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A
group-theoretic framework for low-dimensional topology, or:
how not to study low-dimensional topology?
Sarah Blackwell, Robion Kirby, Michael Klug, Vincent Longo and Benjamin Ruppik |
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Algebraic & Geometric Topology 25 (2025) 4667–4718
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Abstract |
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A correspondence, by way of Heegaard splittings, between closed, oriented 3-manifolds and pairs of surjections from a surface group to a free group has been studied by Stallings, Jaco and Hempel. This correspondence, by way of trisections, was recently extended by Abrams, Gay and Kirby to the case of smooth, closed, connected, oriented 4-manifolds. We unify these perspectives and generalize this correspondence to the case of links in closed, oriented 3-manifolds and links of knotted surfaces in smooth, closed, connected, oriented 4-manifolds. The algebraic manifestations of these four subfields of low-dimensional topology (3-manifolds, 4-manifolds, knot theory and knotted surface theory) are all strikingly similar, and this correspondence perhaps elucidates some unique character of low-dimensional topology. |
Keywords
splitting homomorphisms, group trisections, tangles in
handlebodies, links in 3-manifolds, knotted surfaces in
4-manifolds, surface groups, free groups, Stallings folding
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Mathematical Subject Classification
Primary: 57K40
Secondary: 20F05, 57M05
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References |
Publication
Received: 9 August 2023
Revised: 24 April 2024
Accepted: 26 May 2024
Published: 20 November 2025
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Authors |
| Sarah Blackwell | |
| Department of Mathematics University of Virginia Charlottesville, VA United States |
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| https://seblackwell.com | |
| Robion Kirby | |
| Department of Mathematics University of California, Berkeley Berkeley, CA United States |
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| https://math.berkeley.edu/~kirby | |
| Michael Klug | |
| Department of Mathematics University of Chicago Chicago, IL United States |
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| https://mathematics.uchicago.edu/people/profile/michael-klug | |
| Vincent Longo | |
| Department of Mathematics University of Connecticut Storrs, CT United States |
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| https://math.uconn.edu/person/vincent-longo | |
| Benjamin Ruppik | |
| Heinrich-Heine-Universität Düsseldorf Germany |
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| https://bruppik.de | |
| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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