Chern-Simons Functional and the Homology Cobordism Group

Abstract

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 Frøyshov 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.

44 pages. Comments are welcome! v2: Remark 1.6 (due to Levin and Lidman) added to the paper, references updated