I attended a talk recently and the speaker said, essentially, that gauge theory invariants are expected to never be able to detect exotic 4-spheres because they always vanish, for a reason related to something called "neck cutting." I'm not sure what this means specifically, but I'm aware that Seiberg-Witten theory, for example, would not detect an exotic 4-sphere. My question is: how do we know that other potential gauge theories (SW with two spinors, Vafa-Witten, etc.) would never produce a nontrivial invariant for homotopy 4-spheres?

  • $\begingroup$ @ChrisGerig Okay, this makes sense - I was just looking at this paper the other day, actually! $\endgroup$ Feb 11, 2023 at 5:41

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We cannot go as far as to say all of SW theory is insufficient, but the invariants that exist now are insufficient, via neck 'stretching' and 'pinching'. If you're aware of the proof (such as No homotopy 4-sphere invariants using ECH=SWF), then I believe similar gauge theories fall to the same logic if they have some sort of "independence of metric and perturbation to equations".

To slightly clarify, there are multiple factors: the trivial 2nd cohomology (vaguely 2nd cohomology "rules all" in 4d gauge theories), the lack of stability of moduli of gauge-theoretic solutions when equations are perturbed, and being able to stretch (using metrics) a standard ball out of the sphere (to consider "Floer homology cobordism maps").

In contrast, w.r.t. Gluck's construction to attempt to get exotic spheres from special subsets of the standard sphere, it fails with SW (and tentatively other gauge theories). This latter fact (applied to 4-manifolds with $b^2_+>0$) hasn't been documented in the literature I believe but here is my proof:

On any 4-manifold with $b^2_+>0$ the Gluck construction does not alter the SW invariants, because the induced $\mathbb{Z}$-action on monopole Floer homology of $S^1\times S^2$ (the boundary of the tubular neighborhood of the surgery 2-sphere) is trivial. Indeed, Gluck's action $R(\theta,x)=\big(\theta,r_\theta(x)\big)\in\text{Diff}_+(S^1\times S^2)$ preserves the unique torsion spin-c structure on $S^1\times S^2$ and $R_*=\text{id}_*$ on $H_*(S^1\times S^2;\mathbb{Z})$, so we're done by Propositions 9.7.1+36.1.3 of Kronheimer--Mrowka's bible.


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