Exact sequences for the homology of the matching complex

Jönsson, Jakob

doi:10.1016/j.jcta.2008.03.001

Cited by 12 publications

(18 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We proceed to construct a long exact sequence that relates the reduced homology groups of C N to those of complexes C N ′ , for N ′ ≤ N . Such long exact sequences have been previously studied in the case of matching complexes by [Bou92,SW07,Jon08], and in that of chessboard complexes by [BLVŽ94,SW07].…”

Section: Inductive Approach To Computing the Homology Of Packing Compmentioning

confidence: 99%

Representation stability for syzygies of line bundles on Segre–Veronese varieties

Raicu¹

2016

J. Eur. Math. Soc.

View full text Add to dashboard Cite

The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre-Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology groups of packing complexes. This allows us to deduce stability properties for the syzygies of line bundles on Segre-Veronese varieties. We provide bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties.As a motivation for our investigation, we show in an appendix that Ein and Lazarsfeld's conjecture on the asymptotic vanishing of syzygies of coherent sheaves on arbitrary projective varieties reduces to the case of line bundles on a product of (at most three) projective spaces.where given a partition δ = (δ 1 , δ 2 , · · · ) of some integer r we write δ[m] for the partition (m − r, δ 1 , δ 2 , · · · ). S δ denotes the Schur functor associated to δ, and we make the convention that S δ[m] is identically zero when m − r < δ 1 .Note that the conclusion of the theorem remains true in the case n = 1 if we replace K p,0 (B a ) with the p-th syzygy module of m a , where m is the homogeneous maximal ideal in the polynomial ring S = Sym(V ): it is well-known (see [BE75, Cor. 3.2] or [Gre84b, (1.a.10)]) that the minimal free resolution of m a is given byTheorem 6.1 was known in the case n = 2 where in fact all the modules K p,q (B a ) can be described explicitly (see [FH98,RR00] or [Wey03, Chapter 6] for a more general story). We will prove Theorem 6.1 by applying the techniques of [FH98] involving combinatorial Laplacians.The description of syzygies in Theorem 6.1 is fairly explicit, the only mystery being the calculation of the multiplicities m λ . This is known to be a complicated plethysm problem, and our theorem is meant to illustrate that the problem of computing syzygies even for simple modules supported on a product of projective spaces is in some sense equally difficult. An asymptotic measure of the complexity of the syzygies in the linear and quadratic strands (K p,0 and K p,1 ) for the Veronese varieties has been obtained by Fulger and Zhou [FZ12] by analyzing the number of distinct irreducible representations appearing in these syzygy modules, as well as the sum of their multiplicities. In Theorem 6.4 we provide a concrete illustration of their theory by describing the linear syzygies of O(1) under a Veronese embedding.We view Theorem 6.1 as a stabilization result in the following way, which we'll be able to generalize further: for a large enough (a ≥ p) the number of irreducible representations (counted with multipli...

show abstract

Section: Inductive Approach To Computing the Homology Of Packing Compmentioning

confidence: 99%

Representation stability for syzygies of line bundles on Segre–Veronese varieties

Raicu¹

2016

J. Eur. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Specifically, the bottom nonvanishing homology group of M n is known to be an elementary 3-group for almost all n [3,12]. In fact, there is a nearly complete characterization of all (n, d) such that the homology groupH d (M n ; Z) contains elements of order 3 [6]; see Proposition 1.4 (4) and (5) for a summary. As for the existence of elements of order p for primes different from 3, nothing is known besides the recent discovery [7] thatH 4 (M 14 ; Z) contains elements of order 5.…”

Section: Introductionmentioning

confidence: 99%

“…By the following proposition, the result thatH 6 (M 19 ; Z) andH 8 (M 24 ; Z) contain elements of order 5 turns out to be of particular importance. Proposition 1.2 (Jonsson [6]) For q ≥ 3, ifH 2q (M 5q+4 ; Z) contains elements of order 5, then so doesH 2q+u (M 5q+4+2u ; Z) for each u ≥ 0. Theorem 1 and Proposition 1.2 imply the following.…”

Section: Introductionmentioning

confidence: 99%

More Torsion in the Homology of the Matching Complex

Jönsson¹

2010

Experimental Mathematics

Self Cite

View full text Add to dashboard Cite

A matching on a set $X$ is a collection of pairwise disjoint subsets of $X$ of size two. Using computers, we analyze the integral homology of the matching complex $M_n$, which is the simplicial complex of matchings on the set $\{1, >..., n\}$. The main result is the detection of elements of order $p$ in the homology for $p \in \{5,7,11,13\}$. Specifically, we show that there are elements of order 5 in the homology of $M_n$ for $n \ge 18$ and for $n \in {14,16}$. The only previously known value was $n = 14$, and in this particular case we have a new computer-free proof. Moreover, we show that there are elements of order 7 in the homology of $M_n$ for all odd $n$ between 23 and 41 and for $n=30$. In addition, there are elements of order 11 in the homology of $M_{47}$ and elements of order 13 in the homology of $M_{62}$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $H_d(M_n;Z)$ for $13 \le n \le 16$; a complete description of the homology already exists for $n \le 12$. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of $M_n$ obtained by letting certain groups act on the chain complex.Comment: 35 pages, 10 figure

show abstract

“…A previous paper [12] contains a number of results about the integral homology of the matching complex M n . The purpose of the present paper is to extend a few of these results to the chessboard complex M m,n .…”

Section: Introductionmentioning

confidence: 99%

On the 3-Torsion Part of the Homology of the Chessboard Complex

Jönsson

2010

Ann. Comb.

Self Cite

View full text Add to dashboard Cite

Let 1 ≤ m ≤ n. We prove various results about the chessboard complex M m, n , which is the simplicial complex of matchings in the complete bipartite graph K m, n . First, we demonstrate that there is nonvanishing 3-torsion inH d (M m, n ; Z) whenever m+n−4 3 ≤ d ≤ m−4 and whenever 6 ≤ m < n and d = m − 3. Combining this result with theorems due to Friedman and Hanlon and to Shareshian and Wachs, we characterize all triples (m, n, d) satisfying H d (M m, n ; Z) = 0. Second, for each k ≥ 0, we show that there is a polynomial f k (a, b) of degree 3k such that the dimension ofH k+a+2b−2 M k+a+3b−1, k+2a+3b−1 ; Z 3 , viewed as a vector space over Z 3 , is at most f k (a, b) for all a ≥ 0 and b ≥ k + 2. Third, we give a computer-free proof thatH 2 M 5, 5 ; Z ∼ = Z 3 . Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of M m, n to the homology of M m−2, n−1 and M m−2, n−3 .

show abstract

Exact sequences for the homology of the matching complex

Cited by 12 publications

References 16 publications

Representation stability for syzygies of line bundles on Segre–Veronese varieties

Representation stability for syzygies of line bundles on Segre–Veronese varieties

More Torsion in the Homology of the Matching Complex

On the 3-Torsion Part of the Homology of the Chessboard Complex

Contact Info

Product

Resources

About