In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

In the paper we give explicit formulae for operations in Hochschild cohomology which are analogous to the operations in the homology of double loop spaces. As a corollary we obtain that any brace algebra in finite characteristics is always a restricted Lie algebra.

We study the Madsen-Tillmann spectrum CP∞ −1 as a quotient of the Mahowald pro-object CP∞ −∞ , which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten.

1. Cohomology theories: definition, examples including ordinary cohomology, K-theory and cobordism. 2. Products and operations: motivation, definitions and examples (especially K-theory). 3. New theories from old: splittings, Bousfield localization, formal group laws and the Landweber exact functor theorem. 4. Relationships between theories: some natural transformations, the chromatic viewpoint...