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CYCLIC BRANCHED COVERINGS OF BRIESKORN SPHERES BOUNDING ACYCLIC 4-MANIFOLDS

Part of: Low-dimensional topology Differential topology Topological manifolds

Published online by Cambridge University Press:  03 July 2020

NIMA ANVARI  and
IAN HAMBLETON
NIMA ANVARI
Affiliation:
Department of Mathematics, McMaster University L8S 4K1, Hamilton, ON, Canada, e-mails: anvarin@math.mcmaster.ca; hambleton@mcmaster.ca
IAN HAMBLETON
Affiliation:
Department of Mathematics, McMaster University L8S 4K1, Hamilton, ON, Canada, e-mails: anvarin@math.mcmaster.ca; hambleton@mcmaster.ca
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Abstract

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic 4-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well-known invariants of homology spheres that obstruct acyclic bounding 4-manifolds and then use a method based on equivariant Yang–Mills moduli spaces to rule out extensions of the actions.

MSC classification

Primary: 57M60: Group actions in low dimensions
Secondary: 57N13: Topology of $E^4$, $4$-manifolds 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants 57S17: Finite transformation groups

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 400 - 413
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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