K(n) local duality for finite groups and groupoids

Neil P. Strickland

55P42; 55P60; 55R40

math.AT/0011109

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

Included postscript file: st-kld.eps

We define an inner product (suitably interpreted) on the K(n)-local
spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid.
This gives an inner product on E^*BG_+ for suitable K(n)-local ring
spectra E.  We relate this to the usual inner product on the
representation ring when n=1, and to the Hopkins-Kuhn-Ravenel
generalised character theory.  We show that LG is a Frobenius algebra
object in the K(n)-local stable category, and we recall the connection
between Frobenius algebras and topological quantum field theories to
help analyse this structure.  In many places we find it convenient to
use groupoids rather than groups, and to assist with this we include a
detailed treatment of the homotopy theory of groupoids.  We also
explain some striking formal similarities between our duality and
Atiyah-Poincare duality for manifolds.