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Exact Sequences for the Homology of the Matching Complex
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 , which is the simplicial complex of matchings in the complete
graph . Combining these sequences in different ways, we prove several
results about the 3-torsion part of the homology of . First, we
demonstrate that there is nonvanishing 3-torsion in whenever
\nu_n \le d \le (n-6}/2, where . By results due
to Bouc and to Shareshian and Wachs, is a nontrivial
elementary 3-group for almost all and the bottom nonvanishing homology
group of for all . Second, we prove that is a
nontrivial 3-group whenever . Third, for each , we show that there is a polynomial of degree 3k such that the
dimension of , viewed as a vector space over ,
is at most for all .Comment: 31 page
Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree
Given a link in the three-sphere, Ozsv\'ath and Szab\'o showed that there is
a spectral sequence starting at the Khovanov homology of the link and
converging to the Heegaard Floer homology of its branched double cover. The aim
of this paper is to explicitly calculate this spectral sequence in terms of
bordered Floer homology. There are two primary ingredients in this computation:
an explicit calculation of bimodules associated to Dehn twists, and a general
pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on
computing the bimodules; this part focuses on the pairing theorem for polygons,
in order to prove that the spectral sequence constructed in the previous part
agrees with the one constructed by Ozsv\'ath and Szab\'o.Comment: 85 pages, 19 figures, v3: Version to appear in Journal of Topolog
Five-Torsion in the Homology of the Matching Complex on 14 Vertices
J. L. Andersen proved that there is 5-torsion in the bottom nonvanishing
homology group of the simplicial complex of graphs of degree at most two on
seven vertices. We use this result to demonstrate that there is 5-torsion also
in the bottom nonvanishing homology group of the matching complex on
14 vertices. Combining our observation with results due to Bouc and to
Shareshian and Wachs, we conclude that the case is exceptional; for all
other , the torsion subgroup of the bottom nonvanishing homology group has
exponent three or is zero. The possibility remains that there is other torsion
than 3-torsion in higher-degree homology groups of when and .Comment: 11 page
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