Non-trivial involutions on Floer Homology (Q1396491)

This paper provides the first examples of irreducible 3-manifolds having the homology of \(S^1\times S^2\) and admitting an involution such that the induced action on the Floer Homology is not the identity. Such examples were obtained by \(0\)-surgery along certain composite amphicheiral knots whose \(SU(2)\)-representation variety satisfies a non-degeneracy condition. The first example of irreducible homology spheres \(\Sigma\) admitting an involution acting non-trivially on \(HF_\ast(\Sigma)\) was constructed by \textit{N. Saveliev} in [J. Knot Theory Ramifications 9, 543-556 (2000; Zbl 0999.57033)].