Algebraic & Geometric Topology

On the third homotopy group of Orr's space

Emmanuel Dror Farjoun and Roman Mikhailov

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K Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space K ω . The problem of nontriviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.

Article information

Algebr. Geom. Topol., Volume 18, Number 1 (2018), 569-582.

Received: 30 March 2017
Revised: 2 July 2017
Accepted: 13 July 2017
First available in Project Euclid: 1 February 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q52: Homotopy groups of special spaces 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

link invariants homotopy groups Orr invariant


Dror Farjoun, Emmanuel; Mikhailov, Roman. On the third homotopy group of Orr's space. Algebr. Geom. Topol. 18 (2018), no. 1, 569--582. doi:10.2140/agt.2018.18.569.

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  • G Baumslag, U Stammbach, On the inverse limit of free nilpotent groups, Comment. Math. Helv. 52 (1977) 219–233
  • A K Bousfield, Homological localization towers for groups and $\Pi $–modules, Mem. Amer. Math. Soc. 186, Amer. Math. Soc., Providence, RI (1977)
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)
  • T D Cochran, Link concordance invariants and homotopy theory, Invent. Math. 90 (1987) 635–645
  • T D Cochran, Derivatives of links: Milnor's concordance invariants and Massey's products, Mem. Amer. Math. Soc. 427, Amer. Math. Soc., Providence, RI (1990)
  • E Dror, W G Dwyer, D M Kan, An arithmetic square for virtually nilpotent spaces, Illinois J. Math. 21 (1977) 242–254
  • S Eilenberg, S Mac Lane, On the groups $H(\Pi,n)$, II: Methods of computation, Ann. of Math. 60 (1954) 49–139
  • N Habegger, X-S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389–419
  • K Igusa, K E Orr, Links, pictures and the homology of nilpotent groups, Topology 40 (2001) 1125–1166
  • S O Ivanov, R Mikhailov, On lengths of HZ-localization towers, preprint (2016)
  • J P Levine, Link concordance and algebraic closure of groups, Comment. Math. Helv. 64 (1989) 236–255
  • J Milnor, Isotopy of links, from “Algebraic geometry and topology: a symposium in honor of S Lefschetz” (R Fox, D Spencer, A Tucker, editors), Princeton Univ. Press (1957) 280–306
  • K E Orr, New link invariants and applications, Comment. Math. Helv. 62 (1987) 542–560
  • K E Orr, Homotopy invariants of links, Invent. Math. 95 (1989) 379–394
  • J H C Whitehead, A certain exact sequence, Ann. of Math. 52 (1950) 51–110