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Orientation-reversing involutions on homology 3-spheres

Published online by Cambridge University Press:  24 October 2008

David E. Galewski  and
Ronald J. Stern
David E. Galewski
Affiliation:
University of Georgia and University of Utah
Ronald J. Stern
Affiliation:
University of Georgia and University of Utah
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Extract

An important problem in geometric topology is to find a homology 3-sphere M with Rohlin invariant μ(M) = ½ such that M # M bounds an acyclic 4-manifold. If such an M exists then, for instance, all closed topological manifolds of dimension ≥ 5 support a polyhedral structure(3, 4). One way of producing such examples is to find a homology 3-sphere M with μ(M) = ½ such that M admits an orientation-reversing diffeomorphism h, for then M # M would be diffeomorphic to M # − M which bounds an acyclic manifold. In this paper we observe by elementary means that if h is further assumed to be an involution, then this is not possible, namely:

Theorem. Let M be a homology 3-sphere which admits an orientation reversing involution. Then M bounds a parallelizable rational 4-ball. In particular, μ(M) = 0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 3 , May 1979 , pp. 449 - 451
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

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