Involutions and the Chern-Simons filtration in instanton Floer homology

TL;DR

This paper introduces a new instanton-theoretic invariant using the Chern-Simons filtration to study corks and equivariant bounding, providing novel examples and insights in 4-manifold topology.

Contribution

It develops a distinct Floer-theoretic method based on the Chern-Simons filtration for analyzing corks, leading to new examples and applications in 4-manifold theory.

Findings

Constructed a cork with boundary involution not extending over certain 4-manifolds.

Identified strong corks surviving stabilization by complex projective planes.

Showed that linear combinations of specific surgeries on a knot form strong corks.

Abstract

Building on the work of Nozaki, Sato and Taniguchi, we develop an instanton-theoretic invariant aimed at studying strong corks and equivariant bounding. Our construction utilizes the Chern-Simons filtration and is qualitatively different from previous Floer-theoretic methods used to address these questions. As an application, we give an example of a cork whose boundary involution does not extend over any 4-manifold $X$ with $H_1(X, \mathbb{Z}_2) = 0$ and $b_2(X) \leq 1 $, and a strong cork which survives stabilization by either of $n\smash{\mathbb{CP}^2}$ or $n\smash{\overline{\mathbb{CP}}}^2$. We also prove that every nontrivial linear combination of $1/n$-surgeries on the strongly invertible knot $\smash{\overline{9}_{46}}$ constitutes a strong cork. Although Yang-Mills theory has been used to study corks via the Donaldson invariant, this is the first instance where the critical values of the Chern-Simons functional have been utilized to produce such examples. Finally, we discuss the geography question for nonorientable surfaces in the case of extremal normal Euler number.