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Constructing free resolutions of cohomology algebras

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thesis
posted on 03.03.2017 by Jaleel, Ahsan Ahmed
The H(R)-algebra of a space is defined as the algebraic object consisting of the graded cohomology groups of the space with coefficients in a general ring R, together with all primary cohomology operations on these groups, subject to the relations between the operations.This structure can be encoded as a functor from the category H(R) containing products of Eilenberg-Mac Lane spaces over R to the category of pointed sets. The free H(R)-algebras are the H(R)-algebras of a product of Eilenberg-Mac Lane spaces. In this thesis we show how to construct free simplicial resolutions of H(R)-algebras using the free and underlying functors. Given a space X, we also construct a cosimplicial space such that the cohomology of this cosimplicial space is a free simplicial resolution of the H(R)-algebra of X. For R = Fp, the finite field on p elements, this cosimplicial resolution fits the E2 page of a spectral sequence and give convergence results under certain finiteness restrictions on X. For R = Z, the integers, a similar result is not obtained and the reasons for this are given.

History

Campus location

Australia

Principal supervisor

Andrew Percy

Year of Award

2016

Department, School or Centre

School of Applied Sciences and Engineering (Gippsland)

Degree Type

DOCTORATE

Faculty

Faculty of Science

Exports