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Functoriality of group trisections

Michael Klug [email protected]Authors Info & Affiliations
Edited by David Gabai, Princeton University, Princeton, NJ, and approved August 15, 2018 (received for review October 30, 2017)
October 22, 2018
115 (43) 10875-10879
https://doi.org/10.1073/pnas.1717167115
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Significance

In three dimensions, it has been known for some time that, by using the fact that all three-manifolds admit Heegaard splittings, three-manifold topology can be understood to be in some sense equivalent to understanding maps from surface groups to free groups. Recently, a decomposition for smooth four-manifolds analogous to Heegaard splittings has been discovered and used to establish a similar group-theoretic framework for studying smooth four-manifolds. We review these constructions and show how they are in fact functorial.

Abstract

Building on work by Stallings, Jaco, and Hempel in three dimensions and a more recent four-dimensional analog by Abrams, Kirby, and Gay, we show how the splitting homomorphism and group trisection constructions can be extended to functors between appropriate categories. This further enhances the bridge between smooth four-dimensional topology and the group theory of free and surface groups.

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Acknowledgments

The author wishes to thank Rob Kirby, Abby Thompson, and Julian Chaidez for many helpful conversations.

References

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Information & Authors

Information

Published in

Go to Proceedings of the National Academy of Sciences
Proceedings of the National Academy of Sciences
Vol. 115 | No. 43
October 23, 2018
PubMed: 30348776

Classifications

  1. Physical Sciences
  2. Mathematics

Copyright

© 2018. Published under the PNAS license.

Submission history

Published online: October 22, 2018
Published in issue: October 23, 2018

Keywords

  1. four-manifolds
  2. trisections
  3. free groups
  4. surface groups

Acknowledgments

The author wishes to thank Rob Kirby, Abby Thompson, and Julian Chaidez for many helpful conversations.

Notes

This article is a PNAS Direct Submission.

Authors

Affiliations

Michael Klug1 [email protected]
Department of Mathematics, University of California, Berkeley, CA 94720-3840
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Notes

Author contributions: M.K. performed research and wrote the paper.

Competing Interests

The author declares no conflict of interest.

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Cite this article

M. Klug, Functoriality of group trisections, Proc. Natl. Acad. Sci. U.S.A.

115 (43) 10875-10879,

https://doi.org/10.1073/pnas.1717167115 (2018).

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    Proceedings of the National Academy of Sciences
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