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3-torsion in the Homology of Complexes ofGraphs of Bounded Degree

Published online by Cambridge University Press:  20 November 2018

Jakob Jonsson
Jakob Jonsson*
Affiliation:
Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden, e-mail: jakobj@math.kth.se
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Abstract

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For $\delta \ge 1$ and $n\ge 1$, consider the simplicial complex of graphs on $n$ vertices in which each vertex has degree at most $\delta$; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When $\delta =1$, we obtain the matching complex, for which it is known that there is 3-torsion in degree $d$ of the homology whenever $\left( n-4 \right)/3\le d\le \left( n-6 \right)/2$. This paper establishes similar bounds for $\delta \ge 2$. Specifically, there is 3-torsion in degree $d$ whenever

$$\frac{\left( 3\delta -1 \right)n-8}{6}\le d\le \frac{\delta \left( n-1 \right)-4}{2}.$$

The procedure for detecting torsion is to construct an explicit cycle $z$ that is easily seen to have the property that $3z$ is a boundary. Defining a homomorphism that sends $z$ to a non-boundary element in the chain complex of a certain matching complex, we obtain that $z$ itself is a non-boundary. In particular, the homology class of $z$ has order 3.

Keywords

05E4555U1005C0720K10simplicial complexsimplicial homologytorsion groupvertex degree
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 4 , 01 August 2013 , pp. 843 - 862
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Andersen, J. L., Determinantal rings associated with matrices: a counterexample. Ph.D. Dissertation, University of Minnesota, 1992.Google Scholar
[2] Bouc, S., Homologie de certains ensembles de 2sous- groupes des groupes symétriques. J. Algebra 150(1992), no. 1, 158–186. http://dx.doi.org/10.1016/S0021-8693(05)80054-7 Google Scholar
[3] Dong, X. and L.Wachs, M., Combinatorial Laplacian of the matching complex. Electron. J. Combin. 9(2002), no. 1, R17.Google Scholar
[4] Jonsson, J., Simplicial complexes of graphs. Lecture Notes in Mathematics, 1928, Springer-Verlag, Berlin, 2008.Google Scholar
[5] Jonsson, J., Five-torsion in the homology of the complex on 14 vertices. J. Algebraic Combin. 29(2009),no. 1, 81–90. http://dx.doi.org/10.1007/s10801-008-0123-6 Google Scholar
[6] Jonsson, J., More torsion in the homology of the matching complex. Experiment. Math. 19(2010), no. 3, 363–383. http://dx.doi.org/10.1080/10586458.2010.10390629 Google Scholar
[7] Reiner, V. and Roberts, J., Minimal resolutions and homology of chessboard and matching complexes. J. Algebraic Combin. 11(2000), no. 2, 135–154. http://dx.doi.org/10.1023/A:1008728115910 Google Scholar
[8] Shareshian, J. and L.Wachs, M., Torsion in the matching and chessboard complexes. Adv. Math. 212(2007), no. 2, 525–570. http://dx.doi.org/10.1016/j.aim.2006.10.014 Google Scholar
[9] Stanley, R. P., Combinatorics and commutative algebra. Second ed., Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996.Google Scholar