Constructing free resolutions of cohomology algebras

2017-03-03T01:23:00Z (GMT) 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.