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.

arXiv comment44 pages. Comments are welcome! v2: Remark 1.6 (due to Levin and Lidman) added to the paper, references updated

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• A survey of the homology cobordism group
• Almost simple linear graphs, homology cobordism and connected Heegaard Floer homology
• Equivariant aspects of singular instanton Floer homology
• Instantons, special cycles, and knot concordance
• 3-manifolds without any embedding in symplectic 4-manifolds
• Instanton knot invariants with rational holonomy parameters and an application for torus knot groups