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On branched coverings of singular $(G,X)$-manifolds

Léo Brunswic

Abstract

Branched covering have a long history from ramification of Riemann surfaces to realization of 3-manifolds as covering ramified over a knots; from geometrical topology to algebraic geometry. The present work investigates a notion of branched covering "à la Fox" which is particularly natural for (G,X)-manifolds. The work is two fold. First, we recall and enrich the current state of the art (based upon Montesinos) and develop a Galois theory for such branched covering together with a description of the fiber above branching points. As a consequence, we solve two open questions of Montesinos and construct an example related to another open question. Second, we present a theory of singular (G,X)-manifolds and apply the theory of branched covering we developped to extend the usual framework of (G,X)-manifolds to singular (G,X)-manifolds; in particular, we construct a developping map for such singular manifolds. An application to singular locally Minkowski manifolds is given.

On branched coverings of singular $(G,X)$-manifolds

Abstract

Branched covering have a long history from ramification of Riemann surfaces to realization of 3-manifolds as covering ramified over a knots; from geometrical topology to algebraic geometry. The present work investigates a notion of branched covering "à la Fox" which is particularly natural for (G,X)-manifolds. The work is two fold. First, we recall and enrich the current state of the art (based upon Montesinos) and develop a Galois theory for such branched covering together with a description of the fiber above branching points. As a consequence, we solve two open questions of Montesinos and construct an example related to another open question. Second, we present a theory of singular (G,X)-manifolds and apply the theory of branched covering we developped to extend the usual framework of (G,X)-manifolds to singular (G,X)-manifolds; in particular, we construct a developping map for such singular manifolds. An application to singular locally Minkowski manifolds is given.

Paper Structure

This paper contains 12 sections, 43 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Branched Covering à la Fox from spreads to Galois correspondance
  3. Preliminaries on spreads
  4. Primer on Branched covering
  5. Paths and universal branched covering
  6. Galoisian branched covering
  7. Singular (G,X)-manifolds
  8. Preliminaries on singular $(G,X)$-manifolds
  9. Developping map
  10. Branched covering of Cauchy-compact ${\mathbb E}^{1,2}_{0}$-manifolds
  11. The branched covering of the BTZ model space
  12. The universal branched covering of ${\mathbb E}^{1,2}_{0}$-manifolds

Key Result

Lemma 2.3

Let $X_i\xrightarrow{p_i} Y_i$ be spreads for $i\in \{1,2\}$ and let $(f,g)$ be a spread isomorphism from $X_1\rightarrow Y_1$ to $X_2\rightarrow Y_2$. Then $g(\mathrm{Ord}(X_1)) = \mathrm{Ord}(X_2)$ and $f(\mathrm{Ord}(Y_1)) = \mathrm{Ord}(Y_2)$.

Theorems & Definitions (119)

  • Definition 2.1
  • Remark 1
  • Definition 2.2
  • Remark 2
  • Lemma 2.3
  • proof
  • Definition 2.4: Portly/skeletal subset
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 109 more