Turaev's conjecture relating Seiberg–Witten invariants and torsion

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Let XX be a closed 33-manifold with b1(X)>0b_1(X)>0, let oo be a homology orientation, and let Eul⁡(X)\operatorname{Eul}(X) be the set of Euler structures. Turaev’s torsion defines a map T(X;o) ⁣:Eul⁡(X)→ZT(X;o)\colon \operatorname{Eul}(X)\to\mathbb Z by evaluating the appropriately normalized torsion at 0∈H10\in H_1; when b1(X)=1b_1(X)=1, this depends on the sign of [df][df]. Let ı ⁣:Spin⁡c(X)→Eul⁡(X)\imath\colon \operatorname{Spin}^c(X)\to\operatorname{Eul}(X) be the natural isomorphism. Turaev’s conjecture. The Seiberg–Witten invariant agrees with the Turaev torsion:

SW⁡X,o=T(X;o)∘ı ⁣:Spin⁡c(X)→Z.\operatorname{SW}_{X,o}=T(X;o)\circ\imath\colon \operatorname{Spin}^c(X)\to\mathbb Z.

This conjecture gives a combinatorial description of the Seiberg–Witten invariant in terms of Turaev torsion. The source presents it as a conjecture and does not state a resolution.

References

Primary source

Michael Hutchings and Yi-Jen Lee, “Circle-valued Morse theory and Reidemeister torsion”, arXiv:dg-ga/9706012 (1999).

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