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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Group actions on aspherical $ A\sb{k}(N)$-manifolds

Authors: Hsü Tung Ku and Mei Chin Ku
Journal: Trans. Amer. Math. Soc. 278 (1983), 841-859
MSC: Primary 57S15
MathSciNet review: 701526
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Abstract: By an aspherical $ {A_k}(N)$-manifold, we mean a compact connected manifold $ M$ together with a map $ f$ from $ M$ into an aspherical complex $ N$ such that $ {f^{\ast}}: H^k(N;Q)\to H^k(M;Q)$ is nontrivial. In this paper we shall show that if $ {S^1}$ acts effectively and smoothly on a smooth aspherical $ {A_k}(N)$-manifold, $ k > 1$, $ N$ a closed oriented Riemannian $ k$-manifold, with strictly negative curvature, and the $ K$-degree $ K(f) \ne 0$, then the fixed point set $ F$ is not empty, and at least one component of $ F = { \cup_{j}}{F_j}$ is an aspherical $ {A_k}(N)$-manifold. Moreover, $ {\operatorname{Sign}}(f) = {\Sigma_j}\,{\operatorname{Sign}}(f\vert{F_j})$. We also study the degree of symmetry and semisimple degree of symmetry of aspherical $ {A_k}(N)$-manifolds.

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Keywords: $ {A_k}$-manifold, aspherical $ {A_k}(N)$-manifold, $ \hat A$-degree, $ K$-degree, aspherical complex, Euler characteristic, degree of symmetry, semisimple degree of symmetry, complex manifold, Riemannian manifold, negative curvature, Pontrjagin number, Postnikov system, signature, Thom class
Article copyright: © Copyright 1983 American Mathematical Society

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