Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Noninjectivity of the monodromy of certain equicritical strata

Peter Huxford, Nick Salter

Abstract

An equicritical stratum is the locus of univariate monic squarefree complex polynomials where the critical points have prescribed multiplicities. Tracking the positions of both roots and critical points, there is a natural ``monodromy map'' taking the fundamental group into a braid group. We show here that when there are exactly two critical points, this monodromy map is noninjective.

Noninjectivity of the monodromy of certain equicritical strata

Abstract

An equicritical stratum is the locus of univariate monic squarefree complex polynomials where the critical points have prescribed multiplicities. Tracking the positions of both roots and critical points, there is a natural ``monodromy map'' taking the fundamental group into a braid group. We show here that when there are exactly two critical points, this monodromy map is noninjective.

Paper Structure

This paper contains 13 sections, 6 theorems, 28 equations, 13 figures.

Table of Contents

  1. Introduction
  2. A global description of $\mathop{\mathrm{Poly}}\nolimits_n(\mathbb{C})[p,q]$
  3. Bundle structure
  4. Explicit free generators
  5. Explicit description of the monodromy homomorphism
  6. Description of $f\colon\mathbb{CP}^1\to\mathbb{CP}^1$ as a branched cover
  7. Identification with the braid group
  8. Construction of non-trivial elements in the kernel
  9. Products of consecutive standard generators
  10. Dehn twists
  11. Point pushes
  12. Rewriting $\rho(yx^py^{-1}xy)$
  13. Proof that $\rho(x^{p+1})$ and $\rho(yx^py^{-1}xy)$ commute

Key Result

Theorem 1.1

For all $n\geq3$ and all $p,q\geq1$ such that $p+q=n-1$, the monodromy map is not injective.

Figures (13)

  • Figure 1: Patches of the surfaces $S_1,\ldots,S_n$
  • Figure 2: A patch of the glued surface $S\cong\mathbb{CP}^1$
  • Figure 3: The monodromy $\rho(x)$
  • Figure 4: The monodromy $\rho(y)$
  • Figure 5: The sequence of arcs $\ell_1^0, \ell_2^0, \alpha_2, \ldots, \alpha_p, \ell_{p+1}^1, \ell_{p+2}^1, \beta_{p+2}, \ldots, \beta_{n-1}$.
  • ...and 8 more figures

Theorems & Definitions (15)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 3.1
  • ...and 5 more