## Constructing free resolutions of cohomology algebras

thesis

posted on 03.03.2017 by Jaleel, Ahsan Ahmed#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

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.