Surgeries on periodic links and homology of periodic 3-manifolds (Q2760483)

A (framed) link \(L\) in the standard 3-sphere \(S^3\) is said to be \(p\)-periodic if there is a \(\mathbb{Z}_p\)-action on \(S^3\), with a circle as a fixed point set, which maps \(L\) onto itself, and such that \(L\) is disjoint from the fixed point set. A 3-manifold \(M\) is called \(p\)-periodic if it admits an orientation preserving action of the cyclic group \(\mathbb{Z}_p\) with a circle as a fixed point set, and the action is free outside the circle. The authors prove that \(p\)-periodic 3-manifolds (where \(p\) is a fixed prime integer, \(p\geq 2)\) are obtained by surgeries on \(p\)-periodic links. As a consequence, they prove that for any \(p\)-periodic 3-manifold \(M\), where \(p\) is an odd prime integer, \(p\geq 3\), the first homology group of \(M\) with \(\mathbb{Z}_p\)-coefficients is different from \(\mathbb{Z}_p\). Finally, a similar criterion of 2-periodicity for rational homology 3-spheres completes this nice paper.