My thesis consists of two parts. The first part studies the deformations of formal groups. These are described in my paper "Deformation of formal groups and stable homotopy theory". In this paper I construct a certain deformation complex whose second cohomology group classifies the infinitesimal deformations of a specified formal group. It turns out that all the obstructions vanish, so the main problem is to compute the cohomology of the deformation complex. This is done in a number of cases, namely for the additive formal group and for the standard height n formal group that corresponds to the integral version of Morava K-theory $K(n).$ An explicit description of classes of infinitesimal deformations is given. The group of infinitesimal deformations of the formal group corresponding to the connective Morava K-theory is also computed.
Using these results together with Landweber's Exact Functor Theorem we construct an uncountable family of deformations of cohomology theories that might be considered as deformations of complex K-theory. After suitable completions all these spectra split as products of some well-known spectra.
The second part which represent joint work with J. Block is concerned with stable homotopy theory of sheaves. It is proved that the category of sheaves of spectra and a topological space admits (under some mild restrictions on the space) the structure of a closed model category in the sense of Quillen. The central result is a very general form of Verdier duality for sheaves of spectra.
