In this thesis, we study the homotopy theory of fixed points using methods from equivariant homotopy theory. Given a compact topological space with the action of a p-group, the fixed points and their cohomological properties are studied via the so-called Smith theory. We indicate several different categorifications of this theory.

The thesis consists of four main parts:

In the first part (Chapter 3), we investigate the relation between genuine fixed points of a finite G-space and power operations. We analyse the theory of perfect E8 k-algebras for k a characteristic p field and the perfection functor called tilting. Using this theory, we recover the homotopy type of genuine fixed points from the Borel equivariant cohomology.

In the second part (Chapter 4), we study the Segal conjecture for Z{p; more precisely, given a spectrum X, when is the Tate construction with respect to trivial Z{p is the pcompletion. For X finite spectra, this is the celebrated theorem of Lin and Gunawardena.In this chapter, we give examples of several non-finite spectra that satisfy the Segal conjecture and extend this result to a larger class of spectra.

In the third part (Chapter 5), we compute homotopy fixed points of certain actions of based loops on a compact Lie group and certain p-compact groups coming from geometric representation theory. In certain cases, we also compare the homotopy fixed points to the genuine fixed points.

In the fourth and final part (Chapters 6 and 7), we study a categorification of Smith theory for sheaf cohomology building on the works of D. Treumann.