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How to Depict 5-Dimensional Manifolds

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Abstract

We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional manifolds. However, one can in fact encode the topology of a surface in a 1-dimensional picture. By analogy, one can draw 2-dimensional pictures of 3-manifolds (Heegaard diagrams), and 3-dimensional pictures of 4-manifolds (Kirby diagrams). With the help of open books one can likewise represent at least some 5-manifolds by 3-dimensional diagrams, and contact geometry can be used to reduce these to drawings in the 2-plane.

In this paper, I shall explain how to draw such pictures and how to use them for answering topological and geometric questions. The work on 5-manifolds is joint with Fan Ding and Otto van Koert.

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Notes

  1. “Der Raum ist kein empirischer Begriff, der von äußeren Erfahrungen abgezogen worden. [...] Der Raum ist eine notwendige Vorstellung, a priori, die allen äußeren Anschauungen zum Grunde liegt. [...] Der Raum wird als eine unendliche gegebene Größe vorgestellt.” [13], translation from [14].

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Acknowledgements

I am grateful to Peter Albers and Marc Kegel for their comments on a draft version of this paper. Special thanks to Guido Sweers for creating Figs. 2, 14 and 31. The research of the author is supported by the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the Deutsche Forschungsgemeinschaft.

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Geiges, H. How to Depict 5-Dimensional Manifolds. Jahresber. Dtsch. Math. Ver. 119, 221–247 (2017). https://doi.org/10.1365/s13291-017-0167-4

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