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A characterisation of S³ among homology spheres

Michel Boileau, Luisa Paoluzzi and Bruno Zimmermann

Geometry & Topology Monographs 14 (2008) 83–103

DOI: 10.2140/gtm.2008.14.83

arXiv: math/0606220


We prove that an integral homology 3–sphere is S3 if and only if it admits four periodic diffeomorphisms of odd prime orders whose space of orbits is S3. As an application we show that an irreducible integral homology sphere which is not S3 is the cyclic branched cover of odd prime order of at most four knots in S3. A result on the structure of finite groups of odd order acting on integral homology spheres is also obtained.

To the memory of Heiner Zieschang


homology sphere, geometric structure, 3-manifold, cyclic branched covers, finite group action

Mathematical Subject Classification

Primary: 57M40

Secondary: 57M12, 57M50, 57M60, 57S17


Received: 8 June 2006
Accepted: 11 April 2007
Published: 29 April 2008

Michel Boileau
Laboratoire Emile Picard (UMR 5580 du CNRS)
Université Paul Sabatier
118 route de Narbonne
31062 Toulouse CEDEX 4
Luisa Paoluzzi
IMB (UMR 5584 du CNRS)
Université de Bourgogne
BP 47870
9 av Alain Savary
21078 Dijon CEDEX
Bruno Zimmermann
Dipartimento di Matematica e Informatica
Università degli Studi di Trieste
via Valerio, 12/b
34127 Trieste