Vanishing Theorem of Pin(2)-equivariant Bauer-Furuta Invariants as Stable Sections

arXiv:2611.13193v1 Announce Type: new Abstract: We develop a viewpoint of Pin(2)-equivariant Bauer-Furuta invariant: It can be converted to a section over a finite dimensional Pin(2)-orbit, and it takes values in ordinary stable homotopy class. This viewpoint reveals a topological limitation of the invariant: If the fiber's connectivity exceeds the dimension of the base space, then all sections are homotopic to the trivial section. We use it to recover Lin-Mukherjee vanishing theorem. Furthermore, the family Seiberg-Witten invariant can be obtained from the pullback of such sections to $S^1$-orbit. We use this fact to prove a vanishing theorem of the Pin(2)-equivariant family Bauer-Furuta invariant after two stabilizations.

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