Vol. 282, No. 1, 2016

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Extending smooth cyclic group actions on the Poincaré homology sphere

Nima Anvari

Vol. 282 (2016), No. 1, 9–25
Abstract

Let X0 denote a compact, simply connected, smooth 4-manifold with boundary the Poincaré homology 3-sphere Σ(2,3,5) and with even negative definite E8 intersection form. We obtain constraints on the rotation data if a free p-action on Σ(2,3,5) extends to a smooth, homologically trivial action on X0 with isolated fixed points, for any odd prime p 7. The approach is to study the equivariant Yang–Mills instanton-one moduli space for cylindrical-end 4-manifolds. As an application we show that a smooth, homologically trivial 7-action on #8S2 × S2 with isolated fixed points does not equivariantly split along a free action on Σ(2,3,5).

Keywords
Yang Mills moduli spaces, gauge theory, group actions, Poincaré homology sphere
Mathematical Subject Classification 2010
Primary: 57S17, 58D19
Secondary: 70S15
Milestones
Received: 19 March 2014
Revised: 5 November 2015
Accepted: 6 November 2015
Published: 24 February 2016
Authors
Nima Anvari
Mathematics
University of Miami
1365 Memorial Drive
Coral Gables, FL 33146
United States