Approach To Actions On Differentiable Manifolds By Finite Groups

This dissertation consists of three chapters.In the first chapter, we mainly discuss free actions on closed orientable surfaces by finite groups. For a closed orientable surface T^nr+1(n, r are nonnegative integers), let G be a finite group of order 2n and act freely on T^nr+1. By making use of computer, we determine upper bounds for the number of orientation-reversing free actions of the finite group on the closed orientable surface T^nr+1 and all finite groups which act orientation-reversing freely on the closed orientable surface T^nr+1 with low genus. We also prove that every dihedral group acts orientation-reversing freely on closed orientable surfaces T^"r+1 such that its orbit space T^nr+1/D_n ≌ U^r+2 and determine the classification of finite groups which act orientation-reversing freely on a closed orientable surface T^p such that p — 1 is a prime.In the second chapter, we discuss the involutions fixing the disjoint union of the real projective space with Dold manifold. Let (M, T) be a smooth closed maniflod with a smooth involution T whose fixed point set is F. For F = RP(2^m) U P(2^m, 2n -1), we give the all bordism classes of (M, T) with m = 1, 2, 3 and m ≥ 4, n = 1 or 2n≥ 2^m.In the third chapter, we discuss the involutions fixing products of real projective spaces. When F = RP(2m + 1) x RP(2n)(m — 1,2), we determine the existence and the representative up to bordism of (M, T). For 2n = 2^s, we give the complete rang of possible dimensions of nonbounding involutions (M, T).