Actions of finite groups on ${f R}sp {n+k}$ with fixed set ${f R}sp k$

In this paper we study the existence problem for topological actions of finite groups on euclidean spaces R n+k which are free outside a fixed point set R k (embedded as a vector subspace). We refer to such an action as a semi-free action on ( R n+k , R n ) and note that all our actions will be assumed orientation-preserving. Suppose the finite group π acts semi-freely on ( R n+k , R n ), then it acts freely on ( R n+k – R n ) = S n –l × R k +1 . Since this space is homotopy equivalent to S n –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 π 1 W = π and as considered by Swan [ 41 ].