TORSION INVARIANTS OF Spin c - STRUCTURES ON 3 - MANIFOLDS

Recently there has been a surge of interest in the Seiberg-Witten invariants of 3-manifolds, see [3], [4], [7]. The Seiberg-Witten invariant of a closed oriented 3-manifold M is a function SW from the set of Spin-structures on M to Z. This function is defined under the assumption b1(M) ≥ 1 where b1(M) is the first Betti number of M ; in the case b1(M) = 1 the function SW depends on the choice of a generator of H(M ;Z) = Z. The definition of SW runs parallel to the definition of the SW-invariant of 4-manifolds: one counts the gauge equivalence classes of solutions to the Seiberg-Witten equations. It was observed by Meng and Taubes [4] that the function SW (M) is closely related to a Reidemeister-type torsion of M . The torsion in question was introduced by Milnor [5]; the refined version used by Meng and Taubes is due to the author [12]. Considered up to sign, this torsion is equivalent to the Alexander polynomial of the fundamental group of M , see [5], [8]. The aim of this paper is to discuss relationships between Spin-structures and torsions. We use the torsions introduced by the author in [9], [12], [13] to define a numerical invariant of Spin-structures on closed oriented 3-manifolds. Presumably, in the case b1 ≥ 1, this invariant is equivalent to the one arising in the Seiberg-Witten theory. A related question of finding topological invariants of Spin-structures on 3manifolds was studied in [11] in connection with a classification problem in the knot theory. It was observed in [11] that an orientation of a link in the 3sphere S induces a Spin-structure on the corresponding 2-sheeted branched covering of S. To distinguish Spin-structures on 3-manifolds one can use torsions, see [13]. As a specific application, note the homeomorphism classification of Spin-structures on 3-dimensional lens spaces: a lens space L(p, q) with even p admits an orientation-preserving self-homeomorphism permuting the two Spinstructures on L(p, q) if and only if q = p + 1(mod 2p), see [13], Theorem C.3.1. This implies (the hard part of) the classification of oriented links with two bridges in S first established by Schubert in a different way.