Birman-Hilden theory for 3-manifolds
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.07798
- arXiv:
- arXiv:2408.07798
- Bibcode:
- 2024arXiv240807798L
- Keywords:
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- Mathematics - Geometric Topology
- E-Print:
- 34 pages, 3 figures. Comments welcome. Version 2: new application for bundles, minor edits