Fixed points of Anosov maps of certain manifolds

Abstract:

Lemma. If $H$ is a graded exterior algebra on odd generators with augmentation ideal $J$ and $h:H \to H$ is an algebra homomorphism inducing $J/{J^2} \to J/{J^2}$ with eigenvalues $\{ {\lambda _i}\}$, then the Lefschetz number $L(h) = \Pi (1 - {\lambda _i})$. The lemma is applied to an Anosov map or diffeomorphism of a compact manifold with real cohomology $H$ to give sufficient conditions that none of the eigenvalues ${\lambda _i}$ be a root of unity and that there exist a fixed point. In particular, every Anosov diffeomorphism of a compact connected Lie group has a fixed point.