K-theory for spherical space forms

Abstract

We relate the representation theory of the fundamental group of a spherical space form to the corresponding real, complex, and quaternionic K-theory and show that except for exceptional Z₂ factors coming from the K-theory of a sphere, the complete K-theory arises from the representation theory. The main result (Theorem 1) has been considered as ‘folklore’ for some years and has been proved or has been of interest for particular cases in the work of various people (Fujii [3, 4], Gilkey [5, 6], Mahammed [9, 10], Pitt [11], Yasuo [13]). This result is also intimately linked with the study of equivariant cobordism (Bahri [2]). However, this result has never, to our knowledge, appeared in this generality in the literature.