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ALGORITHMS FOR UPPER BOUNDS OF LOW DIMENSIONAL GROUP HOMOLOGY

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.

Identiferoai:union.ndltd.org:uky.edu/oai:uknowledge.uky.edu:gradschool_diss-1107
Date01 January 2010
CreatorsRoberts, Joshua D.
PublisherUKnowledge
Source SetsUniversity of Kentucky
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceUniversity of Kentucky Doctoral Dissertations

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