Shellability and homology of q-complexes and q-matroids
Sudhir R. Ghorpade, Rakhi Pratihar, Tovohery H. Randrianarisoa
aa r X i v : . [ m a t h . C O ] F e b SHELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS SUDHIR R. GHORPADE, RAKHI PRATIHAR, AND TOVOHERY H. RANDRIANARISOA
Abstract.
We consider a q -analogue of abstract simplicial complexes, called q -complexes, and discuss the notion of shellability for such complexes. It isshown that q -complexes formed by independent subspaces of a q -matroid areshellable. We also outline some partial results concerning the determinationof homology of shellable q -complexes. Introduction
Shellability is an important and useful notion in combinatorial topology andalgebraic combinatorics. Recall that an (abstract) simplicial complex ∆ is said tobe shellable if it is pure (i.e., all its facets have the same dimension) and there is alinear ordering F , . . . , F t of its facets such that for each j “ , . . . , t , the complex x F j y X x F , . . . F j ´ y is generated by a nonempty set of maximal proper faces of F j . Here for i “ , . . . , t , by x F , . . . , F i y we denote the complex generated by F , . . . , F i , i.e., the smallest simplicial complex containing F , . . . , F i .From a topological point of view, a shellable simplicial complex is like a wedge ofspheres. In particular, the reduced homology groups are well understood. Shellablesimplicial complexes are of importance in commutative algebra partly because theirStanley-Reisner rings (over any field) are Cohen-Macaulay. Gr¨obner deformationsof coordinate rings of several classes of algebraic varieties can be viewed as Stanley-Reisner rings of some simplicial complexes. Thus showing that these complexesare shellable becomes an effective way of establishing Cohen-Macaulayness of thecorresponding coordinate rings. Important classes of simplicial complexes thatare known to be shellable include boundary complex of a convex polytope, ordercomplex of a “nice” poset (or more precisely, a bounded, locally upper semimodularposet), and matroid complexes, i.e., complexes formed by the independent subsets Sudhir Ghorpade is partially supported by DST-RCN grant INT/NOR/RCN/ICT/P-03/2018from the Dept. of Science & Technology, Govt. of India, MATRICS grant MTR/2018/000369from the Science & Eng. Research Board, and IRCC award grant 12IRAWD009 from IIT Bombay.During the course of this work, Rakhi Pratihar was supported by a doctoral fellowship at IITBombay from the University Grant Commission, Govt. of India (Sr. No. 2061641156). Currently,she is supported by Grant 280731 from the Research Council of Norway.During the course of this work Tovohery Randrianarisoa was supported by a postdoc fellowshipat IIT Bombay from the Swiss National Science Foundation Grant No. 181446. of matroids. For relevant background and proofs of these assertions, we refer to themonographs [16, 5, 7] and the survey article of Bj¨orner [4].We are interested in a q -analogue of some of these notions and results, whereinfinite sets are replaced by finite-dimensional vector spaces over the finite field F q .One of our motivation comes from the recent work of Jurrius and Pellikaan [10]where the notion of a q -matroid is introduced and several of its properties arestudied. The notion of a simplicial complex admits a straightforward q -analogue,and this goes back at least to Rota [14]. Alder [1] studied q -complexes in histhesis and defined when a q -complex is shellable. A natural question therefore iswhether the q -complex of independent subspaces of a q -matroid is shellable. We willshow in this paper that the answer is affirmative. Next, we consider the questionof determining the homology of shellable q -complexes. This appears to be muchharder than the classical case, and we are able to make partial progress here by wayof explicitly determining the homology of q -spheres and describing the homologyof a shellable q -complex provided it satisfies an additional hypothesis. A basicstumbling block (pointed out in [10] already) is that the notions of difference (oftwo sets) and complement (of a subset of a given set) do not have an obvious andunique analogue in the context of subspaces.Our other motivation is from coding theory and the work of Johnsen and Verdure[9] where to a q -ary linear code (or more generally, to a matroid), one can associatea fine set of invariants, called its Betti numbers. These are obtained by looking ata minimal graded free resolution of the Stanley-Reisner ring of a simplicial complexthat corresponds to the vector matroid associated to the parity check matrix ofthe given linear code. The question that arises naturally is whether something likeBetti numbers can be defined in the context of rank metric codes, or more generally,for q -matroids as in [10] or going even further, for the p q, m q -polymatroids studiedin [15, 6, 8]. We were led to the study of shellability and homology of q -complexes,and especially, complexes associated to q -matroids with a view toward a possibletopological approach to the above question. However, the question of arriving ata suitable notion of Betti numbers of rank metric codes is very far from beinganswered and at the moment the musings above are more like a pie in the sky.This paper is organized as follows. In the next section, we collect some prelimi-naries and recall definitions of basic concepts such as q -complexes and q -matroids.In Section 3, we outline a procedure called “tower decomposition” that providesa useful way to order subspaces in a q -complex. The notion of shellability for q -complexes is reviewed in Section 4 and the shellability of q -matroid complexes isalso established in this section. Finally, our results on the homology of q -spheresand more generally, homology of shellable q -complexes are described in Section 5. HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 3 Preliminaries
Throughout this paper q denotes a power of a prime number and F q the finitefield with q elements. We fix a positive integer n , and denote by E the n -dimensionalvector space F nq over F q . By Σ p E q we denote the set of all subspaces of E . Givenany y , . . . , y r P E , we denote by h y , . . . , y r i the F q -linear subspace of E generatedby y , . . . , y r . Also, for U, V, W P Σ p E q , we often write U “ V ‘ W to mean that U “ V ` W and V X W is the space t u consisting of the zero vector in E . In otherwords, all direct sums considered in this paper are internal direct sums. We denoteby N the set of all nonnegative integers, and by N ` the set of all positive integers.Basic definitions and results concerning simplicial complexes and matroids willnot be reviewed here. These are not formally needed, but they motivate the notionsand results discussed below. If necessary, one can refer to [16] or [7] for simplicialcomplexes, shellability, etc. and to [18] for basics (and more) about matroids. Definition 2.1.
By a q -complex on E : “ F nq we mean a subset ∆ of Σ p E q satisfyingthe property that for every A P ∆, all subspaces of A are in ∆.Let ∆ be a q -complex. Elements of ∆ are called faces , and maximal faces (w.r.t.inclusion) are called facets . The dimension of ∆ is max t dim A : A P ∆ u , and it isdenoted by dim ∆. We say that ∆ is pure if all its facets have the same dimension. Example 2.2. (i) Clearly, Σ p E q is a pure q -complex of dimension n . Also,∆ : “ t A P Σ p E q : A ‰ E u is a pure q -complex of dimension n ´
1; wedenote it by S n ´ q and call it the q -sphere of dimension n ´ A is any subset of Σ p E q , then t B P Σ p E q : B Ď A for some A P A u is a q -complex, called the q -complex generated by A , and denoted by x A y . Incase A “ t A , . . . , A r u , we often write x A y as x A , . . . , A r y . By convention,if A is the empty set, then x A y is defined to be the empty set.We now recall the definition of a q -matroid, as given by Jurrius and Pellikaan [10]. Definition 2.3. A q -matroid on E is a pair M “ p E, ρ q , where ρ : Σ p E q Ñ N is afunction (called the rank function of M ) satisfying the following properties.(r1) 0 ď ρ p A q ď dim A for all A P Σ p E q ,(r2) If A, B P Σ p E q with A Ď B , then ρ p A q ď ρ p B q ,(r3) ρ p A ` B q ` ρ p A X B q ď ρ p A q ` ρ p B q for all A, B P Σ p E q . Definition 2.4.
Let M “ p E, ρ q be a q -matroid, We call ρ p E q the rank of M . Let A P Σ p E q . Then A is said to be independent (in M ) if ρ p A q “ dim A ; otherwise it iscalled dependent . Further, A is a basis (of M ) if A is independent and ρ p A q “ ρ p E q . Example 2.5.
Given a positive integer k ď n , consider ρ : Σ p E q Ñ N defined by ρ p A q “ $&% dim A if dim A ď k,k if dim A ą k. SUDHIR R. GHORPADE, RAKHI PRATIHAR, AND TOVOHERY H. RANDRIANARISOA
Then it is easily seen that p E, ρ q is a q -matroid of rank k ; this is called the uniformmatroid on E of rank k , and it is denoted by U q p k, n q .Important properties of independent spaces in a q -matroid (which, in fact, char-acterize a q -matroid) are proved in [10, Thm. 8] and recalled below. Proposition 2.6.
Let M “ p E, ρ q be a q -matroid, and let I be the family ofindependent subspaces in M . Then I satisfies the following four properties:(i1) I ‰ H .(i2) A P Σ p E q and B P I with A Ď B ñ A P I .(i3) A, B P I with dim A ą dim B ñ there is x P A z B such that B ` h x i P I .(i4) A, B P Σ p E q and I, J are maximal independent subspaces of
A, B , respectively ñ there is a maximal independent subspace K of A ` B such that K Ď I ` J . It is shown in [10] that if I is an arbitrary subset of Σ p E q satisfying (i1)–(i4),then there is a unique q -matroid M I “ p E, ρ I q whose rank function ρ I is given by ρ I p A q “ max t dim B : B P I , B Ď A u for A P Σ p E q ;moreover, I is precisely the family of independent subspaces in M I .We now recall some fundamental properties of bases of a q -matroid, which pro-vide yet another characterization of q -matroids. For a proof, see [10, Thm. 37]. Proposition 2.7.
The set B of bases of a q -matroid on E satisfies the following.(b1) B ‰ H .(b2) If B , B P B are such that B Ď B , then B “ B .(b3) If B , B P B and C P Σ p E q satisfy B X B Ď C Ď B and dim B “ dim C ` ,then there is x P B such that C ` h x i P B .(b4) If A , A P Σ p E q and if I j is a maximal element of t B X A j : B P B u (with respect to inclusion) for j “ , , then there is a maximal element J of t B X p A ` A q : B P B u such that J Ď I ` I . The third property here is called the basis exchange property . It can be usedtogether with (b1) and (b2) to deduce that any two bases of a q -matroid have thesame dimension. See, for example, [10, Prop. 40].As a consequence of Theorem 2.7, we shall derive the following dual basis ex-change property , which will be useful to us in the sequel. Corollary 2.8.
Let M “ p E, ρ q be a q -matroid. Let B , B be bases of M with B ‰ B and let y P B z B . Then there exist U P Σ p E q and x P B z B such that (1) B X B Ď U, B “ U ‘ h x i , and U ‘ h y i is a basis of M .Proof. Let r : “ ρ p M q and s : “ r ´ dim B X B . Note that 1 ď s ď r . We willuse (finite) induction on s . If s “
1, then U : “ B X B and any x P B z B clearly HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 5 satisfy (1). Now suppose s ą s . Thendim B X B ď r ´ A P Σ p E q and y P B z B such that B X B Ď A Ď B and B “ A ‘ h y i ‘ (cid:10) y (cid:11) . Let C : “ A ‘ h y i . Clearly, B X B Ď C Ď B and dim B “ dim C `
1. So by(b3) in Proposition 2.7, there is x P B z B such that C ‘ h x i is a basis of M . Let B : “ C ‘ h x i . Then y P B z B and dim B X B ą dim B X B . Hence by theinduction hypothesis, there is U P Σ p E q and x P B z B such that B X B Ď U, B “ U ‘ h x i , and U ‘ h y i is a basis of M .Now observe that B X B Ď B X A Ď B X C Ď B X B . Consequently, x P B z B and (1) holds. This completes the proof. (cid:3) We end this section by noting that if M “ p E, ρ q is a q -matroid on E “ F nq with ρ p M q “ r , then it follows from Proposition 2.6 that M defines a q -complex∆ M whose faces are precisely the independent subspaces of M , i.e., those F q -linearsubspaces F of F nq such that dim F “ ρ p F q . We will refer to ∆ M as the q -complexassociated to M . Since any two bases of M have the same dimension r , it is clearthat ∆ M is pure of dimension r . By a q -matroid complex on E , we shall mean a q -complex associated to a matroid on E .3. Tower Decompositions
Suppose ∆ is a pure q -complex on F nq of dimension r . Then its facets are certain r -dimensional subspaces of F nq and a priori it is not clear how they can be linearlyordered. In this section, we consider a variant of row reduced echelon forms, calledtower decompositions, which will allow us to put a total order on such subspaces.Fix a positive integer r ď n and let G r p E q denote the Grassmannian consistingof all r -dimensional subspaces of E : “ F nq . Given any U P G r p E q , let M U be agenerator matrix of U in row echelon form, i.e., let M U be a r ˆ n matrix in rowechelon form whose row vectors form a basis of U . We denote by u r , u r ´ , . . . , u the row vectors of M U so that M U “ »——– u r ... u fiffiffifl We define subspaces U , . . . , U r of E and subsets U , . . . , U r of E zt u by(2) U i : “ h u , . . . , u i i and U i : “ U i z U i ´ for i “ , . . . , r, where, by convention U : “ t u . Further, we define τ p U q : “ p U , U , ¨ ¨ ¨ , U r q , SUDHIR R. GHORPADE, RAKHI PRATIHAR, AND TOVOHERY H. RANDRIANARISOA and we shall refer to this as the tower decomposition of U . Observe that although M U (or equivalently, the vectors u , . . . , u r ) need not be uniquely determined by U , the subspaces U i (and hence the subsets U i ) are uniquely determined by U . Tosee this, it suffices to note that there is a unique generator matrix of U , say M ˚ U ,which is in reduced row echelon form, and it is easily seen that the correspondingsubspace U ˚ i is equal to U i for each i “ , . . . , r . Thus the tower decomposition τ p U q of U depends only on U . Moreover, it is obvious that τ p U q determines U ,since U “ U r . Definition 3.1.
Given any nonzero vector u P F nq , the leading index of u , denoted p p u q , is defined to be the least positive integer i such that the i -th entry of u isnonzero. Further, given a subset S of F nq , the profile p p S q of S is defined to be theunion of the leading indices of all of its nonzero elements, i.e., p p S q “ t p p u q : u P S zt uu . Note that the profile of S can be the empty set if S contains no nonzero vector. Lemma 3.2.
Let U P G r p E q and let u r , . . . , u be the rows of a generator matrix M U of U in row echelon form. Then p p u q ą ¨ ¨ ¨ ą p p u r q . Further, given any i P t , . . . , r u , if U i , U i are as in (2) , then p p U i q “ t p p u i qu , and for any u P U zt u , u P U i ðñ p p u q “ p p u i q . Proof.
Since M U has rank r and it is in row-echelon form, it is clear that u , . . . , u r are nonzero and p p u q ą ¨ ¨ ¨ ą p p u r q . Now fix i P t , . . . , r u . Then p p u i q ă p p u j q for1 ď j ă i . Consequently, if u “ c u `¨ ¨ ¨` c i u i for some c , . . . , c i P F q with c i ‰ ,then p p u q “ p p c i u i q “ p p u i q . This shows that p p U i q “ t p p u i qu . The last assertionfollows from this by noting that U zt u is the disjoint union of U , . . . , U r . (cid:3) Fix an arbitrary total order ă on F q such that 0 ă ă α for all α P F q zt , u .This extends lexicographically to a total order on E , which we also denote by ă .For v, w P E “ F nq , we may write v ĺ w if v ă w or v “ w . Lemma 3.3.
Let v, w be nonzero vectors in E “ F nq . If p p v q ă p p w q , then w ă v .Proof. Let i P t , . . . , n u be such that p p v q “ i . Suppose p p w q ą i . Then the j thcoordinate w j of w is 0 for 1 ď j ď i , whereas the i -th coordinate of v is nonzero.Hence it is clear from the definition of ă that w ă v . (cid:3) We are now ready to define a nice total order on G r p E q . In what follows, forany nonempty subset S of E “ F nq , we denote by min S the least element of S withrespect to the total order ă on E defined above. We remark that if S is closed withrespect to multiplication by nonzero scalars (e.g., if S “ U i for some i , where U i are as in (2) for some subspace U P G r p E q ) and if u “ min S , then the first nonzeroentry of the vector u in F nq is necessarily 1. HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 7 Definition 3.4.
Let
U, V P G r p E q and let τ p U q “ p U , . . . , U r q and τ p V q “p V , . . . , V r q be the tower decompositions of U and V , respectively. Define U V if either U “ V or if there exists a positive integer e ď r such that U j “ V j for 1 ď j ă e, U e ‰ V e , and min U e ă min V e . Lemma 3.5.
The relation defined in Definition 3.4 is a total order on G r p E q .Proof. Clearly, is reflexive. Next, let U, V P G r p E q and let τ p U q “ p U , . . . , U r q and τ p V q “ p V , . . . , V r q be their tower decompositions. If U ‰ V , then there existsa unique positive integer e ď r such that U j “ V j for 1 ď j ă e and U e ‰ V e . Let u : “ min U e and v : “ min V e . Observe that U e “ U e ´ ‘ h u i and V e “ V e ´ ‘ h v i .Since U e ´ “ V e ´ and U e ‰ V e , it follows that u ‰ v . Hence either u ă v or v ă u .This shows that any two elements of G r p E q are comparable with respect to .It remains to show the transitivity of . To this end, suppose U V and V W for some W P G r p E q . Let τ p W q “ p W , . . . , W r q be the tower decomposition of W .If U “ V or if V “ W , then clearly U W . Suppose U ‰ V and V ‰ W . Thenthere are unique integers e, d P t , . . . , r u such that U j “ V j for 1 ď j ă e, U e ‰ V e , and min U e ă min V e . and V j “ W j for 1 ď j ă d, V d ‰ W e , and min V d ă min W d . First, suppose e ă d . Then it is clear that U j “ V j “ W j for 1 ď j ă e, U e ‰ V e “ W e , and min U e ă min V e “ min W e . Hence U W . Likewise if we suppose d ă e , then U j “ V j “ W j for 1 ď j ă d, U d “ V d ‰ W d , and min U d “ min V d ă min W d . So, we again obtain U W . Finally, if e “ d , then the transitivity of ă on E isreadily seen to imply that U W . Thus is a total order on G r p E q . (cid:3) Shellability of q -matroid complexes In this section, we begin with the definition of shellability of a q -complex and anequivalent formulation of it. Next, we shall use the results of the previous sectionto obtain a shelling of q -matroid complexes.The following definition is a straightforward analogue of the notion of shellabilityfor q -complexes recalled in the Introduction. A slightly different, but obviouslyequivalent, definition was given by Alder [1, Definition 1.5.1], Definition 4.1.
Let ∆ be a pure q -complex of dimension r . A shelling of ∆ is alinear order on F , . . . , F t of facets of ∆ such that for each j “ , . . . , t , the complex x F j y X x F , . . . F j ´ y is generated by a nonempty set of maximal proper faces of F j .We say that a q -complex is called shellable if it is pure and admits a shelling. SUDHIR R. GHORPADE, RAKHI PRATIHAR, AND TOVOHERY H. RANDRIANARISOA
Example 4.2. (Alder [1, Example 1.5.1]) A q -sphere S n ´ q is a shellable q -complexon E : “ F nq of rank n ´
1. Indeed, its facets are the p n ´ q -dimensional subspaces of E , and if F , . . . , F t is an arbitrary listing of these facets, then it is easily seen fromthe formula for the dimension of the sum of two subspaces, that dim F i X F j “ n ´ ď i ă j ď t . Hence for any j “ , . . . , t , we see that t F i X F j : 1 ď i ă j u is anonempty set of maximal proper faces of F j , which generates x F j y X x F , . . . F j ´ y .Thus F , . . . , F t is a shelling of S n ´ q .The following characterization is analogous to the corresponding result in theclassical case (see, e.g., [7, Proposition 5.4]) and it will be useful to us in the sequel. Lemma 4.3.
Let ∆ be a pure q -complex of rank r , and let F , . . . , F t be a listing ofthe facets of ∆ . Then F , . . . , F t is a shelling of ∆ if and only if for every i, j P N ` with i ă j ď t , there exists k P N ` with k ă j such that (3) F i X F j Ď F k X F j and dim p F k X F j q “ r ´ . Proof.
Suppose F , . . . , F t is a shelling of ∆. Let i, j P N ` with i ă j ď t . Then F i X F j P x F j yXx F , . . . F j ´ y . Hence F i X F j Ď G j , where G j P x F j yXx F , . . . F j ´ y is a maximal proper face of F j . Since G j P x F , . . . F j ´ y , there exists k P N ` with k ă j such that G j Ď F k . Thus, G j Ď F k X F j and moreover, dim G j “ dim F j ´ “ r ´
1. Now, F k ‰ F j , since k ă j . Also, dim F k “ dim F j “ r . It follows thatdim p F k X F j q ď r ´
1. This implies that G j “ F k X F j , and so (3) is proved.Conversely, suppose for every i ă j ď t , there exists k ă j such that (3) holds.Let j P t , . . . , t u and let F be a face of x F j y X x F , . . . F j ´ y . Then F is a face of F j as well as F i for some i ă j . For these i, j , there exists k P N ` with k ă j suchthat (3) holds. Now F Ď F i X F j Ď F k X F j and so F is a face of F k X F j . It followsthat t F k X F j : 1 ď k ă j and dim p F k X F j q “ r ´ u constitutes a nonempty setof maximal proper faces, which generates x F j y X x F , . . . F j ´ y . (cid:3) We are now ready to prove the main result of this section. Here we will makeuse of the total order given in Definition 3.4. As usual, for any U, V P Σ p E q ofthe same dimension, we will write U ă V to mean that U V and U ‰ V . Theorem 4.4.
Let M be a q -matroid on E : “ F nq of rank r . Then the q -complex ∆ M associated to M is shellable. In fact, if F , . . . , F t is an ordering of the facetsof ∆ M such that F i ă F j for ď i ă j ď t , then this defines a shelling of ∆ M .Proof. We have seen already ∆ M is a pure q -complex of dimension r . Let F , . . . , F t be an ordering of the facets of ∆ M such that F ă ¨ ¨ ¨ ă F t . Fix integers i, j with1 ď i ă j ď t . We need to show that there is a positive integer k ă j such that F i X F j Ď F k X F j and dim F k X F j “ r ´ . HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 9 This will be done in several steps. First, let us denote the tower decompositions of F i and F j by τ p F i q “ p W , . . . , W r q and τ p F j q “ p V , . . . , V r q . Since F i ă F j , there is a unique positive integer e ď r such that W “ V , . . . , W e ´ “ V e ´ , W e ‰ V e , and min W e ă min V e . Write w : “ min W e and v : “ min V e . We claim that w P F i z F j . Clearly, w P F i and w ‰
0. Suppose if possible w P F j . Since w ă v , by Lemma 3.3, p p v q ď p p w q .Further, if p p v q “ p p w q , then by Lemma 3.2, p p V e q “ t p p v qu “ t p p w qu , and since w P F j zt u , it follows from Lemma 3.2 that w P V e . But this contradicts theminimality of v in V e since w ă v . Thus p p v q ă p p w q . Now w P F j zt u with p p w q ą p p v q and p p V e q “ t p p v qu . Hence it follows from Lemma 3.2 that w P V s for some positive integer s ă e . But then w P W s and so by Lemma 3.2, p p W s q “t p p w qu “ p p W e q , which is a contradiction. This proves the claim.Since w P F i z F j , we use the dual basis exchange property (Corollary 2.8) toobtain U P Σ p E q and x P F j z F i such that F i X F j Ď U, F j “ U ‘ h x i , and U ‘ h w i is a basis of M. The last condition implies that U ‘ h w i “ F k for a unique positive integer k ď t .Now it is clear that F i X F j Ď U Ď F k X F j . Further, if we show that k ă j , then F k X F j would be a proper subspace of F k and hence dim F k X F j ď r ´
1. On theother hand, since dim U “ r ´ U Ď F k X F j , we see that dim F k X F j “ r ´ k ă j , we consider the tower decompositions of U and F k , say, τ p U q “ p U , . . . , U r ´ q and τ p F k q “ p V ˚ , . . . , V ˚ r q . Recall that W s “ V s for 1 ď s ă e . We now claim that U s “ V s for 1 ď s ă e .To see this, let d be the least least positive integer such that U d ‰ V d . Suppose, ifpossible d ă e . Let α : “ min U d and β : “ min V d . Note that α P U zt u Ď F j zt u .Now if p p α q “ p p β q , then from Lemma 3.2 we see that α P V d . Consequently, V d “ V d ´ ‘ h α i “ U d ´ ‘ h α i “ U d , which is a contradiction. Also, if p p α q ą p p β q ,then from Lemma 3.2 we see that α P V s for some positive integer s ă d . Butthen α P V s “ U s Ď U d ´ , which is a contradiction since α P U d “ U d z U d ´ . Itfollows that p p α q ă p p β q . Finally, if p p α q ă p p β q , then from Lemma 3.2 we see that β P U s for some positive integer s ă d . But then β P U s “ V s Ď V d ´ , which is acontradiction since β P V d “ V d z V d ´ . This proves that d ě e and so the last claimis proved.Now let ℓ be the least positive integer such that V ℓ ‰ V ˚ ℓ . We shall show that k ă j , or equivalently, F k ă F j . by considering separately the following two cases. Case 1. ℓ ă e . Let v ℓ : “ min V ℓ and v ˚ ℓ : “ min V ˚ ℓ . Note that if p p v ˚ ℓ q ą p p v ℓ q , then byLemma 3.3, v ˚ ℓ ă v ℓ , and so F k ă F j . Thus to complete the proof in this caseit suffices to show that p p v ˚ ℓ q ď p p v ℓ q leads to a contradiction.First suppose p p v ˚ ℓ q ă p p v ℓ q . Since ℓ ă e ď d , we find v ℓ P V ℓ “ U ℓ Ď F k and v ℓ ‰
0. Thus, from Lemma 3.2 we see that v ℓ P V ˚ s for some positive integer s ă ℓ .But then V ˚ s “ V s Ď V ℓ ´ and so v ℓ P V ℓ ´ , which is a contradiction.Next, suppose p p v ˚ ℓ q “ p p v ℓ q . In this case, if v ˚ ℓ P F j , then we must have v ˚ ℓ P V ℓ ,thanks to Lemma 3.2. But then V ˚ ℓ “ V ˚ ℓ ´ ‘ h v ˚ ℓ i “ V ℓ , which is a contradiction.Thus v ˚ ℓ R F j . In particular, if y : “ v ˚ ℓ ´ v ℓ , then y ‰
0. Moreover, as remarkedjust before Definition 3.4, the first nonzero entry in v ˚ ℓ as well as v ℓ is 1. Hence p p y q ą p p v ˚ ℓ q “ p p v ℓ q . Also, y P F k , since v ˚ ℓ P F k and v ℓ P V ℓ “ U ℓ Ď F k . Thusfrom Lemma 3.2, we see that y P V ˚ s for some positive integer s ă ℓ . But then y P V s , and so y P F j , which is a contradiction. This completes the proof in Case 1. Case 2. ℓ ě e .Here V ˚ s “ V s “ W s for 1 ď s ă e . Also w ă v , where w : “ min W e and v : “ min V e . So by Lemma 3.3, p p v q ď p p w q . Now pick any z P V ˚ e so that V ˚ e “ V ˚ e ´ ‘ h z i “ V e ´ ‘ h z i and, by Lemma 3.2, p p V ˚ e q “ t p p z qu . Now w P F k zt u and so w P V ˚ s for a unique positive integer s ď r . Also since w P W e , wesee that w R W e ´ “ V ˚ e ´ . Thus s ě e and therefore, in view of Lemma 3.2, p p v q ď p p w q ď p p z q . Now if p p v q “ p p z q , then p p w q “ p p z q , and so w P V ˚ e .Consequently, min V ˚ e ĺ w ă v “ min V e , which implies that F k ă F j . On the other hand, if p p v q ă p p z q , then by Lemma 3.3, z ă v , and hence min V ˚ e ĺ z ă v “ min V e , which implies once again that F k ă F j , as desired. (cid:3) We remark that the shellability of the q -sphere S n ´ q is a trivial consequence ofTheorem 4.4, because S n ´ q is precisely the q -matroid complex corresponding tothe uniform matroid U q p n ´ , n q .5. Homology of q -Spheres and Shellable q -Complexes This section is divided into three subsections. In § q -spheres and explicitly determine their reduced homology groups in § § q -complexes that are shellable and outline an attemptto determine their homology. HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 11 Topological Preliminaries.
Finite topological spaces, or in short, finitespaces, are simply topological spaces having only a finite number of points. Incase they are T , the topology is necessarily discrete and not so interesting. Rathersurprisingly, finite spaces that are T (but not T ) have a rich structure and a closeconnection with finite posets. The study of finite spaces goes back to Alexandroff[2] and had important contributions by Stong [17] and McCord [12]. Good expo-sitions of the theory of finite spaces are given by May [11] and Barmak [3]. Still,the theory is not as widely known as it should, and so for the convenience of thereader, we provide here a quick review of the relevant notions and results.Let X be a finite T space. Then for each x P X , the intersection, say U x , of allopen sets of X containing x is open. Clearly t U x : x P X u is a basis for (the topologyon) X . For x, y P X , define x ď y if x P U y . Then this defines a partial order on X (since X is T ); moreover U y becomes the “basic down-set” t x P X : x ď y u .On the other hand, suppose X is a finite poset (with the partial order denotedby ď ). We call a subset U of X a down-set (resp. up-set ) if whenever y P U and x P X satisfy x ď y (resp. y ď x ), we must have x P U . We can define a topologyon X by declaring that the open sets in X are precisely the down-sets in X (orequivalently, the closed sets in X are precisely the up-sets in X ). This is called the order topology on X , and it makes X a finite T space.Let X, Y be finite posets, both regarded as finite topological spaces with theorder topology. Then it can be shown (cf. [3, Proposition 1.2.1]) that a function f : X Ñ Y is continuous if and only if it is order-preserving. Further, if we let Y X denote the set of all continuous functions from X to Y , then Y X is a poset withthe pointwise partial order defined (for any f, g P Y X ) by f ď g if f p x q ď g p x q for every x P X . Thus Y X can also be regarded as a finite topological space withthe order topology. Moreover f, g P Y X are homotopic (which means, as usual,that there is a continuous map h : X ˆ r , s Ñ Y such that h p x, q “ f p x q and h p x, q “ g p x q for all x P X ) if and only there is a continuous map α : r , s Ñ Y X such that α p q “ f and α p q “ g . We write f » g if f, g P Y X are homotopic.Also, X and Y are said to be homotopy equivalent if there are f, g P Y X such that f ˝ g » Id Y and g ˝ f » Id X . Finally, recall that X is said to be contractible if itis homotopy equivalent to a point. Note that the homotopy groups as well as thereduced (singular) homology groups of contractible spaces are all trivial.We now recall some known basic results for which a reference is given. Thesewill be useful to us later. Unless mentioned otherwise, the topology on finite posetsis assumed to be the order topology and topological notions such as continuity,contractibility are considered with respect to this topology. Proposition 5.1 ([3, Corollary 1.2.6]) . Let
X, Y be finite posets and let f, g P Y X .Then f » g if and only if there is a finite sequence f , f , . . . , f t in Y X such that f “ f ď f ě f ď ¨ ¨ ¨ f t “ g . Proposition 5.2 ([11, Corollary 2.3.4]) . Let X be a finite poset such that X has aunique maximal element or a unique minimal element. Then X is contractible. A finer version of Proposition 5.2 for posets that are of interest to us in thisarticle is the following.
Lemma 5.3.
Let ∆ be a nonempty collection of subspaces of E : “ F nq . Call theelements of ∆ as the faces of ∆ and those faces of ∆ that are maximal with respectto inclusion as the facets of ∆ . Assume that any finite intersection of facets of ∆ that contain a fixed face of ∆ is necessarily a face of ∆ . Suppose there is A P ∆ such that A Ď B for every facet B of ∆ . Then ∆ is contractible.Proof. Fix any C P ∆. Consider f : ∆ Ñ t C u and g : t C u Ñ ∆ defined by f p U q : “ C for all U P ∆ and g p C q : “ A. Clearly, f and g are continuous and f ˝ g “ Id t C u . We will show that g ˝ f » Id ∆ .To this end, define, for any U P ∆, the set V U to be the intersection of all facets of∆ containing U . Let h : ∆ Ñ ∆ be defined by h p U q : “ V U for U P ∆. Observe thatif U , U P ∆ with U Ď U , then any facet of ∆ containing U must contain U ,and therefore, V U Ď V U . Thus h is order-preserving and hence it is continuous.By our hypothesis, A Ď V U for every U P ∆. Hence g ˝ f ď h . Also, since U Ď V U for any U P ∆, we obtain Id ∆ ď h . Thus it follows from by Proposition 5.1 that ∆is homotopy equivalent to t C u . This proves that ∆ is contractible. (cid:3) Definition 5.4.
A set ∆ of satisfying the hypothesis in Proposition 5.3 is called a cone with apex A .5.2. Homology of q -Spheres. If ∆ is a q -complex on E : “ F nq , then ∆ is a finitetopological space with the order topology corresponding to the partial order givenby inclusion. As a topological space, it is contractible because it has a uniqueminimal element, namely, the zero space t u and so Proposition 5.2 applies. Thus,the homology (as well as homotopy) groups of ∆ are trivial. With this in view, andas in the classical case, we will replace ∆ by the punctured q -complex ∆ : “ ∆ z tt uu obtained by removing the zero subspace from ∆. Thus, when we speak of thehomology of ∆, we shall in fact mean the homology of ∆. In this subsection, wewill outline how the (reduced) homology of q -spheres can be computed explicitly.Recall that the q -sphere S n ´ q is the q -complex formed by all the subspaces of E : “ F nq other than E itself. So the punctured q -sphere S n ´ q consists of all thesubspaces of E other than E and t u . It is equipped with the order topologyw.r.t. inclusion. In particular, S n ´ q is the empty set if n “
1. When n “
2, thepunctured q -sphere S n ´ q consists of q ` F q ,which form connected components with respect to the order topology. Thus the HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 13 h x, y, z ih x, y i h x, z i h y, z i h x, y ` z i h y, x ` z i h x ` y, x ` z i h z, x ` y ih x i h y i h z i h x ` y i h x ` z i h y ` z i h x ` y ` z ih , , i Figure 1.
Illustration of the punctured q -sphere S q when q “ n “ n “
2. But the poset structureand the homology becomes a little more difficult to determine when n ě
3. Forexample, the poset structure of the punctured q -sphere S q when q “ x, y, z denote linearly independentelements of F . It is seen here that unlike in the case n “
2, the q -sphere is aconnected space when n “ q -spheres is the following lemma. Here,and hereafter, for a F q -vector space F , we denote by Σ p F q the set of all nonzerosubspaces of F . Lemma 5.5.
Assume that n ě . Then there exists a shelling F , . . . , F t of the q -sphere S n ´ q and a positive integer ℓ ď t such that if for ď i ď t , we let ∆ i : “ x F , . . . , F i y , then the punctured q -complex ∆ ℓ is contractible and moreover, (4) ∆ i X Σ p F i ` q “ Σ p F i ` qzt F i ` u for ℓ ď i ă t, that is, ∆ i X Σ p F i ` q is the punctured q -sphere S n ´ q for each i “ ℓ, . . . , t ´ .Proof. We have seen in Example 4.2 that any ordering of the facets of S n ´ q gives ashelling of S n ´ q . To obtain a shelling with the additional two properties asserted inthe lemma, we proceed as follows. Fix an arbitrary nonzero vector a in F nq . Suppose F , . . . , F ℓ are all the facets of S n ´ q containing a . In other words t F , . . . , F ℓ u isthe set of all p n ´ q -dimensional subspaces of F nq codimension which contain a .Also, let F ℓ ` , . . . , F t denote all the facets of S n ´ q , which do not contain a . Write∆ i : “ x F , . . . , F i y for 1 ď i ď t . Then h a i is the unique minimal element of ∆ ℓ ,and hence by Proposition 5.3, ∆ ℓ is contractible. To prove that F , . . . F t also satisfies (4), first suppose n “
2. Then it is clearthat t “ ℓ ` F t “ h a i . So in this case, Σ p F ℓ ` q “ t h a i u “ t F ℓ ` u and since F ℓ ` R ∆ ℓ , it is clear that the two sets on either sides of the equality in (4) areboth empty. Now suppose n ě
3. Fix i P N such that ℓ ď i ă t . Since F i ` R ∆ i ,it is clear that ∆ i X Σ p F i ` q Ď Σ p F i ` qzt F i ` u . To prove the other inclusion, itsuffices to show that every facet of Σ p F i ` qzt F i ` u is in ∆ i . Let G be a facet ofΣ p F i ` qzt F i ` u . Since i ě ℓ , we see that a R G . Hence G ‘ h a i is a facet of S n ´ q containing a , and therefore, G ‘ h a i “ F k for some positive integer k ď ℓ . Inparticular, G Ď F k and so G P ∆ k Ď ∆ i . (cid:3) Remark 5.6.
It is possible to describe the positive integers t and ℓ in Lemma 5.5explicitly. Indeed, t is the number of subspaces of F nq of dimension n ´
1. Also, theproof of Lemma 5.5 shows that we can take ℓ to be the number of subspaces of F nq of dimension n ´ a . Consequently, both t and ℓ can be described in terms of Gaussian binomial coefficients as follows. t “ „ nn ´ q “ q n ´ q ´ ℓ “ „ n ´ n ´ q “ q n ´ ´ q ´ . Observe that t ´ ℓ “ q n ´ .Let us recall that as per standard conventions in topology, if X is the emptyset, then its reduced homology group r H p p X q is Z if p “ ´ . Ingeneral, the homology groups of (punctured) q -spheres are given by the following. Theorem 5.7.
Let c n : “ q n p n ´ q{ . Then the reduced homology groups of thepunctured q -sphere S n ´ q are given by Ă H p p S n ´ q q “ $&% Z c n if p “ n ´ , otherwise . Proof.
We use induction on n . If n “
1, then the desired result follows from thestandard conventions about the reduced homology of the empty set.Now suppose n ě n smaller than the givenone. Let F , . . . , F t be a shelling of S n ´ q as in Lemma 5.5, and let ℓ be the positiveinteger as in Lemma 5.5 and Remark 5.6. Also let ∆ i , for 1 ď i ď t , be as inLemma 5.5. In the first step, we take X : “ ∆ ℓ and X : “ Σ p F ℓ ` q . Note that both X and X are down-sets, and thus they are open subsets of S n ´ q .Moreover, X Y X “ ∆ ℓ ` , and by Lemma 5.5, X X X can be identified with the Indeed, a p -simplex is the convex hull of p ` p “ ´
1, then this is the empty set,while the singular p -simplex in X consists precisely of the empty function, and the free abeliangroup C p p X q generated by it is Z . On the other hand, all other chain complexes are 0. HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 15 punctured q -sphere S n ´ q . Let us apply the Mayer-Vietoris sequence for reducedhomology: Ă H p p X q‘ Ă H p p X q ÝÑ Ă H p p X Y X q ÝÑ r H p ´ p X X X q ÝÑ r H p ´ p X q‘ r H p ´ p X q and observe that by Lemma 5.5, X is contractible , and since X has a uniquemaximal element (viz., F ℓ ` ), by Proposition 5.2 , X is also contractible. Thusboth the direct sums in the above exact sequence are 0, and we obtain Ă H p p ∆ ℓ ` q “ Ă H p p X Y X q – r H p ´ p X X X q “ r H p ´ p S n ´ q q . So by the induction hypothesis, Ă H p p ∆ ℓ ` q is equal to Z c n ´ if p ´ “ n “
3, i.e., p “ n ´
2, and 0 otherwise. In the next step, we take X : “ ∆ ℓ ` and X : “ Σ p F ℓ ` q , and note that X , X are open subsets of S n ´ q such that X Y X “ ∆ ℓ ` , and byLemma 5.5, X X X can be identified with the punctured q -sphere S n ´ q . Let usapply (a slightly longer) Mayer-Vietoris sequence for reduced homology: Ă H p p X X X q ÝÑ Ă H p p X q ‘ Ă H p p X q ÝÑ Ă H p p X Y X q §§đr H p ´ p X X X q ÝÑ r H p ´ p X q ‘ r H p ´ p X q This time X is contractible, whereas the homology of X is determined in theprevious step, while that of X X X is known, as before, by the induction hypothesis.Using this for p “ n ´
2, we obtain0 ÝÑ Z c n ´ ÝÑ r H n ´ p ∆ l ` q ÝÑ Z c n ´ ÝÑ . The short exact sequence above splits (since Z c n ´ is a projective Z -module, beingfree), and therefore r H n ´ p ∆ l ` q “ Z c n ´ ‘ Z c n ´ . Moreover, r H p p ∆ l ` q “ p ‰ p n ´ q . Now if ℓ ` ă t , we can proceed as before, and we shall obtain that r H p p ∆ l ` q is Z c n ´ ‘ Z c n ´ ‘ Z c n ´ if p “ n ´
2, and 0 otherwise. Continuing in thisway, we see that r H p p ∆ t q is the direct sum of p t ´ ℓ q copies of Z c n ´ if p “ n ´ t “ S n ´ q and in view of Remark 5.6, p t ´ ℓ q c n ´ “ q n ´ q p n ´ qp n ´ q{ “ q n p n ´ q{ “ q c n . This yields the desired result. (cid:3)
Homology of Shellable q -Complexes. We shall now attempt to determinethe homology of a shellable q -complex. We proceed in a manner analogous to theclassical case of simplicial complexes. But as we shall see, there are some difficultiesin obtaining results analogous to those in the classical case.In the classical case, the notion of restriction R p F q of a face F plays an importantrole in the determination of the homology of a shellable simplicial complex; see, e.g.,[4, § q -complexes, a straightforward analogue is not possible because the complement of an element (or even of a one-dimensional subspace) ina F q -linear subspace need not be a subspace. However, the sets I i defined belowprovide an analogue of the intervals r R p F i q , F i s in the classical case. Definition 5.8.
Let F , . . . , F t be a shelling of a shellable q -complex ∆ on E : “ F nq .For 1 ď j ă i ď t , we define R i,j : “ t x P F i : x x y ‘ p F i X F j q “ F i u . and for 1 ď i ď t , we define I i : “ t A P x F i y : A X R i,j ‰ H whenever 1 ď j ă i and R i,j ‰ Hu for 1 ď i ď t. Remark 5.9. If i, j and F , . . . , F t are as in Definition 5.8 and if F i X F j is not ahyperplane in F i , i.e., if dim p F i X F j q ă dim F i ´
1, then R i,j “ H . Also note thatby Lemma 4.3, I i is nonempty for each i “ , . . . , t .The following result gives a partition of a shellable q -complex and it is analogousto [4, Proposition 7.2.2]. Theorem 5.10.
Let F , . . . , F t be a shelling of a shellable q -complex ∆ . For any i “ , , . . . , t , let ∆ i denote the subcomplex x F , . . . , F i y of ∆ generated by F , . . . , F i (in particular, ∆ “ H , as per our convention). Then for any i P N with ď i ď t , (5) ∆ i “ I i Y ∆ i ´ and I i X ∆ i ´ “ H . Consequently, we obtain a partition of ∆ as a disjoint union of “intervals”: (6) ∆ “ t ž i “ I i . Proof.
Clearly, I “ ∆ , and so (5) holds when i “
1. Now suppose 2 ď i ď t . Theinclusion I i Y ∆ i ´ Ď ∆ i is obvious. To prove the other inclusion, let A P ∆ i z I i .Then there is j P N ` with j ă i such that R i,j ‰ H and R i,j X A “ H . Hence any x P R i,j has the property that x R A but x x y ‘ p F i X F j q “ F i . On the other hand,if A * F i X F j , then A would contain an element of R i,j which is impossible by ourassumption. It follows that A Ď F i X F j , and therefore A P ∆ i ´ . This proves that∆ i Ď I i Y ∆ i ´ . Thus ∆ i “ I i Y ∆ i ´ .To show that I i X ∆ i ´ “ H , assume the contrary. Thus, suppose there exists A P I i X ∆ i ´ . Let S : “ t j P N ` : j ă i and R i,j ‰ Hu . Then A X R i,j ‰ H for all j P S , and so we can choose x j P A X R i,j for each j P S . Define G : “ h t x j : j P S u i . Now G P I i and G Ď A Ď F k for some k ă i (because A P ∆ i ´ ).Thus G Ď F k X F i . By Lemma 4.3, there exists ℓ ă i such that F k X F i Ď F ℓ X F i and dim p F ℓ X F i q “ dim F i ´
1. Consequently, R i,ℓ ‰ H , and so ℓ P S . But then x x ℓ y ‘ p F ℓ X F i q “ F i (by the definition of R i,ℓ ), which is a contradiction because x ℓ P G Ď F ℓ X F i . This shows that I i X ∆ i ´ “ H and thus (5) is proved.Finally, (6) follows from (5) by noting that ∆ “ ∆ t and ∆ “ I . (cid:3) HELLABILITY AND HOMOLOGY OF q -COMPLEXES AND q -MATROIDS 17 Recall that for a vector space F , we use Σ p F q to denote the punctured q -complexformed by all the nonzero subspaces of F . Lemma 5.11.
Let F be a vector space of dimension r over F q . Let ℓ P N ` and let G , . . . , G ℓ be subspaces of F of dimension r ´ . For i P N ` with ď i ď l , define U i : “ t x P F : h x i ‘ G i “ F u and I : “ t A P Σ p F q : A X U i ‰ H for i “ , . . . , ℓ u . Then Σ p F qz I “ l ď i “ Σ p G i q . Proof.
Suppose A P Σ p F qz I . Then A X U i “ H for some i P N ` with 1 ď i ď ℓ .We claim that A Ď G i . Indeed, if there is x P A z G i , then h x i ‘ G i “ F . Butthen x P A X U i which is a contradiction. Therefore A P Σ p G i q .On the other hand, if A is a nonzero subspace of G i for some i P t , . . . , ℓ u , thenany element x of A cannot be in U i because h x i ` G i “ G i . Thus A X U i “ H .Hence A R I . This proves the lemma. (cid:3) The above lemma says that Σ p F qz I is a pure q -complex with G i ’s as its facets.We show below that the corresponding punctured q -complex is particularly nice. Corollary 5.12.
Let the notations and hypothesis be as in Lemma 5.11. Furtherlet U : “ U Y ¨ ¨ ¨ Y U ℓ . If U ‰ p F zt uq and if x is any nonzero element of F z U ,then Σ p F qz I is a cone with apex x . Consequently, Σ p F qz I is contractible.Proof. If x R U , we see that G i Ď h x i ` G i ( F for 1 ď i ď ℓ . Hence G i “ G i ` h x i and so x P G i for each i “ , . . . , ℓ . In view of Lemma 5.11, we see that h x i iscontained in every facet of Σ p F qz I . Thus Σ p F qz I is a cone with apex x . The lastassertion follows from Lemma 5.3. (cid:3) Corollary 5.13.
Let F , . . . , F t be a shelling of a shellable q -complex ∆ . Supposethere is i P N ` with ď i ď t such that i ´ ď j “ R i,j ‰ F i zt u . Then Σ p F i qz I i is contractible.Proof. If we take F “ F i in Corollary 5.12, then we see that the set U i is pre-cisely the set t R i,j : 1 ď j ď i ´ , R i,j ‰ Hu and moreover, I “ I i . Thus thecontractibility of Σ p F i qz I i follows from Corollary 5.12. (cid:3) Recall that a topological space is acyclic if all of its reduced homology groupsare trivial. It is well-known that a contractible space is acyclic, but the converse isnot true, in general.
Theorem 5.14.
Suppose F , . . . , F t is a shelling of a shellable q -complex ∆ andthere is a positive integer ℓ ď t such that i ´ ď j “ R i,j ‰ F i zt u for all ď i ď ℓ .Then ∆ ℓ is acyclic, where ∆ ℓ : “ h F , . . . , F ℓ i .Proof. We prove by induction on i (1 ď i ď ℓ ) that each ∆ i is acyclic. Since ∆ “ Σ p F q , has a unique maximal element, by Lemma 5.3 we see that it iscontractible, and therefore acyclic. Now assume that 1 ď i ď t and ∆ i is acyclic.We want to show that ∆ i ` is also acyclic. Now Σ p F q i ` is contractible, and henceacyclic, while ∆ i is acyclic by the induction hypothesis. Moreover, by Lemma 5.5, ∆ i X Σ p F i ` q “ Σ p F i ` qz I i ` , and this is contractible by Corollary 5.13. Thus byapplying Mayer-Vietoris sequence, we see that ∆ i ` “ ∆ i Y Σ p F q i ` is acyclic.Thus the result is proved by induction. (cid:3) We conclude this section by some remarks about our attempt to determine ho-mology for arbitrary shellable q -complexes by adapting the method for shellablesimplicial complexes.In Theorem 5.14, we have seen that from a shellable complex ∆ if we removethe facets F i with Y i ´ j “ R i,j “ F i zt u , then the resulting subcomplex ∆ is acyclic.This result proves a somewhat similar fact for shellable simplicial complex provedin [4, Lemma 7.7.1]. In the inductive proof of [7, Theorem 6.9] about the homologyof shellable complexes, we attach one face F with R p F q “ F and use the Mayer-Vietoris sequence to calculate the singular homology of ∆ Y Σ p F q . Applying Mayer-Vietoris was effective because for a facet F of a simplicial complex with R p F q “ F ,the intersection ∆ X Σ p F q is the boundary complex of F . And the boundarycomplex being a sphere, we knew its homology. But in the case of q -complexes, forany F i with Y i ´ j “ R i,j “ F i zt u , we do not know whether the intersection ∆ X Σ p F i q is a (punctured) q -sphere or not.At any rate, we summarize the observations in the next result and give an ex-pression for the homology of certain shellable q -complexes in terms of homologiesof q -spheres. Theorem 5.15.
Let ∆ be a pure q -complex of dimension d . Assume that F , . . . , F t is a shelling on ∆ and there is an integer ℓ with ď l ď t such that if ∆ is thesubcomplex generated by the facets F , . . . , F ℓ , then ∆ is contractible. Also assumethat Σ p F i q X ∆ i ´ is the q -sphere S d ´ q for all i P N with ℓ ă i ď t . Then Ă H p p ∆ q “ $&% Ć H p ´ p S d ´ q q ‘p t ´ l q if p “ d ´ , if p ‰ . Proof.
Follows using similar arguments as in Theorem 5.7. (cid:3) herefore the determination of singular homology of arbitrary shellable q -complexesstill remains open. Other than that, the previous theorem also leads to the follow-ing natural question: What are the classes of q -complexes for which the hypothesisof Theorem 5.15 is satisfied? We have seen that q -spheres do satisfy them. But arethey all? References [1] S. J. Alder, On q –Simplicial Posets , Ph.D. Thesis, Univ. of East Anglia, UK, 2010.[2] P. Alexandroff, Diskrete r¨aume, Mat. Sb. (N.S.) (1937), 501–518.[3] J. A Barmak, Algebraic Topology of Finite Topological Spaces and Applications , Lect.Notes in Math. , Springer, Heidelberg, 2011.[4] A. Bj¨orner, The homology and shellability of matroids and geometric lattices, in:
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Theory of Matroids , Encyclopedia Math. Appl. , , Cambridge Univ.Press, Cambridge, 1986. Department of Mathematics, Indian Institute of Technology Bombay,Powai, Mumbai 400076, India
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