An atomic approach to Wall-type stabilization problems

TL;DR

This paper introduces an elementary construction of exotic 4-manifolds and knotted surfaces that potentially remain exotic after stabilization, using Khovanov homology and Floer-theoretic tools to detect and compare these phenomena.

Contribution

It provides a new, elementary approach to wall-type stabilization problems, constructing examples that challenge existing expectations about stabilization invariance in 4-manifold topology.

Findings

Exotic surfaces in the 4-ball remain exotic after stabilization.

Khovanov homology detects exoticity in stabilized surfaces.

Comparison with Floer homology suggests a deeper connection between invariants.

Abstract

Wall-type stabilization problems investigate the collapse of exotic 4-dimensional phenomena under stabilization operations (e.g., taking connected sums with $S^2 \times S^2$). We propose an elementary approach to these problems, providing a construction of exotic 4-manifolds and knotted surfaces that are candidates to remain exotic after stabilization -- including examples in the setting of closed, simply connected 4-manifolds. As a proof of concept, we show this construction yields exotic surfaces in the 4-ball that remain exotic after (internal) stabilization, detected by the cobordism maps on universal Khovanov homology. We conclude by comparing these Khovanov-theoretic obstructions for surfaces to the Floer-theoretic counterparts for exotic 4-manifolds obtained as their branched covers, suggesting a bridge via Lin's spectral sequence from Bar-Natan homology to involutive monopole Floer homology.