Extending smooth cyclic group actions on the Poincaré homology sphere

Nima Anvari

Vol. 282 (2016), No. 1, 9–25

DOI: 10.2140/pjm.2016.282.9

Abstract

Let $X_0$ denote a compact, simply connected, smooth $4$-manifold with boundary the Poincaré homology $3$-sphere $\Sigma \left(2,3,5\right)$ and with even negative definite $E_8$ intersection form. We obtain constraints on the rotation data if a free $\mathbb{Z}∕p$-action on $\Sigma \left(2,3,5\right)$ extends to a smooth, homologically trivial action on $X_0$ with isolated fixed points, for any odd prime $p\ge 7$. 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 $\mathbb{Z}∕7$-action on ${\#}^8{S}^2\times S^2$ with isolated fixed points does not equivariantly split along a free action on $\Sigma \left(2,3,5\right)$.

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