Cyclic group actions on contractible 4-manifolds (Q280921)

There are known infinite families of Brieskorn homology \(3\)-spheres which can be realized as boundaries of smooth contractible \(4\)-manifolds. The Brieskorn homology spheres \(\Sigma (a,b,c)\) provide important examples of Seifert fibered \(3\)-manifolds, and have been extensively studied as test cases for questions about smooth \(4\)-manifolds and gauge theory invariants.NEWLINENEWLINERecall that the Brieskorn homology spheres for \(a, b, c\) pairwise relatively prime can be realized as the link of a complex surface singularity NEWLINE\[NEWLINE \Sigma (a,b,c) = \{(x,y,z)\in \mathbb C^3\mid x^a + y^b + z^c = 0 \}\cap S^5 NEWLINE\]NEWLINE with its induced orientation. As a Seifert fibered homology sphere it admits a smooth fixed-point free circle action with three orbits of finite isotropy.NEWLINENEWLINEIn this paper the authors give an answer to a well-known question, asked by Allan Edmonds at Oberwolfach in 1988, about extending smooth free cyclic group actions on \(\Sigma (a,b,c)\) to certain smooth \(4\)-manifolds which they bound. They show that smooth free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible \(4\)-manifold. Also they give a new infinite family of examples in which the actions extend locally linearly but not smoothly.