We construct compatible symplectic structures on manifolds Q3 × S1 for Waldhausen Q3 and show it is possible iff Q3 is a Stallings fibrations over S1. We describe bordisms of Lagrangian T 2-bundles over S1 × I .

The theory of abelian categories proved very useful, providing an ax-iomatic framework for homology and cohomology of modules over a ring (in particular, abelian groups) [5]. A similar framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of groups and Lie algebras etc. The point of my thesis is that semi-abelian categories (in the sense of Janeli...

Homotopy Theory: The Interaction of Category Theory and Homotopy theory A revised version of the 2001 article

A homotopy action of a group G on a space X is a homomorphism from G to the group HAUT(X) of homotopy classes of homotopy equivalences of X. George Cooke developed an 'obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of G and maps between them that are...

We prove an analogue of the de Rham theorem for the extended L 2-cohomology introduced by M.Farber [Fa]. This is done by establishing that the de Rham complex over a compact closed manifold with coefficients in a flat Hilbert bundle E of A-modules over a finite von Neumann algebra A is chain-homotopy equivalent in the sense of [GS] (i.e. with bounded morphisms and homotopy operators) to a combi...

Given a connected space X, we consider the effect of Quillen’s plus construction on the homotopy groups of X in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequence AX → X → X+, we explain the contribution of πnX to πnX +, πn+1X + and πnAX, πn+1AX explicitly in terms of the low dimensional homology of πnX regarded as a module over π1X. Key ing...