Algebraic & Geometric Topology

Homotopy nilpotent groups

Abstract

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define homotopy $n$–nilpotent groups as homotopy algebras over certain simplicial algebraic theories. This notion interpolates between infinite loop spaces and loop spaces, but backwards. We study the relation to ordinary nilpotent groups. We prove that $n$–excisive functors of the form $ΩF$ factor over the category of homotopy $n$–nilpotent groups.

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 1 (2010), 33-61.

Dates
Revised: 2 October 2009
Accepted: 7 October 2009
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882307

Digital Object Identifier
doi:10.2140/agt.2010.10.33

Mathematical Reviews number (MathSciNet)
MR2580428

Zentralblatt MATH identifier
1329.55008

Citation

Biedermann, Georg; Dwyer, William G. Homotopy nilpotent groups. Algebr. Geom. Topol. 10 (2010), no. 1, 33--61. doi:10.2140/agt.2010.10.33. https://projecteuclid.org/euclid.agt/1513882307

References

• G Arone, M Ching, Operads and chain rules for the calculus of functors
• G Arone, W G Dwyer, K Lesh, Loop structures in Taylor towers, Algebr. Geom. Topol. 8 (2008) 173–210
• G Arone, M Kankaanrinta, A functorial model for iterated Snaith splitting with applications to calculus of functors, from: “Stable and unstable homotopy (Toronto, ON, 1996)”, (W G Dwyer, S Halperin, R Kane, S O Kochman, M E Mahowald, P S Selick, editors), Fields Inst. Commun. 19, Amer. Math. Soc. (1998) 1–30
• G Arone, M Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999) 743–788
• B Badzioch, Algebraic theories in homotopy theory, Ann. of Math. $(2)$ 155 (2002) 895–913
• B Badzioch, K Chung, A A Voronov, The canonical delooping machine, J. Pure Appl. Algebra 208 (2007) 531–540
• G Biedermann, B Chorny, O R öndigs, Calculus of functors and model categories, Adv. Math. 214 (2007) 92–115
• G Biedermann, W G Dwyer, Homotopy nilpotent spaces, work in progress
• A K Bousfield, E M Friedlander, Homotopy theory of $\Gamma$–spaces, spectra, and bisimplicial sets, from: “Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II”, Lecture Notes in Math. 658, Springer, Berlin (1978) 80–130
• M Ching, A chain rule for Goodwillie derivatives of functors from spectra to spectra, to appear in Trans. Amer. Math. Soc.
• M Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005) 833–933
• E B Curtis, Some relations between homotopy and homology, Ann. of Math. $(2)$ 82 (1965) 386–413
• A Dold, Homology of symmetric products and other functors of complexes, Ann. of Math. $(2)$ 68 (1958) 54–80
• W G Dwyer, Localizations, from: “Axiomatic, enriched and motivic homotopy theory”, (J P C Greenlees, editor), NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer Acad. Publ., Dordrecht (2004) 3–28
• W G Dwyer, D M Kan, Homotopy theory and simplicial groupoids, Nederl. Akad. Wetensch. Indag. Math. 46 (1984) 379–385
• W G Dwyer, D M Kan, Equivalences between homotopy theories of diagrams, from: “Algebraic topology and algebraic $K$–theory (Princeton, N.J., 1983)”, (W Browder, editor), Ann. of Math. Stud. 113, Princeton Univ. Press (1987) 180–205
• G Ellis, R Steiner, Higher-dimensional crossed modules and the homotopy groups of $(n+1)$–ads, J. Pure Appl. Algebra 46 (1987) 117–136
• P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser Verlag, Basel (1999)
• T G Goodwillie, Calculus. II. Analytic functors, $K$–Theory 5 (1992) 295–332
• T G Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645–711
• M Hall, Jr, A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950) 575–581
• P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003)
• J F Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987) 35–87
• B Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc. 347 (1995) 1295–1321
• A Joyal, Letter to A Grothendieck
• N J Kuhn, Goodwillie towers and chromatic homotopy: an overview, from: “Proceedings of the Nishida Fest (Kinosaki 2003)”, (M Ando, N Minami, J Morava, W S Wilson, editors), Geom. Topol. Monogr. 10, Geom. Topol. Publ., Coventry (2007) 245–279
• F W Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963) 869–872
• A Mauer-Oats, Goodwillie calculi, PhD thesis, University of Illinois Urbana-Champaign (2002)
• A Mauer-Oats, Algebraic Goodwillie calculus and a cotriple model for the remainder, Trans. Amer. Math. Soc. 358 (2006) 1869–1895
• C Reedy, Homology of algebraic theories, PhD thesis, University of California, San Diego (1974)
• S Schwede, Stable homotopy of algebraic theories, Topology 40 (2001) 1–41
• J-P Serre, Lie algebras and Lie groups, second edition, Lecture Notes in Math. 1500, Springer (1992) 1964 lectures given at Harvard University
• E H Spanier, Algebraic topology, McGraw-Hill, New York (1966)