FIXED POINTS OF UNITARY Z/p -MANIFOLDS

Let G = Z/ps ( p an odd prime). We show that restricting the local representations in a unitary G-manifold M with isolated fixed points results in severe restrictions on the number of fixed points (counted with the sign of their orientation), paralleling results obtained by Conner and Floyd in the case G = Z/p . Specifically, the number of noncancelling fixed points is either zero or divisible by p" , where n —» oo as the dimension of M —» oo . This result also parallels phenomena in framed G-manifolds, as discussed by the first author in a previous paper. Introduction Conner and Floyd proved the following result in [CF, 40.1]. Let G — Z/p ( p an odd prime) and let M be a smooth unitary «-dimensional G-manifold with isolated fixed points. Assume also that the local representations normal to the fixed points coincide. Then if we denote the collection of fixed points, counted with orientation, by Y, one has [Y] e p Q0 , where Q0 = Z is zero dimensional unitary bordism, and where a(n) —* oo as n —> oo . In particular, there cannot exist any unitary G-manifold with a single fixed point. Here we show that the Conner and Floyd result generalizes directly to the case G = Zjps for arbitrary s. (The precise result is stated in §2.) Note that the generalization follows easily by induction on the order of G if either the local representation possesses a nonzero fixed subspace by some nontrivial subgroup K or if the manifold M in question contains only isolated fixed points by G ; in the first instance one can restrict to the fixed set by K, and in the second instance one can regard M as a Af-manifold. The general case, in which M has isolated G-orbits of the form G/K for K ^ G, is far less tractable. The analogous result for framed G-manifold appears in fact to require the full force of the Segal conjecture [Wl]. In the absence of an analogous result for unitary G-bordism, our proof here makes use of the eta invariant of Atiyah-PatodiSinger [APS], and uses the combinatorial formulas of Gilkey [Gl]. Received by the editors July 17, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P10, 57S25. ©1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page