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Finding Non-orientable Surfaces in 3-Manifolds

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Abstract

We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.

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Notes

  1. On the other hand this problem is not even known to be decidable. This places it in the same complexity limbo as testing embeddability of 2-complexes into \(\mathbb {R}^4\) [28].

  2. Note that the proof of co-NP membership for 3-sphere recognition [9] assumes the Generalized Riemann Hypothesis.

  3. In parallel to this work, new NP-hardness results have appeared very recently, for Heegaard genus [2] and for the Sublink problem and the Upper bound for the Thurston complexity of an unoriented classical link [23].

  4. Since P is not orientable, this is of course not well-defined. We mean an orientation “in the northern hemisphere” of P in Fig. 1. Up to homeomorphism, it does not change anything, but this will be useful for the surgery arguments used throughout the proof.

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Acknowledgements

We would like to thank Saul Schleimer and Eric Sedgwick for stimulating discussions, and the anonymous reviewers for helpful comments.

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Authors and Affiliations

  1. School of Mathematics and Physics, The University of Queensland, Brisbane, QLD, 4072, Australia

    Benjamin A. Burton

  2. INRIA Sophia-Antipolis and Université Paris-Est Marne-la-Vallée, 77454, Marne-la-Vallée, France

    Arnaud de Mesmay

  3. IST Austria, 3400, Klosterneuburg, Austria

    Uli Wagner

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  1. Benjamin A. Burton
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  2. Arnaud de Mesmay
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  3. Uli Wagner
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Correspondence to Arnaud de Mesmay.

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Editor in Charge: Kenneth Clarkson

The second author has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n\(^\circ \) [291734]. The first author is supported by the Australian Research Council (Project DP140104246).

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Burton, B.A., de Mesmay, A. & Wagner, U. Finding Non-orientable Surfaces in 3-Manifolds. Discrete Comput Geom 58, 871–888 (2017). https://doi.org/10.1007/s00454-017-9900-0

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