  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-Stern's $R$-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.
