Chern-Simons Functional and the Homology Cobordism Group

TL;DR

The paper introduces a new integer-valued function for integral homology spheres that depends only on their homology cobordism class, relates to existing invariants, and has applications in topological classification.

Contribution

It constructs the function Y, proves its dependence on homology cobordism, and explores its connections to the Frf8yshov and Fintushel-Stern invariants, with applications to cobordism properties.

Findings

Y depends only on homology cobordism class.

Y recovers the Frf8yshov invariant.

If Y is trivial, no nontrivial simply connected homology cobordism exists from Y to itself.

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 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.