Workshop on 4-manifolds and algorithms

In this workshop, we aim to bring together researchers and students to discuss topics around algorithms in 4-manifold topology. The goal is to understand recent developments in the field and discuss some open problems. We would like to have every day an introductory talk in the morning, a lot of time to discuss and work in small groups in the afternoon, and perhaps a research talk to cap off the day. 

The talks and topics discussed in the afternoon should ideally be accessible to advanced master students. 

The deadline to apply for participation has now passed.

Article:

In the workshop we discussed various topics about algorithms in 4-manifold topology. We have summarized our findings in the following article. 

• 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, Magdalina von Wunsch-Rolshoven

• arXiv:2411.08775 (November 2024)

• Abstract: We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-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 4-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 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.