Lescop's invariant and gauge theory

Poudel, Prayat

doi:10.1142/s0218216515500509

Cited by 1 publication

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“…In what follows, we describe how to deduce Theorem 2 using Theorem 1 and related results of Poudel [Pou15] and Turaev [Tur83]. By Poudel [Pou15], the Casson invariant λ(Y, E) may be identified with Lescop's invariant of [Les96], slightly modified. The proof utilizes Floer's exact triangle for instanton homology and Dehn surgery techniques à la Lescop [Les96].…”

Section: Introductionmentioning

confidence: 99%

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Klein-four connections and the Casson invariant for nontrivial admissible U(2) bundles

Scaduto

Stoffregen

2017

Algebr. Geom. Topol.

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Given a rank 2 hermitian bundle over a 3-manifold that is non-trivial admissible in the sense of Floer, one defines its Casson invariant as half the signed count of its projectively flat connections, suitably perturbed. We show that the 2-divisibility of this integer invariant is controlled in part by a formula involving the mod 2 cohomology ring of the 3-manifold. This formula counts flat connections on the induced adjoint bundle with Klein-four holonomy. arXiv:1605.01016v1 [math.GT] 3 May 2016Note that Y supports a non-trivial admissible bundle if and only if b 1 (Y ) ≥ 1, where b 1 (Y ) denotes the rank of H 1 (Y ; Z). In general we have b 1 (2) ≥ b 1 (Y ), with strict inequality if and only if H 1 (Y ; Z) has 2-torsion. Theorem 1 and its proof are generalizations of a rather simple idea due to Ruberman and Saveliev [RS04]. Their result is the case of Theorem 1 when H 1 (Y ; Z) is free abelian of rank 3, i.e., when Y is a homology 3-torus. To obtain their statement, one identifies v Y (E) with the triple cup product modulo 2, which for a homology 3-torus is a simple computation. (More generally, see the corollary below.) Our adaptation of Ruberman and Saveliev's argument is summarized, modulo perturbations, as follows.The invariant λ(Y, E) is one half of a signed count of projectively flat connections on the bundle E. There is an action of H 1 (Y ; Z 2 ) on this set of connections, and the quotient is identified with flat connections on the adjoint SO(3) bundle induced by E. The only possible stabilizers of this action are {1}, Z 2 , and V 4 , the Klein-four group isomorphic to Z 2 × Z 2 . Further, the connections with stabilizer V 4 are flat connections with holonomy group V 4 . The number v Y (E) is the number of connections on the induced SO(3) bundle with holonomy V 4 , up to gauge equivalence. The proof of Theorem 1 follows from counting the H 1 (Y ; Z 2 )-orbits with stabilizer V 4 .Vanishing conditions, and relation to Lescop's invariant. The quantity v Y (E) (mod 2) of congruence (1) is often, but not always, equal to zero. The parity also turns out to be independent of our choice of non-trivial admissible bundle E. To state the result,Here β 1 is the Bockstein homomorphism defined on H 1 (Y ; Z 2 ) associated to the coefficient exact sequence 0 → Z 2 → Z 4 → Z 2 → 0. As is well-known, β 1 (a) = a 2 . We note that if H 1 (Y ; Z) is written as a direct sum of prime-power order cyclic summands and copies of Z, then k(Y ) is just the number of Z 2 k summands with k > 1, plus the number of Z summands. In particular, k(Y ) ≥ b 1 (Y ).Theorem 2. Let Y be a closed, oriented and connected 3-manifold with k(Y ) ≥ 1. Let x ∈ H 2 (Y ; Z 2 ) be any element that is not a cup-square. Then v Y (x) (mod 2) is independent of the choice of such x. If furthermore k(Y ) ≥ 4 then we have v Y (x) ≡ 0 (mod 2).Note that the statement holds for a larger class of elements x ∈ H 2 (Y ; Z 2 ) than those just coming from admissible bundles. The conditions are best understood through examples. The simplest interesting examples are ...

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Section: Introductionmentioning

confidence: 99%

“…As a result, the parity of v Y (E) is independent of E, the choice of non-trivial admissible bundle. After some substitutions, the congruences resulting from Theorem 1 and [Pou15] may be summarized as follows.…”

Section: Introductionmentioning

confidence: 99%

Klein-four connections and the Casson invariant for nontrivial admissible U(2) bundles

Scaduto

Stoffregen

2017

Algebr. Geom. Topol.

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Lescop's invariant and gauge theory

Cited by 1 publication

References 15 publications

Klein-four connections and the Casson invariant for nontrivial admissible U(2) bundles

Klein-four connections and the Casson invariant for nontrivial admissible U(2) bundles

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