ALGORITHMS FOR UPPER BOUNDS OF LOW DIMENSIONAL GROUP HOMOLOGY

• Main Author: Roberts, Joshua D.
• Format: Others
• Published: UKnowledge 2010
• Subjects: homology; linear groups; Quillen conjecture; finitely-presented groups; GAP; Geometry and Topology; Mathematics; Physical Sciences and Mathematics
• Online Access: http://uknowledge.uky.edu/gradschool_diss/104; http://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1107&context=gradschool_diss

A motivational problem for group homology is a conjecture of Quillen that states, as reformulated by Anton, that the second homology of the general linear group over R = Z[1/p; ζp], for p an odd prime, is isomorphic to the second homology of the group of units of R, where the homology calculations are over the field of order p. By considering the group extension spectral sequence applied to the short exact sequence 1 → SL2 → GL2 → GL1 → 1 we show that the calculation of the homology of SL2 gives information about this conjecture. We also present a series of algorithms that finds an upper bound on the second homology group of a finitely-presented group. In particular, given a finitely-presented group G, Hopf's formula expresses the second integral homology of G in terms of generators and relators; the algorithms exploit Hopf's formula to estimate H2(G; k), with coefficients in a finite field k. We conclude with sample calculations using the algorithms.