Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Klein-four connections and the Casson invariant for non-trivial admissible $U(2)$ bundles

Christopher Scaduto, Matthew Stoffregen

Abstract

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.

Klein-four connections and the Casson invariant for non-trivial admissible $U(2)$ bundles

Abstract

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.

Paper Structure

This paper contains 8 sections, 12 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-trivial admissible bundles
  3. Configuration spaces and stabilizers
  4. Klein-four connections
  5. The argument modulo perturbations
  6. Including holonomy perturbations
  7. Establishing the vanishing result
  8. Examples and properties

Key Result

Theorem 1

Suppose $E$ is a non-trivial admissible $U(2)$ bundle over a closed, oriented, connected 3-manifold $Y$ with $b_1(2)\geq 3$. Then $\lambda(Y,E)$ is divisible by $2^{b_1(2)-3}$. Furthermore, we have If $b_1(2)=2$, this congruence also holds, implying that $v_Y(E)$ is even. If $b_1(2)=1$, then the integer $v_Y(E)$ is zero. In these two cases $v_Y(E)$(mod 2) yields no information about $\lambda(Y,E)

Figures (2)

  • Figure 1: Surgery on the Borromean rings with framings $(j,k,l)$ on the three components. When $j,k,l$ are either 0 or various powers of 2, these surgeries yield non-vanishing examples of the congruence in Theorem \ref{['thm:main']}, in which $v_Y(E)\equiv 1$ (mod 2) and $k(Y)=1,2,3$.
  • Figure 2: The link $L=$L8n8 with its four components labelled by $\{1,2,3,4\}$. This link has determinant zero and thus its branched double cover supports non-trivial admissible bundles.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3: rs1
  • Lemma 4
  • ...and 9 more