Five-torsion in the homology of the matching complex on 14 vertices
Jakob Jonsson
KTH Dept. of Mathematics Stockholm 100 44 Sweden
DOI: 10.1007/s10801-008-0123-6
Abstract
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 M 14 \mathsf {M}_{14} on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we conclude that the case n=14 is exceptional; for all other n, 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 M n \mathsf {M}_{n} when n\geq 13 and n\neq 14.
Pages: 81–90
Keywords: keywords matching complex; simplicial homology; torsion subgroup
Full Text: PDF
References
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2. Athanasiadis, C.A.: Decompositions and connectivity of matching and chessboard complexes. Discrete Comput. Geom. 31(3), 395-403 (2004)
3. Babson, E., Björner, A., Linusson, S., Shareshian, J., Welker, V.: Complexes of not i-connected graphs. Topology 38(2), 271-299 (1999)
4. Björner, A., Lovász, L., Vrećica, S.T., Živaljević, R.T.: Chessboard complexes and matching complexes. J. London Math. Soc. 49(2), 25-39 (1994)
5. Bouc, S.: Homologie de certains ensembles de 2-sous-groupes des groupes symétriques. J. Algebra 150, 187-205 (1992)
6. Bredon, G.E.: Introduction to Compact Transformation Groups. Academic Press, San Diego (1972)
7. Dong, X.: The topology of bounded degree graph complexes and finite free resolutions. Ph.D. Thesis, University of Minnesota (2001)
8. Dong, X., Wachs, M.L.: Combinatorial Laplacian of the matching complex. Electronic J. Combin. 9(1), R17 (2002)
9. Friedman, J., Hanlon, P.: On the Betti numbers of chessboard complexes. J. Algebraic Combin. 8, 193-203 (1998)
10. Garst, P.F.: Cohen-Macaulay complexes and group actions. Ph.D. Thesis, University of Wisconsin- Madison (1979)
11. Dumas, J.-G., Heckenbach, F., Saunders, B.D., Welker, V.: Simplicial Homology, a share package for GAP (2000)
12. Jonsson, J.: Simplicial complexes of graphs. Doctoral Thesis, KTH (2005)
13. Jonsson, J.: Simplicial Complexes of Graphs. Lecture Notes in Mathematics, vol.
1928. Springer, New York (2008)
14. Jonsson, J.: Exact sequences for the homology of the matching complex (2008, submitted)
15. Karaguezian, D.B.: Homology of complexes of degree one graphs. Ph.D. Thesis, Stanford University (1994)
16. Karaguezian, D.B., Reiner, V., Wachs, M.L.: Matching complexes, bounded degree graph complexes and weight spaces of GLn-complexes. J. Algebra 239, 77-92 (2001)
17. Ksontini, R.: Propriétés homotopiques du complexe de Quillen du groupe symétrique. Ph.D. Thesis, Université de Lausanne (2000) J Algebr Comb (2009) 29: 81-90
18. Pilarczyk, P.: Computational Homology Program (2004)
19. Reiner, V., Roberts, J.: Minimal resolutions and homology of chessboard and matching complexes. J. Algebraic Combin. 11, 135-154 (2000)
20. Shareshian, J., Wachs, M.L.: Torsion in the matching and chessboard complexes. Adv. Math. 212(2), 525-570 (2007)
21. Wachs, M.L.: Topology of matching, chessboard and general bounded degree graph complexes. Alg.