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Actions of Finite Groups on Rn+k with Fixed Set Rk

Published online by Cambridge University Press:  20 November 2018

Ian Hambleton
Affiliation:
McMaster University, Hamilton, Ontario
Ib Madsen
Affiliation:
Universitet sparken, 8000 Aarhus C, Denmark
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In this paper we study the existence problem for topological actions of finite groups on euclidean spaces Rn+k which are free outside a fixed point set Rk (embedded as a vector subspace). We refer to such an action as a semi-free action on (Rn+k, Rn) and note that all our actions will be assumed orientation-preserving.

Suppose the finite group π acts semi-freely on (Rn+k, Rn), then it acts freely on (Rn+kRn) = Sn–l × Rk+1. Since this space is homotopy equivalent to Sn–l, π will have periodic integral cohomology and n will be a multiple of the period. In fact the orbit space

is a finitely-dominated Poincaré complex of formal dimension n – 1 with π1W = π and as considered by Swan [41].

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1986

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