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Smith equivalence of representations

Published online by Cambridge University Press:  24 October 2008

Ted Petrie
Ted Petrie
Affiliation:
Rutgers University, New Brunswick, New Jersey
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Extract

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 ∪ 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 ∪ q. In this case we write V ≈ W.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 1 , July 1983 , pp. 61 - 99
Copyright
Copyright © Cambridge Philosophical Society 1983

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