ALGORITHMS FOR UPPER BOUNDS OF LOW DIMENSIONAL GROUP HOMOLOGY

Author

Joshua D. Roberts, University of KentuckyFollow

Date Available

5-6-2011

Year of Publication

2010

Degree Name

Doctor of Philosophy (PhD)

Document Type

Dissertation

College

Arts and Sciences

Department

Mathematics

First Advisor

Dr. Marian Anton

Second Advisor

Dr. Edgar Enochs

Abstract

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; ζ_p], 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.

Recommended Citation

Roberts, Joshua D., "ALGORITHMS FOR UPPER BOUNDS OF LOW DIMENSIONAL GROUP HOMOLOGY" (2010). University of Kentucky Doctoral Dissertations. 104.
https://uknowledge.uky.edu/gradschool_diss/104