p–Modular TQFT's, Milnor torsion and the Casson–Lescop invariant

Abstract

We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev theory. AMS Classification Introduction and summaryInvariants of 3-manifolds that admit extensions to topological quantum field theories (TQFT) are structurally highly organized. Consequently, their evaluations permit an equally deeper insight into the topological structure of the underlying 3-manifolds beyond the mere distinction of their homeomorphism types. Although the notion and many examples of TQFT's have been around for more than a decade there are still surprisingly large gaps in the understanding of the explicit TQFT content of the "classical" invariants such as the Milnor-Turaev torsion or the Casson-Walker-Lescop invariant.In this article we derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold. Specifically, these invariants are expressed in (6) of Theorem 3, in (7) of Theorem 4, in (15) of Theorem 6, and in (18) of Theorem 7 as traces and matrix elements of operators acting on * H 1 (Σ) for a surface Σ. We relate these formulae to previous results in [10],[11], [12] and [5] on the Frohman-Nicas and Reshetikhin-Turaev theories. In the course, we develop the general notion of a q/l-solvable TQFT and consider reductions to the p-modular cases, as needed for the quantum theories. As an

Problems on invariants of knots and 3–manifolds

“…(Note our normalization V p (S 3 ) = 1). As remarked in [212] the identity in (46) is true for p = 5 and general M with…”

“…Recall, that the similar relation (46) for λ CW L for manifolds with b 1 (M ) ≥ 1 is already contained in the information of a homological (L = 0) TQFT, and is indeed a special evaluation of the Turaev-Milnor Torsion, see [212]. Given the richer structure of a 1/1-solvable TQFT we will expect new invariants Ξ that are refinements of λ CW L and the torsion invariants.…”

“…In this way the Frohman Nicas theories V (j) yields the coefficients of the Alexander Polynomial, and, as shown in [212], thus also λ CW L .…”

“…The notion of finite length can be refined into the notion of q/l-solvable introduced in [212], indicating a TQFT over R = M[y]/y l+1 such that the constant order TQFT over the ground ring M is of length q . This, clearly, defines a special case of a TQFT of length ≤ (q · l + q + l) .…”

“…For the modular TQFT over F 5 [y]/y 2 obtained from the Reshetikhin Turaev theory this invariant has already been defined in [212], and we may expect it to lift, similarly, to an invariant Ξ Z over Z. For p > 5 we expect, as in the case of λ CW L , the next order terms in the expansions (46) Preparations for item (1) can be found in [212] in which formulae for the Casson invariant over Z are derived that have the same form as general TQFT formulae.…”