TANGENTIAL REPRESENTATIONS AT ISOLATED FIXED POINTS OF ODD-DIMENSIONAL G-MANIFOLDS

Let G be a compact abelian Lie group, and M an odd-dimensional closed smooth G-manifold. If the fixed point set $M^G\neq\emptyset$ and dim $M^G=0$, then G has a subgroup H with $G/H{\cong}\mathbb{Z}_2$, the cyclic group of order 2. The tangential representation $\tau_x$(M) of G at $x{\in}M^G$ is also regarded as a representation of H by restricted action. We show that the number of fixed points is even, and that the tangential representations at fixed points are pairwise isomorphic as representations of H.