Involutions and the Chern-Simons filtration in instanton Floer homology

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 $H_1\left(X, \mathbb{Z}_2\right)=0$ and $b_2(X) \leq 1$, and a strong cork which survives stabilization by either of $n \mathbb{C P}^2$ or $n \overline{\mathbb{C P}}^2$. We also prove that every nontrivial linear combination of $1 / n$-surgeries on the strongly invertible knot $\overline{9}_{46}$ 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.