Topological obstructions to certain group actions on manifolds

Given a smooth closed S1-manifold M, this dissertation studies the extent to which certain numbers of the form (f* (x) · P · C) [M] are determined by the fixed-point set $MS1$, where $f:M\to Kp1M,1$ classifies the universal cover of $M,xeH*p1M;Q,P$ is a polynomial in the Pontrjagin classes of M, and C is in the subalgebra of $H*M;Q$ generated by $H2M;Q$. When $MS1=\oslash ,$ various vanishing theorems follow, giving obstructions to certain fixed-point-free actions.; Let G be a connected semisimple compact Lie group.; If an S1-action on M extends to a G-action with each component of $MS1$ intersecting MG, then (P · C) [M] is shown to be calculable in terms of the topology of $MS1\hookrightarrow M$ and the isotropy S1-representations. Under the same condition, (L (M) · C) [M] (L (M) being the the Hirzebruch L-class of M) is shown to depend only on the topology of the submanifold of M consisting of those components of $MS1$ with codimensions congruent to 0 mod 4.; These considerations yield some vanishing results. For example, if M admits a G-action with some element $geG$ acting freely, then (f*(x) · P · C) [M] = 0.; If a nontrivial S1-action on a spin manifold M extends to a G-action with $MS1=MG$, then .