Chern–Simons functional and the homology cobordism group

Abstract

For each integral homology sphere $Y$, we construct a function ${\Gamma }_Y$ on the set of integers. We establish that ${\Gamma }_Y$ depends only on the homology cobordism class of $Y$ and that it recovers the Frøyshov invariant. We state a relation between ${\Gamma }_Y$ and Fintushel and Stern’s $R$-invariant. We show that the value of ${\Gamma }_Y$ at each integer is related to the critical values of the Chern–Simons functional, and we give some topological applications of ${\Gamma }_Y$. In particular, we show that if ${\Gamma }_Y$ is trivial, then there is no simply connected homology cobordism from $Y$ to itself.