Extending smooth cyclic group actions on the Poincaré homology sphere

Abstract

Let $X_0$ denote a compact, simply-connected smooth $4$-manifold with boundary the Poincar\'e homology $3$-sphere $\Sigma(2,3,5)$ and with even negative definite intersection form $Q_{X_0}=E_8$. We show that free $\mathbb{Z}/p$ actions on $\Sigma(2,3,5)$ do not extend to smooth actions on $X_0$ with isolated fixed points for any prime $p>7$. The approach is to study the equivariant version of the Yang-Mills instanton-one moduli space for $4$-manifolds with cylindrical ends. As an application we show that for $p>7$ a smooth $\mathbb{Z}/p$ action on $\#^8 S^2 \times S^2$ with isolated fixed points does not split along a free action on $\Sigma(2,3,5)$. The results hold for $p=7$ if the action is homologically trivial.