On the existence of free actions of the groups Z_2, S^1 and S^3 on some finitistic spaces and cohomology of orbit spaces

Abstract

Let G be a compact Lie group and X be a finitistic space. If G acts continuously on X, we can construct the fibration

X \hookrightarrow X_{G} \arrow & B_{G}, (1)

called Borel fibration, where G\hookrightarrow E_{G}\to B_{G} denotes the universal G-bundle and X_{G} is the orbit space (E_{G}\times X)/G, also known as the Borel space.

When the action on G on X is free, there is a homotopy equivalence between the orbit space X/G and the space X_{G}. Therefore, we can use the Leray-Serre spectral sequence {E_{r}^{\ast,\ast},d_{r}}, associated to the fibration (1), which converges to the cohomology of the total space X_{G}, to get the cohomology ring of the orbit space X/G.

In this thesis, we use these tools to investigate the existence of free actions of the compact Lie groups Z_2, S^1 and S^3 on some finitistic spaces. Precisely, we study the existence of free action on finitistic spaces with mod 2 cohomology of a Dold manifold P(m,n), a Wall manifold Q(m,n), a Milnor manifold H(m,n), a product of spheres, the (real, complex or quaternionic) projective spaces and spaces of type (a,b). When the space X admit such such structure, we compute the mod 2 cohomology of the respective orbit space X/G.