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Algorithms in
$4$-manifold topology
Stefan Bastl, Rhuaidi Burke, Rima Chatterjee, Subhankar Dey, Alison Durst, Stefan Friedl, Daniel Galvin, Alejandro García Rivas, Tobias Hirsch, Cara Hobohm, Chun-Sheng Hsueh, Marc Kegel, Frieda Kern, Shun Ming Samuel Lee, Clara Löh, Naageswaran Manikandan, Léo Mousseau, Lars Munser, Mark Pencovitch, Patrick Perras, Mark Powell, José Pedro Quintanilha, Lisa Schambeck, David Suchodoll, Martin Tancer, Annika Thiele, Paula Truöl, Matthias Uschold, Simona Veselá, Melvin Weiss and Magdalina von Wunsch-Rolshoven |
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Algebraic & Geometric Topology 26 (2026) 3017–3049
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DOI: 10.2140/agt.2026.26.3017
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Abstract |
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We show that there exists an algorithm that takes as input two closed, simply connected, topological -manifolds and decides whether or not these -manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological -manifolds can be naturally represented by a Kirby diagram consisting only of -handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby–Siebenmann invariant of a closed, simply connected, topological -manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric. In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth -manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow -subgroup, an infinite cyclic group, or a group of geometric dimension at most (in the latter case we additionally assume that the universal covers of both -manifolds are not spin), and decides whether or not these two -manifolds are orientation-preserving stably diffeomorphic. |
Keywords
$4$-manifolds, algorithms
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Mathematical Subject Classification
Primary: 57K40
Secondary: 57K10, 57R65
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References |
Publication
Received: 22 November 2024
Revised: 4 August 2025
Accepted: 6 September 2025
Published: 13 November 2026
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Authors |
| Stefan Bastl | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Rhuaidi Burke | |
| School of Mathematics and
Physics The University of Queensland Brisbane Australia |
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| Rima Chatterjee | |
| School of Mathematics and
Physics The University of Queensland Brisbane Australia |
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| Subhankar Dey | |
| Department of Mathematics IIT Palakkad Palakkad India |
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| Alison Durst | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Stefan Friedl | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Daniel Galvin | |
| Department of Mathematics The University of Texas at Austin Austin, TX United States |
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| Alejandro García Rivas | |
| Universität Bonn Bonn Germany |
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| Tobias Hirsch | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Cara Hobohm | |
| Universität Bonn Bonn Germany |
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| Chun-Sheng Hsueh | |
| Humboldt Universität zu Berlin Berlin Germany |
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| Marc Kegel | |
| Departamento de Álgebra Universidad de Sevilla Sevilla Spain |
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| Frieda Kern | |
| Universität Bonn Bonn Germany |
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| Shun Ming Samuel Lee | |
| Universität Bonn Bonn Germany |
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| Clara Löh | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Naageswaran Manikandan | |
| Humboldt Universität zu Berlin Berlin Germany |
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| Léo Mousseau | |
| Humboldt Universität zu Berlin Berlin Germany |
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| Lars Munser | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Mark Pencovitch | |
| School of Mathematics and
Statistics University of Glasgow Glasgow United Kingdom |
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| Patrick Perras | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Mark Powell | |
| School of Mathematics and
Statistics University of Glasgow Glasgow United Kingdom |
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| José Pedro Quintanilha | |
| Fakultät für Mathematik und
Informatik Universität Heidelberg Heidelberg Germany |
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| Lisa Schambeck | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| David Suchodoll | |
| Humboldt Universität zu Berlin Berlin Germany |
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| Martin Tancer | |
| Department of Applied
Mathematics Charles University Prague Czech Republic |
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| Annika Thiele | |
| Humboldt Universität zu Berlin Berlin Germany |
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| Paula Truöl | |
| Max Planck Institute for
Mathematics Bonn Germany |
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| Matthias Uschold | |
| Fakultät für Mathematik Universität Regensburg Regensburg Germany |
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| Simona Veselá | |
| Universität Bonn Bonn Germany |
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| Melvin Weiss | |
| Universität Bonn Bonn Germany |
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| Magdalina von Wunsch-Rolshoven | |
| Universität Bonn Bonn Germany |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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