Cohomology ofS 1-orbit spaces of cohomology spheres and cohomology complex projective spaces

Abstract

Let G be a compact Lie group acting on a compact space X and let X/G denote its orbit space. It is of some interest to investigate how the cohomology of X and the orbit structure of the given action are reflected on the cohomology structure of X/G. In this paper, we shall determine the integral cohomology rings H*(X/G) when G is the circle group S ~ or the cyclic group •,, of order m and X is a cohomology sphere or a cohomology complex projective space. If X is a cohomology sphere, then I-I*(X/G) is completely determined by the dimensions of the fixed point sets F(~vr, X ) of restricted Zpr-actions for various prime powers p~ such that ~prcG. In this case, F (Z ; r ,X ) is a cohomology 7Zp-sphere by the Smith theorem. If X is a cohomology complex projective space, then H*(X/S 1) is determined by the number of connected components of the fixed point set F(Sa, X) and their dimensions. It is known that they are also cohomology complex projective spaces. The precise statements of the results are given in Theorem 1.4, 2.2 and 3.4. Prior to our results, the additive structure of H*(X/S 1) were already determined by Conner and Floyd when X is a cohomology sphere [5]. In the case of linear G-actions on the sphere S 2"+t, T. Kawasaki calculated the ring structure of /-/* (S 2" + l/G) [7]. Our main tool is the Leray spectral sequence of the projection re: XG--+X/G , where X G denotes the orbit space (EG x X)/G, EG being the total space of a universal G-bundle. We note that the cohomology ring H*(XG) is fairly well understood. Throughout this paper, p will denote a prime integer and L the ring Z, Q or ~gv=ag/pZ. We use the Alexander-Spanier cohomology. Since all spaces are assumed to be Hausdroff and compact, this coincides with sheaf theoretic cohomology. For spaces X and Y, X ~ Y means that the cohomology rings L H* (X; L) and H* (Y; L) are isomorphic. Section 1 is devoted to the case where X~S" and G=S 1. In w we deal with L