Chern-Simons Functional and the Homology Cobordism Group

For each integral homology sphere $Y$, a function $Gamma_Y$ on the set of integers is constructed. It is established that $Gamma_Y$ depends only on the homology cobordism of $Y$ and it recovers the Fr{o}yshov invariant. A relation between $Gamma_Y$ and Fintushel-Sterns $R$-invariant is stated. It is shown that the value of $Gamma_Y$ at each integer is related to the critical values of the Chern-Simons functional. Some topological applications of $Gamma_Y$ are given. In particular, it is shown that if $Gamma_Y$ is trivial, then there is no simply connected homology cobordism from $Y$ to itself.