Morse theory for group presentations

Fernández, Ximena

doi:10.1090/tran/8958

Morse theory for group presentations
HTML articles powered by AMS MathViewer

by Ximena Fernández;

Trans. Amer. Math. Soc. 377 (2024), 2495-2523

DOI: https://doi.org/10.1090/tran/8958

Published electronically: February 14, 2024

HTML | PDF | Request permission

Abstract:

We introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplified Morse CW-complex, with an explicit description of the attaching maps and bounds on the dimension of the complexes involved in the deformation. We apply this technique to show that some known potential counterexamples to the Andrews–Curtis conjecture do satisfy the conjecture.

References

Selman Akbulut and Robion Kirby, A potential smooth counterexample in dimension $4$ to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture, Topology 24 (1985), no. 4, 375–390. MR 816520, DOI 10.1016/0040-9383(85)90010-2
J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192–195. MR 173241, DOI 10.1090/S0002-9939-1965-0173241-8
Jonathan A. Barmak, Algebraic topology of finite topological spaces and applications, Lecture Notes in Mathematics, vol. 2032, Springer, Heidelberg, 2011. MR 3024764, DOI 10.1007/978-3-642-22003-6
J. A. Barmak, A counterexample to a strong version of the Andrews–Curtis conjecture, Preprint, arXiv:1806.11493, 2018.
R. Sean Bowman and Stephen B. McCaul, Fast searching for Andrews-Curtis trivializations, Experiment. Math. 15 (2006), no. 2, 193–197. MR 2253006, DOI 10.1080/10586458.2006.10128962
Piotr Brendel, PawełDłotko, Graham Ellis, Mateusz Juda, and Marian Mrozek, Computing fundamental groups from point clouds, Appl. Algebra Engrg. Comm. Comput. 26 (2015), no. 1-2, 27–48. MR 3320904, DOI 10.1007/s00200-014-0244-1
P. Brendel, P. Dłotko, G. Ellis, M. Juda, and M. Mrozek, Fundamental group algorithm for low dimensional tessellated CW complexes, Preprint, arXiv:1507.03396, 2015.
M. R. Bridson, The complexity of balanced presentations and the Andrews–Curtis conjecture, Preprint, arXiv:1504.04187, 2015.
Ronald Brown, Coproducts of crossed $P$-modules: applications to second homotopy groups and to the homology of groups, Topology 23 (1984), no. 3, 337–345. MR 770569, DOI 10.1016/0040-9383(84)90016-8
Manoj K. Chari, On discrete Morse functions and combinatorial decompositions, Discrete Math. 217 (2000), no. 1-3, 101–113 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766262, DOI 10.1016/S0012-365X(99)00258-7
Marshall M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, Vol. 10, Springer-Verlag, New York-Berlin, 1973. MR 362320, DOI 10.1007/978-1-4684-9372-6
X. L. Fernández, Combinatorial methods and algorithms in low dimensional topology and the Andrews–Curtis conjecture, Ph.D. Thesis, Universidad de Buenos Aires, 2017.
X. L. Fernández, GitHub repository SageMath: finite topological spaces, 2021, https://github.com/ximenafernandez/Finite-Topological-Spaces/tree/1fdf3af56d0f3103a58d75cce7212372a458e6ba.
X. L. Fernández, K. I. Piterman, and I. Sadofschi Costa, GitHub repository GAP: posets, 2019, https://github.com/isadofschi/posets.
Robin Forman, A discrete Morse theory for cell complexes, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 112–125. MR 1358614
Robin Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1, 90–145. MR 1612391, DOI 10.1006/aima.1997.1650
Robin Forman, A user’s guide to discrete Morse theory, Sém. Lothar. Combin. 48 (2002), Art. B48c, 35. MR 1939695
The GAP Group, GAP – groups, algorithms, and programming, version 4.10.2, 2019.
S. M. Gersten, On Rapaport’s example in presentations of the trivial group, Preprint, 1987.
D. Gillman and D. Rolfsen, The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture, Topology 22 (1983), no. 3, 315–323. MR 710105, DOI 10.1016/0040-9383(83)90017-4
George Havas, P. E. Kenne, J. S. Richardson, and E. F. Robertson, A Tietze transformation program, Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 69–73. MR 760651
George Havas and Colin Ramsay, Breadth-first search and the Andrews-Curtis conjecture, Internat. J. Algebra Comput. 13 (2003), no. 1, 61–68. MR 1970867, DOI 10.1142/S0218196703001365
Cynthia Hog-Angeloni and Wolfgang Metzler (eds.), Two-dimensional homotopy and combinatorial group theory, London Mathematical Society Lecture Note Series, vol. 197, Cambridge University Press, Cambridge, 1993. MR 1279174, DOI 10.1017/CBO9780511629358
Rob Kirby (ed.), Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–473. MR 1470751, DOI 10.1090/amsip/002.2/02
Dmitry Kozlov, Combinatorial algebraic topology, Algorithms and Computation in Mathematics, vol. 21, Springer, Berlin, 2008. MR 2361455, DOI 10.1007/978-3-540-71962-5
Dmitry N. Kozlov, Organized collapse: an introduction to discrete Morse theory, Graduate Studies in Mathematics, vol. 207, American Mathematical Society, Providence, RI, [2020] ©2020. MR 4249617, DOI 10.1090/gsm/207
K. Krawiec and J. Swan, Distance metric ensemble learning and the Andrews–Curtis conjecture, Preprint, arXiv:1606.01412, 2016.
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024, DOI 10.1007/978-3-642-61896-3
Wolfgang Metzler, Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 291–326 (German). MR 564434
Wolfgang Metzler, On the Andrews-Curtis conjecture and related problems, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 35–50. MR 813099, DOI 10.1090/conm/044/813099
Alexei D. Miasnikov, Genetic algorithms and the Andrews-Curtis conjecture, Internat. J. Algebra Comput. 9 (1999), no. 6, 671–686. MR 1727164, DOI 10.1142/S0218196799000370
Alexei D. Miasnikov and Alexei G. Myasnikov, Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 257–263. MR 1829485
Charles F. Miller III and Paul E. Schupp, Some presentations of the trivial group, Groups, languages and geometry (South Hadley, MA, 1998) Contemp. Math., vol. 250, Amer. Math. Soc., Providence, RI, 1999, pp. 113–115. MR 1732210, DOI 10.1090/conm/250/03848
Alexei D. Myasnikov, Alexei G. Myasnikov, and Vladimir Shpilrain, On the Andrews-Curtis equivalence, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 183–198. MR 1921712, DOI 10.1090/conm/296/05074
Alexei D. Myasnikov, Alexei G. Myasnikov, and Vladimir Shpilrain, On the Andrews-Curtis equivalence, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 183–198. MR 1921712, DOI 10.1090/conm/296/05074
The Sage Developers, Sagemath, the Sage Mathematics software system (version 9.1), 2020, https://www.sagemath.org.
Heinrich Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys. 19 (1908), no. 1, 1–118 (German). MR 1547755, DOI 10.1007/BF01736688
J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428. MR 4123, DOI 10.2307/1968907
J. H. C. Whitehead, On incidence matrices, nuclei and homotopy types, Ann. of Math. (2) 42 (1941), 1197–1239. MR 5352, DOI 10.2307/1970465
J. H. C. Whitehead, Simple homotopy types, Amer. J. Math. 72 (1950), 1–57. MR 35437, DOI 10.2307/2372133
Wolfgang Metzler, On the Andrews-Curtis conjecture and related problems, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 35–50. MR 813099, DOI 10.1090/conm/044/813099
Perrin Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208 (1975), 161–169. MR 380813, DOI 10.1090/S0002-9947-1975-0380813-5
E. C. Zeeman, On the dunce hat, Topology 2 (1964), 341–358. MR 156351, DOI 10.1016/0040-9383(63)90014-4