Exact sequences for the homology of the matching complex
2008 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 115, no 8, 1504-1526 p.Article in journal (Refereed) Published
Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex M., which is the simplicial complex of matchings in the complete graph K-n. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of M, First, we demonstrate that there is nonvanishing 3-torsion in (H) over bar (d)(M-n : Z) whenever v(n) <= d <= [n-6/2], where v(n) =[n-4/3]. By results due to Bouc and to Shareshian and Wachs, (H) over bar (d)(M-n : Z) is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of M. for all n 0 2. Second, we prove that (H) over bar (d)(M-n : Z) is a nontrivial 3-group whenever v(n) <= d <= [2n-9/5]. Third, for each k >= 0, we show that there is a polynomial f(k)(r) of degree 3k such that the dimension of (H) over bar (k-1+r) (M2k+1+3r:Z(3)), viewed as a vector space over Z(3), is at most f(k)(r) for all r >= k + 2.
Place, publisher, year, edition, pages
2008. Vol. 115, no 8, 1504-1526 p.
Matching complex, Simplicial homology, Long exact sequence, chessboard complexes
IdentifiersURN: urn:nbn:se:kth:diva-17919DOI: 10.1016/j.jcta.2008.03.001ISI: 000260441300011ScopusID: 2-s2.0-52749092281OAI: oai:DiVA.org:kth-17919DiVA: diva2:335964
QC 201005252010-08-052010-08-05Bibliographically approved