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Counting $SL(2,\mathbb{C})$ connections on Seifert-fibered spaces

Juan Muñoz-Echániz

Abstract

We study the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology $3$-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the $SL(2,\mathbb{C})$ Chern--Simons functional and prove a localisation result: the perturbed critical points either approach a compact subset of the $SL(2, \mathbb{C})$ character variety or else `escape to infinity'. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localising critical points. As an application, we obtain formulae for the Euler characteristic and Poincaré polynomial of the stable locus of the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology $3$-sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by four) of any weighted-homogeneous isolated complete intersection singularity whose link is the given $3$-manifold.

Counting $SL(2,\mathbb{C})$ connections on Seifert-fibered spaces

Abstract

We study the character variety of a Seifert-fibered homology -sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the Chern--Simons functional and prove a localisation result: the perturbed critical points either approach a compact subset of the character variety or else `escape to infinity'. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localising critical points. As an application, we obtain formulae for the Euler characteristic and Poincaré polynomial of the stable locus of the character variety of a Seifert-fibered homology -sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by four) of any weighted-homogeneous isolated complete intersection singularity whose link is the given -manifold.

Paper Structure

This paper contains 50 sections, 41 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. A perturbation of the Chern--Simons functional
  3. Localisation results for Seifert-fibered spaces
  4. Dimensional reduction results
  5. The Milnor number and $SL(2, \mathbb{C})$ representations
  6. Outline and Comments
  7. Stable flat connections on $3$-manifolds
  8. Setup
  9. Spaces of connections and Sobolev completions
  10. The ' $U(2)$--model ' for $SO(3)$ and $PSL(2,\mathbb{C})$ connections
  11. Orbit spaces
  12. The Kähler structure of $\mathscr{A}_{E}^c$
  13. Moment maps and stability
  14. Local structure of the stable orbit space
  15. The variational viewpoint
  16. ...and 35 more sections

Key Result

Theorem A

Let $(Y,\pi , g)$ be a Seifert-fibered homology $3$-sphere equipped with a Seifert metric $g$. Then there exists a compact subset $\mathscr{Z} = \mathscr{Z}(g) \subset \mathscr{M}^\ast$, a collection $q_1 , \ldots , q_N : S^1 \times D^2 \hookrightarrow Y$ of disjoint smooth embeddings, and a smooth there exists a constant $\varepsilon_0 = \varepsilon_0 ( g , q_i , f, \mathscr{U} , C ) > 0$ such t

Theorems & Definitions (97)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary A
  • Corollary B
  • Conjecture : Neumann--Wahl neumann-wahl
  • Conjecture 1.2
  • Proposition 2.1
  • proof
  • ...and 87 more