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

Jonsson, Jakob

Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 843-862

Résumé

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.

Détail

DOI : 10.4153/CJM-2013-008-4

Mots-clés : 05E45, 55U10, 05C07, 20K10, simplicial complex, simplicial homology, torsion group, vertex degree

Bibliographie

[1] [1] Andersen, J. L., Determinantal rings associated with matrices: a counterexample. Ph.D. Dissertation, University of Minnesota, 1992. Google Scholar

[2] [2] Bouc, S., Homologie de certains ensembles de 2sous- groupes des groupes symétriques. J. Algebra 150(1992), no. 1, 158–186. Google Scholar | DOI

[3] [3] Dong, X. and L.Wachs, M., Combinatorial Laplacian of the matching complex. Electron. J. Combin. 9(2002), no. 1, R17. Google Scholar

[4] [4] Jonsson, J., Simplicial complexes of graphs. Lecture Notes in Mathematics, 1928, Springer-Verlag, Berlin, 2008. Google Scholar

[5] [5] Jonsson, J., Five-torsion in the homology of the complex on 14 vertices. J. Algebraic Combin. 29(2009),no. 1, 81–90. Google Scholar | DOI

[6] [6] Jonsson, J., More torsion in the homology of the matching complex. Experiment. Math. 19(2010), no. 3, 363–383. Google Scholar | DOI

[7] [7] Reiner, V. and Roberts, J., Minimal resolutions and homology of chessboard and matching complexes. J. Algebraic Combin. 11(2000), no. 2, 135–154. Google Scholar | DOI

[8] [8] Shareshian, J. and L.Wachs, M., Torsion in the matching and chessboard complexes. Adv. Math. 212(2007), no. 2, 525–570. Google Scholar | DOI

[9] [9] Stanley, R. P., Combinatorics and commutative algebra. Second ed., Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996. Google Scholar