A Twisted Bass-Heller-Swan Decomposition for Localising Invariants and Equivariant Poincaré Duality

Abstract

This thesis consists of two essentially independent parts. <br/> The first part contains a generalisation of the so called Bass-Heller-Swan decomposition in algebraic <em>K</em>-theory, which relates the algebraic <em>K</em>-theory of a Laurent polynomial ring to the algebraic <em>K</em>-theory of its coefficients. We extend it to a splitting for localising invariants of certain categorical mapping tori. This contains most known generalisations of the Bass-Heller-Swan decomposition as a special case. As an application, we obtain splitting results for Waldhausen's <em>A</em>-theory of mapping tori as well as the <em>K</em>-theory of certain localised tensor algebras. <br/> The second part is concerned with the question of what equivariant Poincaré duality for a compact Lie group <em>G</em> is supposed to be, which is the basic homological property of smooth closed <em>G</em>-manifolds. We first introduce the notion of parametrised Poincaré duality in the setting of parametrised category. Specialising this to the equivariant world, we obtain a robust theory of equivariant Poincaré duality for compact Lie groups and we show that this is compatible with various equivariant constructions and operations. Finally, we apply this theory to give, among other things, a new proof of the rigidity theorem of Atiyah-Bott and Conner-Floyd on group actions on manifolds with a single fixed point.