Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Smith equivalence of representations

Ted Petriea1

a1 Rutgers University, New Brunswick, New Jersey

An old question of P. A. Smith asks: If a finite group G acts smoothly on a closed homotopy sphere Σ with fixed set ΣG consisting of two points p and q, are the tangential representations Tp Σ and Tq Σ of G at p and q equal? Put another way: Describe the representations (V, W) of G which occur as (Tp ΣTq Σ) for Σ a sphere with smooth action of G and ΣG = p xs222A q. Under these conditions we say V and W are Smith equivalent (21) and write V ~ W. A stronger equivalence relation is also interesting. We say representations V and W are s-Smith equivalent if (V, W) = (Tp Σ, Tq Σ) and Σ is a semi-linear G sphere (23), i.e. ΣK is a homotopy sphere for all K and ΣG = p xs222A q. In this case we write VW.

(Received June 30 1982)

(Revised March 17 1983)