Equivariant Localization

The purpose of this note is to construct and characterize localizations of G-spaces, where G is a compact Lie group. It was written at the request of Reinhard Schultz, who needed its results in work of his own. When G is finite, the possibility of such a construction is folklore familiar to the experts. That the general case presents little more difficulty comes as something of a surprise. Of course, restriction to integral primes is essential to such a space level, unstable, construction. One cannot expect, for example, to localize general G-spaces (as opposed to infinite loop G-spaces) at prime ideals of the Burnside ring. We are very grateful to Terry Wall for his careful reading of both this paper and its sequel on equivariant completion. We shall work in the category cf based G-spaces (with G-fixed basepoint). Write n(X, Y) G for the set of G-homotopy classes of based G-maps X-> Y. A G-map / : X-> Y is said to be a weak G-equivalence if its restriction f H : X H-> Y H to H-fixed point spaces is an ordinary weak equivalence for each closed subgroup H of G; subgroups are understood to be closed henceforward. If X and Y are G-CW complexes, then / is necessarily a G-homotopy equivalence. More generally, the following result holds [2, 10, 16]. PROPOSITION 1. If X is a G-CW complex and f : Y-* Z i s a weak G-equivalence, then is a bijection. For any G-space X, there is a G-CW complex TX and a weak G-equivalence a: TX-* • X (see [16]). On the homotopy category hG3T of based G-spaces, Y is a functor and a is natural. We define By the proposition, [X, Y~] G ^ n(X, Y) G if X is a G-CW complex. With the obvious composition, [X, Y] c gives the set of morphisms X-> Y in the category HG$~ obtained from hGZT by formally inverting the weak G-equivalences. This is the appropriate category in which to study localizations. We shall need a special case of the following rough analog of Proposition 1, which seems to be new; we defer the proof to the end of the paper.