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Fixed points and homotopy fixed points. (English) Zbl 0656.57023
Let G be a finite group, EG be a free contractible G-space, and define \(X^{hG}=Map_ G(EG,X)\) (equivariant mapping space). The main theorem of this paper proves that the following two statements are equivalent (Theorem A): (1) G is a p-group. (2) For every finite G-simplicial complex X, the fixed point set \(X^ G=\emptyset\) if and only if \(X^{hG}=\emptyset\). This result has been proved earlier by Haeberly and S. Jackowsky using G. Carlsson’s proof of the Segal conjecture [e.g. Proc. Am. Math. Soc. 102, 205-208 (1988)].