A combinatorial formulation for the Seiberg-Witten invariants of 3-manifolds

Abstract

The Seiberg-Witten invariant of a closed connected oriented 3-manifold M is an integer-valued function SW = SW (M) on the set of Spin-structures S(M) on M . This function is defined under the assumption b1(M) ≥ 1 where b1 is the first Betti number; in the case b1(M) = 1 the function SW depends on the choice of a generator of the group H(M ;Z) = Z. The definition of SW runs parallel to the definition of the Seiberg-Witten invariant of 4-manifolds, cf. [Mo], [MT], [MOY]. It was observed by Meng and Taubes [MT] that a weaker function SW (M) is essentially equivalent to the Alexander polynomial of M . Their proof of this remarkable theorem is based on the interpretation of the Alexander polynomial as a Reidemeister-type torsion, see [Mi] for the case of 3-manifolds with boundary and [Tu1] for the case of closed 3-manifolds. In 1989, the author introduced so-called Euler structures on manifolds and their combinatorial torsion invariants, see [Tu4]. In dimension 3, the Euler structures are equivalent to Spin-structures. Combining these facts with the constructions of torsions introduced in the author’s earlier papers, one obtains a combinatorially defined function T = T (M) : S(M) → Z, see [Tu5]. This function is well defined for b1(M) ≥ 2 and depends on the choice of a generator of H(M ;Z) = Z for b1(M) = 1. The function T determines the Alexander polynomial of M . These facts and the Meng-Taubes theorem strongly suggest a close relationship between the functions SW and T . The following theorem is the main result of this paper.