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Almost free actions on manifolds

Published online by Cambridge University Press:  17 April 2009

Philip T. Church  and
Klaus Lamotke
Philip T. Church
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York, USA;
Klaus Lamotke
Affiliation:
Mathematisches Institut der Universitat Köln, Köln, Germany.
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Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp

(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and

(2) the (even) number of fixed points.

Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 2 , April 1974 , pp. 177 - 196
Copyright
Copyright © Australian Mathematical Society 1974

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