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Birman-Hilden theory for 3-manifolds

Trent Lucas

TL;DR

This work extends Birman–Hilden theory to 3-manifolds by studying the lifting map $\mathcal{L}_p$ for finite regular branched covers $p:M\to N$ and showing that, unlike the surface case, $\mathcal{L}_p$ is typically not injective for most covers. The authors introduce the relative fundamental group $\pi_1^{\text{rel}}(M)$ to rephrase the problem and prove that infinite kernels occur in broad reducible cases, while in a key hyperelliptic family $p_n:S^3\to S^3$ branched over an unlink, the kernel is precisely the normal closure of a single symmetry $\rho$ acting on $n$ components. Algebraically, they map $\mathrm{Mod}(S^3,C_n)\cong \mathrm{SymOut}(F_n)$ to $\mathrm{SymOut}(H_n)$ via reduction mod 2, with the kernel generated by the involutions $\rho_i$, yielding a clean description of the lifting kernel in that hyperelliptic case. They further connect the kernel to McCullough–Miller complexes, deriving finite generating sets in some instances and revealing deep links between 3-manifold topology, braid-like groups, and automorphism groups of free products. The results provide both qualitative and quantitative insight into when lifting maps fail to be injective and when kernels admit explicit normal generators, with implications for topological and geometric structures of fiber bundles and equivariant actions.

Abstract

Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism between mapping class groups. Following a question of Margalit-Winarski, we study the injectivity of this lifting map in the case of $3$-manifolds. We show that in contrast to the case of surfaces, the lifting map is generally not injective for most regular branched covers of $3$-manifolds. This includes the double cover of $S^3$ branched over the unlink, which generalizes the hyperelliptic branched cover of $S^2$. In this case, we find a finite normal generating set for the kernel of the lifting map.

Birman-Hilden theory for 3-manifolds

TL;DR

This work extends Birman–Hilden theory to 3-manifolds by studying the lifting map for finite regular branched covers and showing that, unlike the surface case, is typically not injective for most covers. The authors introduce the relative fundamental group to rephrase the problem and prove that infinite kernels occur in broad reducible cases, while in a key hyperelliptic family branched over an unlink, the kernel is precisely the normal closure of a single symmetry acting on components. Algebraically, they map to via reduction mod 2, with the kernel generated by the involutions , yielding a clean description of the lifting kernel in that hyperelliptic case. They further connect the kernel to McCullough–Miller complexes, deriving finite generating sets in some instances and revealing deep links between 3-manifold topology, braid-like groups, and automorphism groups of free products. The results provide both qualitative and quantitative insight into when lifting maps fail to be injective and when kernels admit explicit normal generators, with implications for topological and geometric structures of fiber bundles and equivariant actions.

Abstract

Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism between mapping class groups. Following a question of Margalit-Winarski, we study the injectivity of this lifting map in the case of -manifolds. We show that in contrast to the case of surfaces, the lifting map is generally not injective for most regular branched covers of -manifolds. This includes the double cover of branched over the unlink, which generalizes the hyperelliptic branched cover of . In this case, we find a finite normal generating set for the kernel of the lifting map.
Paper Structure (45 sections, 26 theorems, 77 equations, 3 figures)

This paper contains 45 sections, 26 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivating questions and main results
  3. Application: trivial bundles with a nontrivial fiberwise action
  4. The algebraic perspective of Theorem \ref{['mainthm:hyperelliptic_kernel']}
  5. Key idea: the relative fundamental group
  6. Proof idea of Theorem \ref{['mainthm:BH_counterexamples']}
  7. Proof idea of Theorem \ref{['mainthm:hyperelliptic_kernel']}
  8. The relative fundamental group and braid groups
  9. The irreducible case
  10. The unbranched case
  11. Outline
  12. Acknowledgements
  13. Preliminaries
  14. The branch set
  15. Group actions on handlebodies
  16. ...and 30 more sections

Key Result

Theorem 1

Let $p:M \rightarrow N$ be a finite regular branched cover of closed oriented $3$-manifolds given by a finite group action $G \leq \mathop{\mathrm{Diff}}\nolimits^+(M)$. Assume the branch set $C \subseteq N$ is nonempty, and let $N^\circ$ denote the complement of a regular open neighborhood of $C$. then the kernel of the lifting map $\mathcal{L}_p:\mathop{\mathrm{LMod}}\nolimits_p(N) \rightarrow

Figures (3)

  • Figure 1: The cover $p_n$ in the case $n=3$. The top hemisphere of each sphere in $W_1$ is glued to the bottom hemisphere of the opposing sphere in $W_2$.
  • Figure 2: Two bipartite trees labelled by the basis $\{b_1, b_2, b_3, b_4\}$.
  • Figure 3: A vertex automorphism based at the vertex labelled by $b_3$.

Theorems & Definitions (49)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Proposition 1.4
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem:branch_set_regularity']}.
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 39 more