Dimensions of fixed point sets of smooth actions

Let p be a prime and G = (Z(p))(r) or (S-l)(r). Suppose G acts smoothly on a closed manifold M(n) with nonempty fixed point set F. Let I(M(G)) be the set of integers k such that F has a component of dimension n - k. Let M(G) be the Borel construction and (T) over bar(M(n)) the tangent bundle along the fibres of the fibre bundle q:M(G) --> B-G. In this paper, we study the relations between I(M(G)) and the cohomology of M(n) or M(G). We prove I(M(G)) = {d(x) \ x epsilon F}, where d(x) = max{j \ C-j(rho*(x)((T) over bar(M(n))xC)) not equal 0}, rho(x) is the section of q associated with x epsilon F and C-j(-) is the jth Chern class.