Dimensions of fixed point sets of smooth actions

Let p be a prime and G = (Zp)r or (S1)r. Suppose G acts smoothly on a closed manifold Mn with nonempty fixed point set F. Let I(MG) be the set of integers k such that F has a component of dimension n − k. Let MG be the Borel construction and T̄(Mn) the tangent bundle along the fibres of the fibre bundle q : MG → BG. In this paper, we study the relations between I(MG) and the cohomology of Mn or MG. We prove I(MG) = {dx ¦ x ϵ F}, where dx = max{j ¦ Cj(ρ∗x(T̄(Mn) ⊗ C)) ≠ 0}, ρx is the section of q associated with x ϵ F and Cj(−) is the jth Chern class.