Group actions and generalized periodicity

Abstract

As motivation for this paper consider the sequence of real projective spaces IRP k together with their canonical real line bundles co k. These bundles are compatible with respect to the usual inclusions I R P l c IRP k and there exist homeomorphisms (IR pk)ok ~ IR pk + 1 also compatible with inclusions [1]. Here (IR pk)~ok denotes the Thom complex of co k. Taking the limit as k--~ oo we obtain a homeomorphism ( IRP~)~ ~, where co is the universal real line bundle. If we now apply the cohomology functor and use the Thorn isomorphism theorem we deduce that the cohomology ring H*(7//27/) is periodic. Thus on the one hand we have a geometric statement, namely (IRP~)~~ ~ and on the other hand an algebraic statement, namely the periodicity of H* (7//27/), with the Thorn isomorphism theorem serving as a one way bridge. If the t/k are the usual complex line bundles over the complex projective spaces q2 pk then we again have homeomorphisms (• pk),~ ~ C pk+~ compatible with inclusions [1]. Therefore (CP~~ is homeomorphic to r poo, where t/is the universal complex line bundle, and consequently H * ( C P ~176 is periodic. As one more example consider the standard lens spaces Lk=sZk-~/7: associated to the representation 0: rt = 7//p 7 / ~ U(1), where O(t)= e 2~l/~f/p for some fixed generator t. Then there exist complex line bundles v k over L k and a family of compatible homeomorphisms (I~)*k~Lk+I/La, producing in the limit a homeomorphism (L~)*~-L~ ~ (see [5]). Cupping with a Thom class TeH2(7//pig) then gives isomorphisms for all s > 0