An arithmetic analysis of closed surfaces
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- by Matthew Harrison-Trainor and Alexander Melnikov;
- Trans. Amer. Math. Soc. 377 (2024), 1543-1596
- DOI: https://doi.org/10.1090/tran/8915
- Published electronically: January 18, 2024
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Abstract:
From a computability-theoretic standpoint, we consider the following problem: Given a closed surface, as a topological space, how hard is it to recover an atlas? We prove that every computable Polish space homeomorphic to a closed surface admits an arithmetic atlas, and indeed an arithmetic triangulation. This is as simple as one could reasonably hope for; essentially, the locally Euclidean structure of a surface can be recovered from the topological structure in a first-order way, i.e., without reference to curves or homeomorphisms or other higher-order objects.
It follows that given two computable presentations of the same closed surface, there is an arithmetic homeomorphism between them. Moreover, the homeomorphism problem for closed surfaces, presented as topological spaces, is arithmetic. From the algorithmic and definability-theoretic standpoint, this improves Kline’s conjecture proved by Bing in the 1940s. We also consider $\mathbb {R}^2$ and the closed unit ball.
References
- Marcelo A. Aguilar and Rodolfo Conde, Computable structures on topological manifolds, CoRR, abs/1703.04075, 2017.
- S. I. Adyan, Finitely presented groups and algorithms, Dokl. Akad. Nauk SSSR (N.S.) 117 (1957), 9–12 (Russian). MR 95873
- S. I. Adyan, Unsolvability of some algorithmic problems in the theory of groups, Trudy Moskov. Mat. Obšč. 6 (1957), 231–298 (Russian). MR 95872
- C. J. Ash and J. Knight, Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000. MR 1767842
- Selman Akbulut and John D. McCarthy, Casson’s invariant for oriented homology $3$-spheres, Mathematical Notes, vol. 36, Princeton University Press, Princeton, NJ, 1990. An exposition. MR 1030042, DOI 10.1515/9781400860623
- Vasco Brattka, Peter Hertling, and Klaus Weihrauch, A tutorial on computable analysis, New computational paradigms, Springer, New York, 2008, pp. 425–491. MR 2762094, DOI 10.1007/978-0-387-68546-5_{1}8
- R. H. Bing, The Kline sphere characterization problem, Bull. Amer. Math. Soc. 52 (1946), 644–653. MR 16645, DOI 10.1090/S0002-9904-1946-08614-0
- Jeongseon Baek, Dong-Soo Kim, and Young Ho Kim, A characterization of the unit sphere, Amer. Math. Monthly 110 (2003), no. 9, 830–833. MR 2024752, DOI 10.2307/3647802
- Tyler A. Brown, Timothy H. McNicholl, and Alexander G. Melnikov, On the complexity of classifying Lebesgue spaces, J. Symb. Log. 85 (2020), no. 3, 1254–1288. MR 4231623, DOI 10.1017/jsl.2020.63
- William W. Boone, The word problem, Ann. of Math. (2) 70 (1959), 207–265. MR 179237, DOI 10.2307/1970103
- Vasco Brattka, Recursive and computable operations over topological structures, 1998.
- Barbara F. Csima and Robert I. Soare, Computability results used in differential geometry, J. Symbolic Logic 71 (2006), no. 4, 1394–1410. MR 2275866, DOI 10.2178/jsl/1164060462
- M. Dehn, Transformation der Kurven auf zweiseitigen Flächen, Math. Ann. 72 (1912), no. 3, 413–421 (German). MR 1511705, DOI 10.1007/BF01456725
- Rod Downey, Gregory Igusa, and Alexander Melnikov, On a question of Kalimullin, Proc. Amer. Math. Soc. 146 (2018), no. 8, 3553–3563. MR 3803679, DOI 10.1090/proc/13954
- Rod Downey and Antonio Montalbán, The isomorphism problem for torsion-free abelian groups is analytic complete, J. Algebra 320 (2008), no. 6, 2291–2300. MR 2437501, DOI 10.1016/j.jalgebra.2008.06.007
- Rodney G. Downey and Alexander G. Melnikov, Computable analysis and classification problems, Beyond the horizon of computability, Lecture Notes in Comput. Sci., vol. 12098, Springer, Cham, [2020] ©2020, pp. 100–111. MR 4139528
- S. S. Goncharov and Dzh. Naĭt, Computable structure and antistructure theorems, Algebra Logika 41 (2002), no. 6, 639–681, 757 (Russian, with Russian summary); English transl., Algebra Logic 41 (2002), no. 6, 351–373. MR 1967769, DOI 10.1023/A:1021758312697
- Noam Greenberg, Alexander G. Melnikov, Julia F. Knight, and Daniel Turetsky, Uniform procedures in uncountable structures, J. Symb. Log. 83 (2018), no. 2, 529–550. MR 3835076, DOI 10.1017/jsl.2017.91
- Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Based on the 1981 French original; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 2307192
- Wolfgang Haken, Ein Verfahren zur Aufspaltung einer $3$-Mannigfaltigkeit in irreduzible $3$-Mannigfaltigkeiten, Math. Z. 76 (1961), 427–467 (German). MR 141108, DOI 10.1007/BF01210988
- Geoffrey Hemion, On the classification of homeomorphisms of $2$-manifolds and the classification of $3$-manifolds, Acta Math. 142 (1979), no. 1-2, 123–155. MR 512214, DOI 10.1007/BF02395059
- M. Hoyrup, T. Kihara, and V. Selivanov, Degree spectra of homeomorphism types of Polish spaces, Preprint.
- Matthew Harrison-Trainor, Alexander Melnikov, and Keng Meng Ng, Computability of Polish spaces up to homeomorphism, J. Symb. Log. 85 (2020), no. 4, 1664–1686. MR 4243758, DOI 10.1017/jsl.2020.67
- Greg Kuperberg, Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization, Pacific J. Math. 301 (2019), no. 1, 189–241. MR 4007377, DOI 10.2140/pjm.2019.301.189
- Steffen Lempp, The computational complexity of torsion-freeness of finitely presented groups, Bull. Austral. Math. Soc. 56 (1997), no. 2, 273–277. MR 1470080, DOI 10.1017/S0004972700031014
- A. I. Mal′cev, Constructive algebras. I, Uspehi Mat. Nauk 16 (1961), no. 3(99), 3–60 (Russian). MR 151377
- Ciprian Manolescu, Lectures on the triangulation conjecture, Proceedings of the Gökova Geometry-Topology Conference 2015, Gökova Geometry/Topology Conference (GGT), Gökova, 2016, pp. 1–38. MR 3526837
- Ciprian Manolescu, Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture, J. Amer. Math. Soc. 29 (2016), no. 1, 147–176. MR 3402697, DOI 10.1090/jams829
- A. Markov, The insolubility of the problem of homeomorphy, Dokl. Akad. Nauk SSSR 121 (1958), 218–220 (Russian). MR 97793
- Yuri V. Matiyasevich, Hilbert’s tenth problem, Foundations of Computing Series, MIT Press, Cambridge, MA, 1993. Translated from the 1993 Russian original by the author; With a foreword by Martin Davis. MR 1244324
- Alexander Melnikov, Computable topological groups and Pontryagin duality, Trans. Amer. Math. Soc. 370 (2018), no. 12, 8709–8737. MR 3864392, DOI 10.1090/tran/7355
- John Milnor, Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575–590. MR 133127, DOI 10.2307/1970299
- Alexander G. Melnikov and André Nies, The classification problem for compact computable metric spaces, The nature of computation, Lecture Notes in Comput. Sci., vol. 7921, Springer, Heidelberg, 2013, pp. 320–328. MR 3102033, DOI 10.1007/978-3-642-39053-1_{3}7
- Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 48805, DOI 10.2307/1969769
- Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 488059, DOI 10.1007/978-1-4612-9906-6
- Robert L. Moore, Concerning a set of postulates for plane analysis situs, Trans. Amer. Math. Soc. 20 (1919), no. 2, 169–178. MR 1501119, DOI 10.1090/S0002-9947-1919-1501119-X
- James R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000. Second edition of [ MR0464128]. MR 3728284
- P. Novikov, On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov, 44 pp. 1–143, 1955.
- André Nies and Slawomir Solecki, Local compactness for computable Polish metric spaces is $\Pi ^1_1$-complete, Evolving computability, Lecture Notes in Comput. Sci., vol. 9136, Springer, Cham, 2015, pp. 286–290. MR 3382368, DOI 10.1007/978-3-319-20028-6_{2}9
- Alexander Nabutovsky and Shmuel Weinberger, Algorithmic unsolvability of the triviality problem for multidimensional knots, Comment. Math. Helv. 71 (1996), no. 3, 426–434. MR 1418946, DOI 10.1007/BF02566428
- Alexander Nabutovsky and Shmuel Weinberger, Variational problems for Riemannian functionals and arithmetic groups, Inst. Hautes Études Sci. Publ. Math. 92 (2000), 5–62 (2001). MR 1839486
- Marian B. Pour-El and J. Ian Richards, Computability in analysis and physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989. MR 1005942, DOI 10.1007/978-3-662-21717-7
- Bjorn Poonen, Undecidable problems: a sampler, Interpreting Gödel, Cambridge Univ. Press, Cambridge, 2014, pp. 211–241. MR 3468188
- Michael O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958), 172–194. MR 110743, DOI 10.2307/1969933
- Hans Sagan, Space-filling curves, Universitext, Springer-Verlag, New York, 1994. MR 1299533, DOI 10.1007/978-1-4612-0871-6
- V. L. Selivanov, On degree spectra of topological spaces, Lobachevskii J. Math. 41 (2020), no. 2, 252–259. MR 4131078, DOI 10.1134/s1995080220020146
- Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, 2009. MR 2517689, DOI 10.1017/CBO9780511581007
- Robert I. Soare, Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987. A study of computable functions and computably generated sets. MR 882921, DOI 10.1007/978-3-662-02460-7
- Robert I. Soare, Computability theory and differential geometry, Bull. Symbolic Logic 10 (2004), no. 4, 457–486. MR 2136634, DOI 10.2178/bsl/1102083758
- Carsten Thomassen, A characterization of the $2$-sphere in terms of Jordan curve separation, Proc. Amer. Math. Soc. 115 (1992), no. 3, 863–864. MR 1124153, DOI 10.1090/S0002-9939-1992-1124153-0
- Carsten Thomassen, The Jordan-Schönflies theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992), no. 2, 116–130. MR 1144352, DOI 10.2307/2324180
- William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975, DOI 10.1515/9781400865321
- A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42 (1936), no. 3, 230–265. MR 1577030, DOI 10.1112/plms/s2-42.1.230
- A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction, Proc. London Math. Soc. (2) 43 (1937), no. 7, 544–546. MR 1575661, DOI 10.1112/plms/s2-43.6.544
- I. A. Volodin, V. E. Kuznecov, and A. T. Fomenko, The problem of the algorithmic discrimination of the standard three-dimensional sphere, Uspehi Mat. Nauk 29 (1974), no. 5(179), 71–168 (Russian). Appendix by S. P. Novikov. MR 405426
- Klaus Weihrauch, Computable analysis, Texts in Theoretical Computer Science. An EATCS Series, Springer-Verlag, Berlin, 2000. An introduction. MR 1795407, DOI 10.1007/978-3-642-56999-9
Bibliographic Information
- Matthew Harrison-Trainor
- Affiliation: Department of Mathematics, University of Michigan Ann Arbor, Michigan
- MR Author ID: 977639
- Email: matthhar@umich.edu
- Alexander Melnikov
- Affiliation: School of Mathematics and Statistics, Victoria University of Wellington Wellington, New Zealand
- MR Author ID: 878230
- ORCID: 0000-0001-8781-7432
- Email: alexander.g.melnikov@gmail.com
- Received by editor(s): June 17, 2022
- Received by editor(s) in revised form: December 5, 2022
- Published electronically: January 18, 2024
- Additional Notes: The authors were partially supported by the Marsden Fund of New Zealand. The second author was partially supported by the Rutherford Discovery Fellowship RDF-MAU1905, Royal Society Te Apārangi.
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 1543-1596
- MSC (2020): Primary 03D45, 57K20
- DOI: https://doi.org/10.1090/tran/8915
- MathSciNet review: 4744736