A group-theoretic framework for low-dimensional topology

Sarah Blackwell; Robion Kirby; Michael Klug; Vincent Longo; Benjamin Ruppik

doi:10.2140/agt.2025.25.4667

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arXiv · 2301.05685

A group-theoretic framework for low-dimensional topology

Sarah Blackwell·Robion Kirby·Michael Klug·Vincent Longo·Benjamin Ruppik

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Abstract

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.

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BibTeX RIS

Sarah Blackwell, Robion Kirby, Michael Klug, Vincent Longo, Benjamin Ruppik. 2024-08-02. A group-theoretic framework for low-dimensional topology. https://doi.org/10.2140/agt.2025.25.4667

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