Algebraic & Geometric Topology

Equivariant diagrams of spaces

Emanuele Dotto

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We generalize two classical homotopy theory results, the Blakers–Massey theorem and Quillen’s Theorem B, to G–equivariant cubical diagrams of spaces, for a discrete group G. We show that the equivariant Freudenthal suspension theorem for permutation representations is a direct consequence of the equivariant Blakers–Massey theorem. We also apply this theorem to generalize to G–manifolds a result about cubes of configuration spaces from embedding calculus. Our proof of the equivariant Theorem B involves a generalization of the classical Theorem B to higher-dimensional cubes, as well as a categorical model for finite homotopy limits of classifying spaces of categories.

Article information

Algebr. Geom. Topol., Volume 16, Number 2 (2016), 1157-1202.

Received: 19 June 2015
Accepted: 23 July 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P91: Equivariant homotopy theory [See also 19L47]
Secondary: 55Q91: Equivariant homotopy groups [See also 19L47]

equivariant connectivity homotopy limits


Dotto, Emanuele. Equivariant diagrams of spaces. Algebr. Geom. Topol. 16 (2016), no. 2, 1157--1202. doi:10.2140/agt.2016.16.1157.

Export citation


  • J,F Adams, Prerequisites (on equivariant stable homotopy) for Carlsson's lecture, from “Algebraic topology” (I Madsen, B Oliver, editors), Lecture Notes in Math. 1051, Springer, Berlin (1984) 483–532
  • C Barwick, D,M Kan, Quillen Theorems $B_n$ for homotopy pullbacks of $(\infty, k)$–categories, preprint (2013)
  • M B ökstedt, W,C Hsiang, I Madsen, The cyclotomic trace and algebraic $K\!$–theory of spaces, Invent. Math. 111 (1993) 465–539
  • A,K Bousfield, D,M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer, Berlin (1972)
  • W Chachólski, J Scherer, Homotopy theory of diagrams, Mem. Amer. Math. Soc. 736, Amer. Math. Soc., Providence, RI (2002)
  • E Dotto, Equivariant calculus of functors and $\mathbb{Z}/2$–analyticity of real algebraic $K\!$–theory, J. Inst. Math. Jussieu (2015) 1–55 published online
  • E Dotto, K Moi, Homotopy theory of $G$–diagrams and equivariant excision, Alg. Geom. Topol. 2016 (2016) 325–395
  • W,G Dwyer, D,M Kan, Function complexes for diagrams of simplicial sets, Nederl. Akad. Wetensch. Indag. Math. 45 (1983) 139–147
  • T,G Goodwillie, Calculus, II: Analytic functors, $K\!$–Theory 5 (1991/92) 295–332
  • T,G Goodwillie, J,R Klein, Multiple disjunction for spaces of Poincaré embeddings, J. Topol. 1 (2008) 761–803
  • P,S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)
  • S Jackowski, J Słomińska, $G$–functors, $G$–posets and homotopy decompositions of $G$–spaces, Fund. Math. 169 (2001) 249–287
  • L,G Lewis, Jr, Equivariant Eilenberg–Mac Lane spaces and the equivariant Seifert–van Kampen and suspension theorems, Topology Appl. 48 (1992) 25–61
  • M,G Lydakis, Homotopy limits of categories, J. Pure Appl. Algebra 97 (1994) 73–80
  • M Merling, Equivariant algebraic $K\!$–theory, PhD thesis, The University of Chicago, Ann Arbor, MI (2014) Available at \setbox0\makeatletter\@url {\unhbox0
  • U Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (1983) 431–460
  • D Quillen, Higher algebraic $K\!$–theory, I, from “Algebraic $K\!$–theory, I: Higher $K\!$–theories (Proc. Conf., Battelle Memorial Inst.)” (H Bass, editor), Lecture Notes in Math. 341, Springer, Berlin (1973) 85–147
  • S-i Takayasu, On stable summands of Thom spectra of $B({\bf Z}/2)\sp n$ associated to Steinberg modules, J. Math. Kyoto Univ. 39 (1999) 377–398
  • J Thévenaz, P,J Webb, Homotopy equivalence of posets with a group action, J. Combin. Theory Ser. A 56 (1991) 173–181
  • R,W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109
  • R Villarroel-Flores, The action by natural transformations of a group on a diagram of spaces, preprint (2004)
  • F Waldhausen, Algebraic $K\!$–theory of spaces, from “Algebraic and geometric topology” (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 318–419