arXiv · 1810.05988
We prove that compact topological 4-manifolds can be effectively presented by a finite amount of data.
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Michael Freedman, Daniele Zuddas. 2018-10-14. Certifying a compact topological 4-manifold. https://doi.org/10.4310/mrl.2019.v26.n1.a5
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We completely determine the mod $2$ Seiberg-Witten invariants for any spin structure on any closed, oriented, smooth $4$-manifold $X$. Our computation confirms the validity of the simple type conjecture mod $2$ for spin structures. Our proof also works for families of spin $4$-manifolds and thus computes the mod $2$ Seiberg-Witten invariants for spin families. The proof of our main result uses $Pin(2)$-symmetry to define an enhancement of the mod $2$ Seiberg-Witten invariants. We prove a connected sum formula for the enhanced invariant using localisation in equivariant cohomology. Unlike the usual Seiberg-Witten invariant, the enhanced invariant does not vanish on taking connected sums and by exploiting this property, we are able to compute the enhanced invariant.
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