An atomic approach to Wall-type stabilization problems

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.