Examples of homology 3-spheres whose Chern-Simons function is not Morse-Bott

TL;DR

This paper constructs specific homology 3-spheres where the $SU(2)$ Chern-Simons function fails to be Morse-Bott, revealing complex critical point structures in these topological spaces.

Contribution

It provides explicit examples of homology 3-spheres with non-Morse-Bott Chern-Simons functions, including degenerate critical points and non-manifold critical sets.

Findings

Existence of homology 3-spheres with non-Morse-Bott Chern-Simons functions

Identification of degenerate isolated critical points

Critical sets not homeomorphic to manifolds

Abstract

We construct two homology 3-spheres for which the (unperturbed) $SU(2)$ Chern-Simons function is not Morse-Bott. In one case, there is a degenerate isolated critical point. In the other, a path component of the critical set is not homeomorphic to a manifold. The examples are $+1$ surgeries on connected sums of torus knots.