · ⊆ X r = X such that X is a PL-ball and for 1 ≤ j ≤ r − X j +1 is obtained from X j by adding aPL-handle.The following result first appeared as a remark in Section 6 of [36]. We include it here forcompleteness. Theorem 7.2.
Suppose ∆ is a ( d − -dimensional PL-manifold with boundary, d ≥ , and ¯ h ′′ i (∆) = 0 for some ≤ i ≤ d − . Then k ∆ k has a PL-handle decomposition using handles ofindex less than i .Proof: Let ∆ ′′ be the second barycentric subdivision of ∆ . For each nonempty face F of ∆ , let v F be the vertex in ∆ ′′ which represents F. The star of v F in ∆ ′′ is a PL-ball and every facetof ∆ ′′ is contained in exactly one such star. Now order the interior faces F of ∆ , F , F , . . . , F r so that all of the codimension zero faces (the facets) of ∆ come first, then the interior faces ofcodimension one, etc. Finally, set X j = S jk =1 st ∆ ′′ v F k . Thus, for j ≤ f d − (∆) , X j is a disjointunion of j PL-balls. By [30, Proposition 6.9] and the discussion that precedes it, X ⊆ · · · ⊆ X r is a handle decomposition of k ∆ k with a collar of the boundary removed. Furthermore, the indexof the handle attached to go from X j to X j +1 is the codimension of F j +1 . Since removing acollar does not change the PL-homeomorphism type of a complex, X ⊆ · · · ⊆ X r is the handledecomposition of a PL-manifold which is PL-homeomorphic to k ∆ k . Theorem 7.1 completes theproof. (cid:3)
What about the converse?
Problem 7.3.
Suppose X is a ( d − -dimensional PL-manifold with boundary that has a PL-handle decomposition using handles of index less than i for some ≤ i ≤ d − . Is there aPL-triangulation ∆ of X such that ¯ h ′′ i (∆) = 0?For i = 1 , , and d − X has a PL-handledecomposition involving only handle additions of index zero, then X is a disjoint union of PL-balls. Hence a disjoint union of ( d −
1) simplices triangulates X and has ¯ h ′′ = 0 . For i = 2 we firstobserve that if X has a PL-handle decomposition using handles of index zero or one, then X is ahandlebody and all of these have stacked triangulations, which are precisely triangulations with¯ h ′′ = 0. (This observation is any easy consequence of, say, [11, Theorem 3.11].) For the last casewe first note that any ( d − d − X ′ is obtained from X by adding an s -handle. Then the boundary of X ′ is a ( d − ∂X by removing a copy of ∂B s × B t from ∂X and replacing it with B s × ∂B t along the common boundary ∂B s × ∂B t . Such an operation is18alled an ( s − -surgery on ∂X and we call s − index of the surgery. We denote such a surgeryoperation by ∂X ⇒ ∂X ′ . So, if X has a handle decomposition B d − = X ⊆ X ⊆ · · · ⊆ X r = X, then ∂X has a surgery sequence S d − = ∂X ⇒ · · · ⇒ ∂X r = ∂X. From the g -vector point ofview the connection between these two is given by the following theorem of Murai–Nevo. Notethat if ∆ is a ( d − g r (∆) = g r (∆) − (cid:0) d +1 r (cid:1) P rj =1 ( − r − j ˜ β j − (∆). We use the same equationto define ˜ g r for all ( d − Theorem 7.4. [21]
Let ∆ be a ( d − -dimensional homology manifold and d ≥ .
1. If ∂ ∆ = ∅ and ¯ h ′′ i (∆) = 0 for some i ≤ ( d − / , then ˜ g i ( ∂ ∆) = 0 .
2. If ∂ ∆ = ∅ , the links of the vertices of ∆ have the WLP, and ˜ g i (∆) = 0 for some ≤ i ≤ ( d − / , then ∆ is ( i − -stacked. In combination with Theorem 7.2 two natural questions are:
Problem 7.5.
Let ∆ be a ( d − -dimensional PL-manifold without boundary, d ≥ , and ≤ i ≤ ( d − / .
1. If ˜ g i (∆) = 0 , does k ∆ k have a surgery sequence beginning with S d − and using surgerieswhose indices are less than i −
2. Suppose X is a ( d − –dimensional PL-manifold with a surgery sequence X = X ⇒· · · ⇒ X r = k ∆ k whose indices are less than i − . Does X have a PL-triangulation ∆ with ˜ g i (∆) = 0?Note that for i = 2 the answer to the first part of the problem is yes; see the discussion precedingTheorem 7.11.In [23] Murai and Novik considered a different invariant of the f -vector. Let ∆ be a homologymanifold and define f i (∆ , ∂ ∆) to be the number of interior i -dimensional faces. If ∆ has anonempty boundary, f − (∆ , ∂ ∆) = 0 as the empty set is no longer an interior face. Now defineall of the other invariants, such as h i (∆ , ∂ ∆) and g i (∆ , ∂ ∆) by using f i (∆ , ∂ ∆) instead of f i (∆) . For example, g (∆ , ∂ ∆) = h (∆ , ∂ ∆) − h (∆ , ∂ ∆) = f (∆ , ∂ ∆) − ( d + 1) f − (∆ , ∂ ∆) , and g (∆ , ∂ ∆) = h (∆ , ∂ ∆) − h (∆ , ∂ ∆) = f (∆ , ∂ ∆) − d f (∆ , ∂ ∆) + (cid:18) d + 12 (cid:19) f − (∆ , ∂ ∆) . Among Murai–Novik’s results is the following.
Theorem 7.6. [23]
Let ∆ be a ( d − -dimensional k -homology manifold and d ≥ .
1. For i = 1 or , g i (∆ , ∂ ∆) ≥ (cid:0) d +1 i (cid:1) i X j =1 ( − i − j ˜ β j − (∆ , ∂ ∆) . . If the links of the vertices of ∆ satisfy the WLP and ≤ i ≤ d/ , then g i (∆ , ∂ ∆) ≥ (cid:18) d + 1 i (cid:19) r X j =1 ( − i − j ˜ β j − (∆ , ∂ ∆) . In fact, Theorem 7.6(1) holds for the larger class of normal pseudomanifolds with boundary andBetti numbers replaced with the more subtle µ -invariant of Bagchi and Datta. See [23, Theorem7.3] for details.Now we consider the implications of equality in Theorem 7.6. Suppose ∆ satisfies the hypothe-ses of Theorem 7.6. Then g (∆ , ∂ ∆) = ( d + 1) ˜ β (∆ , ∂ ∆) if and only if all of the vertices of everycomponent of ∆ which has boundary are on the boundary, and every component of ∆ which doesnot have boundary is the boundary of a d -simplex. In particular, if ∆ is also a PL-manifold, thenits components with boundary have no further topological restrictions [9], while the componentswithout boundary must be PL-spheres. The situation for general homology manifolds is less clear.For instance, suppose X is the suspension of R P . Now remove an open ball whose closure doesnot include the suspension points of X and call the resulting space Y. Then Y is a Q -homologyball and excision applied to homology with integer coefficients around the suspension points of X shows that in any triangulation ∆ of Y the suspension points of X must be vertices of ∆ and arenot on the boundary of ∆ . Problem 7.7.
What are the topological restrictions imposed on k -homology manifolds by therelation g (∆ , ∂ ∆) = ( d + 1) ˜ β (∆ , ∂ ∆)?For k -homology manifolds which satisfy equality in Theorem 7.6(1) with i = 2, Murai andNovik gave a local combinatorial description in terms of the links of the vertices. If ∆ does satisfy7.6(1) with equality and i = 2, we say that ∆ has minimal g . Before stating their result wereview the operations and properties of connected sum and handle addition.Let ∆ and ∆ be ( d − F and F are facets of ∆ and ∆ respectively and φ : F → F is a bijection. The connected sum of ∆ and ∆ along φ is the complex obtained by identifying all faces σ ⊆ F with φ ( σ ) ⊆ F andthen removing the identified facet F ≡ F . The resulting complex is denoted by ∆ , or by∆ φ ∆ if we need to specify φ. To define handle addition we suppose F and F are both facetsof a single component of a complex ∆ and φ is still a bijection between them. Now make thesame identifications and facet removal as in the connected sum. As long as the graph distancebetween v and φ ( v ) is at least three for all v ∈ F , the result is a simplicial complex which wedenote by ∆ , or by ∆ φ if we need to specify φ. If F and F are in the same complex, butdistinct components we rename the components as distinct complexes and use the connected sumnotation. Note that if ∆ and ∆ are PL-manifolds without boundary then k ∆ k and k ∆ k are produced from k ∆ ∪ ∆ k and k ∆ k respectively by 0-surgery.As pointed out in [23, Lemma 7.7] the connected sum of a k -homology ball and a k -homologysphere of the same dimension is a k -homology ball whose boundary is the same as the boundaryof the original homology ball. Similarly, the connected sum of two k -homology spheres of thesame dimension is another k -homology sphere. On the other hand, the connected sum of two k -homology balls of the same dimension is neither a k -homology ball nor a sphere. Thus, if ∆ and∆ are k -homology manifolds of the same dimension, then ∆ φ ∆ is a k -homology manifold ifand only if for each vertex v in the identified facet at least one of v or φ ( v ) is an interior vertex.20 similar statement holds for ∆ . Lastly, we observe that the boundary of ∆ is the disjointunion of the boundaries of ∆ and ∆ . Similarly, the boundary of ∆ equals the boundary of ∆ . Both connected sum and handle addition introduce a missing facet into the resulting complex.A missing facet in a ( d − F of cardinality d of the verticessuch that F / ∈ ∆ , but every proper subset of F is a face of ∆ . For future inductive purposes weobserve that connected sum and handle addition strictly increase the number of missing facets. Inhomology manifolds missing facets characterize the connected sum and handle addition operations.
Proposition 7.8.
Suppose ∆ is a ( d − -dimensional homology manifold, d ≥ , and F is amissing facet of ∆ . Then either ∆ is a connected sum of homology manifolds, or ∆ is the resultof a handle addition on a homology manifold.Proof: Consider ˆ∆. In ˆ∆, the links of all of the vertices of F are homology spheres, andso Alexander duality implies that the boundary of F is locally two-sided. The argument of [7,Lemma 3.2] then shows that k ∂F k is two-sided in k ˆ∆ k . Now, if ∂ ∆ = ∅ , in which case ∆ = ˆ∆,and k ∂F k is two-sided in k ∆ k , the above statement is known; for a very detailed treatment see[5, Lemma 3.3]. So assume ∂ ∆ = ∅ . Cut ˆ∆ along the boundary of F and fill in the two missing( d − F . We obtain either a connected complex or two disjoint complexesone of which contains v — the singular vertex of ˆ∆. Thus we can write ˆ∆ = ˆΓ or ˆ∆ = ˆ∆ ,where v is in ˆ∆ . Removing v allows us to write ∆ = Γ or ∆ = ∆ .We consider the case ∆ = ∆ φ ∆ , φ : F → F , as the handle addition case is virtuallyidentical. All that remains is to show that ∆ and ∆ are homology manifolds. The vertices of∆ and ∆ which are not in F or F have links which are simplicially isomorphic to their imagein ∆ , and hence are homology balls or spheres. Now suppose that v ∈ F and let x be its image in F. If the link of x in ∆ was a homology sphere, then the links of v in ∆ and φ ( v ) in ∆ are alsohomology spheres. If the link of x in ∆ was a homology ball, then in ˆ∆ the link of x is a homologysphere Γ which is the link of x in ∆ with its boundary coned off. Since F − x is a missing facet inΓ , we can write Γ = Γ , where each Γ i is a homology sphere and the identified facet is F − x. The link of v in ∆ is then Γ with the vertex v removed and hence is a homology ball, while thelink of φ ( v ) in ∆ is Γ and is therefore a homology sphere. Finally, to see that the boundariesof ∆ and ∆ are (possibly empty) ( d − is equal to the disjoint union of the boundaries of ∆ and ∆ . (cid:3) We now list several procedures which result in a ∆ that has minimal g . All of the proofsare routine applications of the definitions and/or an expected Mayer-Vietoris sequence. For in-stance, the proof of the third part relies on the following observations: ˜ β (cid:0) ∆ , ∂ (∆ ) (cid:1) =˜ β (∆ , ∂ ∆ ) + ˜ β (∆ , ∂ ∆ ), f (cid:0) ∆ , ∂ (∆ ) (cid:1) = f (∆ , ∂ ∆ ) + f (∆ , ∂ ∆ ) − (cid:0) d (cid:1) , and f (cid:0) ∆ , ∂ (∆ ) (cid:1) = f (∆ , ∂ ∆ ) + f (∆ , ∂ ∆ ) − d . Proposition 7.9.
Let ∆ be a ( d − -dimensional k -homology manifold, where d ≥ .1. If ∆ has no interior edges, then ∆ has minimal g . ∆ has minimal g if and only if each component of ∆ has minimal g .
3. If ∆ = ∆ with ∆ and ∆ both k -homology manifolds, then ∆ has minimal g if andonly if ∆ and ∆ have minimal g and at least one of ∆ , ∆ has no boundary. . If ∆ = Γ with Γ a k -homology manifold, then ∆ has minimal g if and only if Γ hasminimal g . Here is the Murai–Novik restriction on links of vertices in complexes with minimal g . Incombination with the previous propositions it allows us to describe a global combinatorial char-acterization of such complexes.
Theorem 7.10. [23, Section 7]
Let ∆ be a ( d − -dimensional k -homology manifold with d ≥ and minimal g . Then the link of every interior vertex is a stacked sphere. Furthermore, for everyboundary vertex v there exists m ≥ (which depends on v ) such that the link of v is of the form T S · · · S m , where T is a homology ball with no interior vertices and each S i is the boundary of a ( d − -simplex. Recall that homology manifolds without boundary and minimal g are well understood: ac-cording to [20, Theorem 5.3] (that built on [28, Theorem 5.2] and [4, Theorem 1.14], as well as onthe notions of σ - and µ -numbers introduced in [6]), they are stacked homology manifolds withoutboundary, which in turn are precisely the elements of the Walkup’s class introduced in [38] (seealso [13, Section 8]). Each such manifold is obtained by starting with several disjoint boundarycomplexes of the d -simplex and repeatedly forming connected sums and/or handle additions. Inparticular, if ∆ is a stacked homology manifold without boundary, then ∆ is PL; furthermore, k ∆ k is a sphere, a sphere bundle over S , or a connected sum of several of these. In view of thisand Proposition 7.9(2), we now concentrate on connected homology manifolds with boundary.Our goal is to prove the following theorem. Theorem 7.11.
Let ∆ be a ( d − -dimensional, connected, k -homology manifold with boundary.Assume further that ∆ has minimal g and d ≥ . Then there is a sequence ∆ → · · · → ∆ r = ∆ such that every ∆ i has boundary, minimal g , and ∆ has no interior edges. Furthermore, forevery ≤ i ≤ r − , ∆ i +1 is equal to ∆ i or ∆ i , where ∂ Γ = ∅ and Γ has minimal g . Proof:
If the link of any vertex is the boundary of a ( d − d -simplex or we can remove the vertex and replace its star with a facet. Repeating thisprocedure as many times as necessary we can assume that there is no vertex whose link is theboundary of a ( d − is the required sequence. Thus let e be an interior edge with endpoints v and w. There are twocases to consider: (i) either v or w is an interior vertex, say v , or (ii) both v and w are boundaryvertices. Theorem 7.10 then shows that in the former case, the link of v must be a stacked spherewhich by our assumption is not the boundary of the simplex; hence, the link of v is of the form S S · · · φ S m , where m ≥ S m is the boundary of the ( d − v is the boundary vertex whose link has the interior vertex w , the link of v is T S · · · φ S m , where m ≥ S m is the boundary of the ( d − v contains a vertex x (e.g., the vertex of S m that is not in the image of φ ) such that the link of the edge f = { v, x } is the boundary of the ( d − G (the facet22f S m opposite to x ). Hence st f is f ∗ ∂G. If G / ∈ ∆ then we retriangulate st f by removing f and inserting two new facets v ∪ G and x ∪ G. (This is usually called a ( d −
2) bistellar move.)The resulting complex is homeomorphic to ∆ but has smaller g . This is impossible, so G ∈ ∆ . However, G ∈ ∆ implies that v ∪ G or x ∪ G is a missing facet of ∆ as otherwise ∆ contains theboundary of the d -simplex { v, x } ∪ G. Once we know that ∆ has at least one missing facet we can write ∆ as ∆ or Γ (seeProposition 7.8) and apply Proposition 7.9 and the induction hypothesis along with the knowncharacterization of stacked homology manifolds without boundary to produce the required se-quence of complexes. (cid:3) There are no immediately obvious Betti number restrictions on ∆ when ∆ has minimal g . However, there are some topological restrictions. For instance, let X be an integral homologysphere with nontrivial fundamental group and let Y be X with a small ball removed. If ∆ is atriangulation of Y, then [23, Theorem 7.3] (see also [22]) can be used to show that even though˜ β (∆ , ∂ ∆) = ˜ β (∆ , ∂ ∆) = 0 , g (∆ , ∂ ∆) > . Problem 7.12.
What topological restrictions does the above combinatorial decomposition implyfor PL-manifolds with boundary that have minimal g ? What about general homology manifoldswith boundary that have minimal g ? Problem 7.13.
Is there a similar decomposition for ∆ when ∆ has minimal g i for i ≥ g ( ˆ∆) ≥ . As noted in Remark 5.4, at least for d ≥ g ( ˆ∆) ≥ g ( ˆ∆) = 0 . When aconnected orientable k -homology manifold ∆ satisfies ˜ g ( ˆ∆) = 0 we will say ˆ∆ has minimal ˜ g . (Note that for homology manifolds without boundary, having minimal ˜ g and having minimal g are equivalent properties.) We begin by noting how connected sum and handle addition interactwith minimal ˜ g . The proofs are the usual applications of Mayer-Vietoris and the definitions.
Proposition 7.14.
Let ∆ , ∆ , and Γ be ( d − -dimensional, connected, orientable k -homologymanifolds with boundary.1. ˆΓ has minimal ˜ g if and only if ˆΓ has minimal ˜ g .
2. Suppose that the connected sum of ∆ and ∆ is a k -homology manifold. Then the comple-tion of ∆ has minimal ˜ g if and only if ˆ∆ and ˆ∆ have minimal ˜ g and at least oneof ∆ or ∆ has no boundary. Like in the previous two cases, the key to analyzing complexes with minimal ˜ g involvesunderstanding the links of vertices. Proposition 7.15.
Let ∆ be a ( d − -dimensional, connected, orientable k -homology manifoldwith boundary such that d ≥ and the completion of ∆ has minimal ˜ g . Then the link of everyinterior vertex of ∆ is a stacked sphere while the link of every boundary vertex is a stacked spherewith one vertex removed. roof: First we consider d ≥ . Since ˜ g ( ˆ∆) = 0, eq. (5.3) implies that h ′′ d − ( ˆ∆) = h ′ d − ( ˆ∆) . Soan argument along the same lines as in [28, Theorem 5.2] (but using Lemma 3.3 instead of [35,Proposition 4.24]) shows that the link of every nonsingular vertex in ˆ∆ is a stacked sphere, andthe result follows. This argument depends on the fact that a ( d − d ≥ h d − = h d − is a stacked sphere. Since vertex links of 3-dimensional homologyspheres are 2-dimensional spheres and h = h for all two-dimensional spheres, stacked or not, weuse a different approach for d = 4 . Thus assume d = 4 . The definition of g i shows that g (∆) + g ( ∂ ∆) = g (∆ , ∂ ∆) + g ( ∂ ∆) . So ˜ g ( ˆ∆) = 0 and (5.2) imply that g (∆ , ∂ ∆) + g ( ∂ ∆) = 6 ˜ β (∆) + 4 ˜ β (∆) . Since ∂ ∆ is anorientable compact surface g ( ∂ ∆) = 3 ˜ β ( ∂ ∆) and hence, g (∆ , ∂ ∆) = 6 ˜ β (∆) − β ( ∂ ∆) + 4 ˜ β (∆) . Now, the long exact sequence of the pair (∆ , ∂ ∆) implies that˜ β (∆ , ∂ ∆) + ˜ β (∆) + ˜ β ( ∂ ∆) + ˜ β (∆ , ∂ ∆) = ˜ β ( ∂ ∆) + ˜ β (∆ , ∂ ∆) + ˜ β (∆) + ˜ β ( ∂ ∆) . Poincar´e-Lefschetz duality applied to ∆ and ∂ ∆ gives us1 + 2 ˜ β (∆ , ∂ ∆) + ˜ β ( ∂ ∆) = 2 ˜ β (∆) + 2 ˜ β ( ∂ ∆) + 1 . Thus, g (∆ , ∂ ∆) = 6 ˜ β (∆ , ∂ ∆) − β ( ∂ ∆) + 4 ˜ β (∆) = 10 ˜ β (∆ , ∂ ∆) − β ( ∂ ∆) . By Theorem 7.6, ∆ has minimal g and ∂ ∆ has only one component. Theorem 7.10 and the factthat triangulations of two-dimensional disks with no interior vertices are stacked spheres with onevertex removed proves that the links of the vertices of ∆ are as claimed. (cid:3) Theorem 7.16.
Let ∆ be a ( d − -dimensional, connected, orientable homology manifold withboundary such that ˆ∆ has minimal ˜ g and d ≥ . Then there exists a sequence of ( d − -dimensional homology manifolds ∆ −→ · · · −→ ∆ r = ∆ such that ∆ is a stacked homologymanifold, and for all ≤ j ≤ r − , ∆ j +1 = ∆ j , where Γ has minimal ˜ g and no boundary, or ∆ j +1 = ∆ j . Proof:
As in the proof of Theorem 7.11 we can assume that there is no vertex whose link is theboundary of a ( d − . If ∆ has a missing facet, then Propositions 7.8 and 7.14 allow us to write ∆ as a connected sumor handle addition as required for the induction step.In preparation for the base case where ∆ has no missing facets, we first show that if the linkof any vertex w has a missing facet F , then { w } ∪ F is a missing facet of ∆ . For this it is sufficientto prove that F ∈ ∆ . To prove that F ∈ ∆ we follow Walkup’s idea in [38] and retriangulate ˆ∆as follows. The previous proposition shows that the link of w in ˆ∆ is a stacked sphere. Remove w from ˆ∆ and insert F. The union of lk ˆ∆ w and F consists of two PL-spheres whose intersectionis F. Now add two new vertices x and y which cone off these two PL-spheres and call the newcomplex ˆ∆ ′ . Counting edges shows that g ( ˆ∆ ′ ) = g ( ˆ∆) − . This is a contradiction since ˆ∆ has24inimal ˜ g and ˆ∆ ′ is homeomorphic to ˆ∆. To see that ˆ∆ ′ is homeomorphic to ˆ∆ we note thatst w and st x ∪ st y are homeomorphic since they are both ( d − w of ∆ can bewritten as ( S · · · S m ) − v, where the S i are boundaries of ( d − v is v — the vertex added to form the completion of ∆ . It must be the case that v is in every S i . Otherwise there would be a missing facet in the link of w . But now the union of (images) of S i − v ( i = 1 , . . . , m ) is a stacking of the link of w which proves that the link of w is a stacked ball. Sinceall of the links of vertices of ∆ are stacked balls, ∆ is a stacked homology manifold. Indeed, if F ∈ ∆ were an interior face of ∆ of codimension ≥
2, then for any w ∈ F , F − w would be aninterior face of codimension ≥ w . (cid:3) Remark 7.17.
All ∆ i in the statement of Theorem 7.16 have a nonempty connected boundary.Theorem 7.16 allows a description of the possible topological types of ∆ such that ˆ∆ has minimal˜ g . Corollary 7.18. If ∆ is a ( d − -dimensional, connected, orientable homology manifold suchthat d ≥ and ˆ∆ has minimal ˜ g , then k ∆ k is a ball, sphere, orientable handlebody with boundary,orientable S d − -bundle over S , or a connected sum of two or more of these which have a (possiblyempty) connected boundary. Problem 7.19.
Is there a similar decomposition for minimal ˜ g i when i ≥ References [1] Karim Adiprasito. Combinatorial Lefschetz theorems beyond positivity. Preprint available at https://arxiv.org/abs/1812.10454 , 2018.[2] Christos A. Athanasiadis. Some combinatorial properties of flag simplicial pseudomanifolds andspheres.
Ark. Mat. , 49(1):17–29, 2011.[3] Eric Babson and Isabella Novik. Face numbers and nongeneric initial ideals.
Electron. J. Combin. ,11(2):Research Paper 25, 23, 2004/06.[4] Bhaskar Bagchi. The mu vector, Morse inequalities and a generalized lower bound theorem for locallytame combinatorial manifolds.
European J. Combin. , 51:69–83, 2016.[5] Bhaskar Bagchi and Basudeb Datta. Lower bound theorem for normal pseudomanifolds.
Expo. Math. ,26(4):327–351, 2008.[6] Bhaskar Bagchi and Basudeb Datta. On stellated spheres and a tightness criterion for combinatorialmanifolds.
European J. Combin. , 36:294–313, 2014.[7] B. Basak and E. Swartz. Three-dimensional pseudomanifolds with relatively few edges. Preprintavailable at https://arxiv.org/pdf/1803.08942.pdf , 2018.[8] Louis J. Billera and Carl W. Lee. A proof of the sufficiency of McMullen’s conditions for f -vectors ofsimplicial convex polytopes. J. Combin. Theory Ser. A , 31(3):237–255, 1981.
9] R. H. Bing. Some aspects of the topology of 3-manifolds related to the Poincar´e conjecture. In
Lectureson modern mathematics, Vol. II , pages 93–128. Wiley, New York, 1964.[10] Ulrich Brehm and Wolfgang K¨uhnel. Combinatorial manifolds with few vertices.
Topology , 26(4):465–473, 1987.[11] Basudeb Datta and Satoshi Murai. On stacked triangulated manifolds.
Electron. J. Combin. , 24(4):Pa-per 4.12, 14, 2017.[12] Hans-Gert Gr¨abe. The canonical module of a Stanley-Reisner ring.
J. Algebra , 86(1):272–281, 1984.[13] Gil Kalai. Rigidity and the lower bound theorem. I.
Invent. Math. , 88(1):125–151, 1987.[14] Wolfgang K¨uhnel. Higher-dimensional analogues of Cz´asz´ar’s torus.
Results Math. , 9(1-2):95–106,1986.[15] Wolfgang K¨uhnel.
Tight polyhedral submanifolds and tight triangulations , volume 1612 of
LectureNotes in Mathematics . Springer-Verlag, Berlin, 1995.[16] Wolfgang K¨uhnel and Gunter Lassmann. Permuted difference cycles and triangulated sphere bundles.
Discrete Math. , 162(1-3):215–227, 1996.[17] Frank H Lutz. Triangulated manifolds with few vertices: Combinatorial manifolds. Preprint availableat http://arxiv.org/pdf/math/0506372v1.pdf , 2005.[18] P. McMullen. The numbers of faces of simplicial polytopes.
Israel J. Math. , 9:559–570, 1971.[19] Satoshi Murai. Algebraic shifting of strongly edge decomposable spheres.
J. Combin. Theory Ser. A ,117(1):1–16, 2010.[20] Satoshi Murai. Tight combinatorial manifolds and graded betti numbers.
Collect. Math. , 2015.[21] Satoshi Murai and Eran Nevo. On r -stacked triangulated manifolds. J. Algebraic Combin. , 39(2):373–388, 2014.[22] Satoshi Murai and Isabella Novik. Face numbers and the fundamental group.
Israel J. Math. ,222(1):297–315, 2017.[23] Satoshi Murai and Isabella Novik. Face numbers of manifolds with boundary.
Int. Math. Res. Not.IMRN , (12):3603–3646, 2017.[24] Satoshi Murai, Isabella Novik, and Ken-ichi Yoshida. A duality in Buchsbaum rings and triangulatedmanifolds.
Algebra Number Theory , 11(3):635–656, 2017.[25] Isabella Novik. Upper bound theorems for homology manifolds.
Israel J. Math. , 108:45–82, 1998.[26] Isabella Novik and Ed Swartz. Applications of Klee’s Dehn-Sommerville relations.
Discrete Comput.Geom. , 42(2):261–276, 2009.[27] Isabella Novik and Ed Swartz. Gorenstein rings through face rings of manifolds.
Compos. Math. ,145(4):993–1000, 2009.[28] Isabella Novik and Ed Swartz. Socles of Buchsbaum modules, complexes and posets.
Adv. Math. ,222(6):2059–2084, 2009.[29] Isabella Novik and Ed Swartz. Face numbers of pseudomanifolds with isolated singularities.
Math.Scand. , 110(2):198–222, 2012.[30] Colin Patrick Rourke and Brian Joseph Sanderson.
Introduction to piecewise-linear topology . SpringerStudy Edition. Springer-Verlag, Berlin-New York, 1982. Reprint.[31] Peter Schenzel. On the number of faces of simplicial complexes and the purity of Frobenius.
Math.Z. , 178(1):125–142, 1981.
32] Richard P. Stanley. The upper bound conjecture and Cohen-Macaulay rings.
Studies in Appl. Math. ,54(2):135–142, 1975.[33] Richard P. Stanley. The number of faces of a simplicial convex polytope.
Adv. in Math. , 35(3):236–238,1980.[34] Richard P. Stanley.
Combinatorics and commutative algebra , volume 41 of
Progress in Mathematics .Birkh¨auser Boston, Inc., Boston, MA, second edition, 1996.[35] Ed Swartz. Face enumeration—from spheres to manifolds.
J. Eur. Math. Soc. (JEMS) , 11(3):449–485,2009.[36] Ed Swartz. Thirty-five years and counting. http://arxiv.org/pdf/1411.0987.pdf , 2014.[37] Tiong-Seng Tay, Neil White, and Walter Whiteley. Skeletal rigidity of simplicial complexes. I.
EuropeanJ. Combin. , 16(4):381–403, 1995.[38] David W. Walkup. The lower bound conjecture for 3- and 4-manifolds.
Acta Math. , 125:75–107, 1970.[39] Walter Whiteley. Vertex splitting in isostatic frameworks.
Structural Topology , (16):23–30, 1990. DualFrench-English text., (16):23–30, 1990. DualFrench-English text.