A FIXED-POINT THEOREM FOR p-GROUP ACTIONS

We prove Sullivan's fixed-point conjecture for fixed-point-free actions of compact Lie groups which are extensions of a p-group by a torus. Moreover, we show that for a finite p-group G and a compact or finitely dimensional paracompact G-space X the fixed point set XG is nonempty iff the induced homomorphism of zero-dimensional stable cohomotopy groups 7r°(SG) -» tí0 (EG xG X) is injective. Let qG : EG —> BG be a universal principal G-bundle. For a G-space X consider the associated bundle q^: EG Xq X —> BG with fiber X. Let jr°(—) denote the zero-dimensional stable cohomotopy functor. THEOREM A. Let G be a finite p-group. Suppose that a G-space X is compact, or is paracompact with finite cohomological dimension. Then the following conditions are equivalent: (a) The fixed point set XG is nonempty, (b) there exists a G-map EG —► X, (c) q°has a section, (d) the induced homomorphism (qx)* '■ n°(BG) —* ir°(EG Xq X) is injective. (For a definition of the cohomological dimension cf. Quillen [1971].) Our theorem generalizes results of W. Y. Hsiang and T. torn Dieck. Hsiang [1975, IV.I] proved it on the assumption that G is an elementary abelian p-group or a torus, and with singular cohomology replacing zero-dimensional stable cohomotopy in (d). T. torn Dieck [1972a] extended Hsiang's result to abelian compact Lie groups G such that the group of components G/Go is a p-group, considering in (d) unitary cobordism theory instead of singular cohomology. In both theorems, in the case of a nondiscrete group G and a noncompact space X, we have to assume additionally that X has only finitely many orbit types. Proofs of those results are based on computations of appropriate cohomology of the classifying space of the acting group. Similary the proof of Theorem A relies on G. Carlsson's work (Carlsson [1984]) describing stable cohomotopy of classifying spaces. Theorem A is related to a conjecture of D. Sullivan [1970, p. 5.118]. Suppose that G and X are as in Theorem A. The (generalized) Sullivan conjecture says that the map XG —► m&pG(EG,X) which assigns to every fixed point x e XG the constant map fx : EG —» X induces an isomorphism of mod p cohomology. For Received by the editors February 6, 1986 and, in revised form, July 2, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55N25, 57S99.