Nicholson, John; (2021) Projective modules over integral group rings and Wall's D2 problem. Doctoral thesis (Ph.D), UCL (University College London).

Text PhD Thesis.pdf - Accepted Version Available under License : See the attached licence file. Download (1MB)

Abstract

In the first part of this thesis, we study the problem of when P ‘ZG – Q‘ZG implies P – Q for projective ZG modules P, Q where ZG is the integral group ring of a finite group G. Our main result is a general condition on G under which cancellation holds. This builds upon the results of R. G. Swan and our condition includes all G for which cancellation was previously known to hold. In the second part of this thesis, we explore applications of these results to Wall’s D2 problem which asks whether every cohomologically 2-dimensional finite complex X is homotopy equivalent to a finite 2-complex. The case where G “ π1pXq has 4-periodic cohomology has been the source of many proposed counterexamples to Wall’s D2 problem and is of special interest due to its implications on the possible cell structures of finite Poincar´e 3-complexes. Our main result is a solution to Wall’s D2 problem for several infinite families of groups with 4-periodic cohomology, building upon the results of F. E. A. Johnson.

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