#### 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 ${X}_{0}$ 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 $ℤ∕p$-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 $ℤ∕7$-action on ${#}^{8}{S}^{2}×{S}^{2}$ 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