Geometry & Topology

Combinatorial group theory and the homotopy groups of finite complexes

Roman Mikhailov and Jie Wu

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Abstract

For n>k3, we construct a finitely generated group with explicit generators and relations obtained from braid groups, whose center is exactly πn(Sk). Our methods can be extended to obtain combinatorial descriptions of homotopy groups of finite complexes. As an example, we also give a combinatorial description of the homotopy groups of Moore spaces.

Article information

Source
Geom. Topol., Volume 17, Number 1 (2013), 235-272.

Dates
Received: 23 September 2011
Revised: 2 October 2012
Accepted: 2 October 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732523

Digital Object Identifier
doi:10.2140/gt.2013.17.235

Mathematical Reviews number (MathSciNet)
MR3035327

Zentralblatt MATH identifier
1270.55011

Subjects
Primary: 55Q40: Homotopy groups of spheres 55Q52: Homotopy groups of special spaces
Secondary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 20F36: Braid groups; Artin groups 55U10: Simplicial sets and complexes 57M07: Topological methods in group theory

Keywords
homotopy groups braid groups free product with amalgamation simplicial groups spheres Moore spaces Brunnian words

Citation

Mikhailov, Roman; Wu, Jie. Combinatorial group theory and the homotopy groups of finite complexes. Geom. Topol. 17 (2013), no. 1, 235--272. doi:10.2140/gt.2013.17.235. https://projecteuclid.org/euclid.gt/1513732523


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References

  • V G Bardakov, R Mikhailov, V V Vershinin, J Wu, Brunnian braids on surfaces, Algebr. Geom. Topol. 12 (2012) 1607–1648
  • G Baumslag, Finitely presented groups, from: “Proc. Internat. Conf. Theory of Groups”, (L G Kovacs, B H Neumann, editors), Gordon and Breach, New York (1967) 37–50
  • A J Berrick, F R Cohen, Y L Wong, J Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006) 265–326
  • A K Bousfield, E B Curtis, D M Kan, D G Quillen, D L Rector, J W Schlesinger, The ${\rm mod}-p$ lower central series and the Adams spectral sequence, Topology 5 (1966) 331–342
  • R Brown, J-L Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311–335
  • W-L Chow, On the algebraical braid group, Ann. of Math. (2) 49 (1948) 654–658
  • F R Cohen, J Wu, On braid groups, free groups, and the loop space of the 2-sphere, from: “Categorical decomposition techniques in algebraic topology”, (G Arone, J Hubbuck, R Levi, M Weiss, editors), Progr. Math. 215, Birkhäuser, Basel (2004) 93–105
  • F R Cohen, J Wu, Artin's braid groups, free groups, and the loop space of the 2-sphere, Q. J. Math. 62 (2011) 891–921
  • D Conduché, Modules croisés généralisés de longueur $2$, from: “Proceedings of the Luminy conference on algebraic $K$-theory”, (E M Friedlander, M Karoubi, editors), volume 34 (1984) 155–178
  • E B Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971) 107–209
  • G Ellis, R Mikhailov, A colimit of classifying spaces, Adv. Math. 223 (2010) 2097–2113
  • A O Houcine, Embeddings in finitely presented groups which preserve the center, J. Algebra 307 (2007) 1–23
  • D M Kan, A combinatorial definition of homotopy groups, Ann. of Math. 67 (1958) 282–312
  • D M Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958) 38–53
  • D M Kan, W P Thurston, Every connected space has the homology of a $K(\pi ,1)$, Topology 15 (1976) 253–258
  • J Y Li, J Wu, On symmetric commutator subgroups, braids, links and homotopy groups, Trans. Amer. Math. Soc. 363 (2011) 3829–3852
  • W Magnus, A Karrass, D Solitar, Combinatorial group theory: presentations of groups in terms of generators and relations, Pure and Applied Mathematics 13, Interscience Publishers (1966)
  • R Mikhailov, I B S Passi, J Wu, Symmetric ideals in group rings and simplicial homotopy, J. Pure Appl. Algebra 215 (2011) 1085–1092
  • J Milnor, On the construction $F[K]$, from: “Algebraic topology–-a student guide”, (J F Adams, editor), London Mathematical Society Lecture Note Series 4, Cambridge Univ. Press (1972) 119–136
  • D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer, Berlin (1967)
  • J H C Whitehead, On the asphericity of regions in a $3$–sphere, Fund. Math. 32 (1939) 149–166
  • J Wu, On combinatorial descriptions of homotopy groups and the homotopy theory of mod 2 Moore spaces, PhD thesis, University of Rochester (1995)
  • J Wu, On fibrewise simplicial monoids and Milnor–Carlsson's constructions, Topology 37 (1998) 1113–1134
  • J Wu, Combinatorial descriptions of homotopy groups of certain spaces, Math. Proc. Cambridge Philos. Soc. 130 (2001) 489–513
  • J Wu, A braided simplicial group, Proc. London Math. Soc. 84 (2002) 645–662
  • H Zhao, X Wang, Combinatorial description of the homotopy groups of wedge of spheres, Proc. Amer. Math. Soc. 137 (2009) 371–380