Algebraic & Geometric Topology

Groups of unstable Adams operations on $p$–local compact groups

Ran Levi and Assaf Libman

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


A p–local compact group is an algebraic object modelled on the homotopy theory associated with p–completed classifying spaces of compact Lie groups and p–compact groups. In particular p–local compact groups give a unified framework in which one may study p–completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and p–compact groups, p–local compact groups admit unstable Adams operations: self equivalences that are characterised by their cohomological effect. Unstable Adams operations on p–local compact groups were constructed in a previous paper by F Junod, R Levi, and A Libman. In the present paper we study groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions, unstable Adams operations are determined by their degree. We also examine a particularly well behaved subgroup of unstable Adams operations.

Article information

Algebr. Geom. Topol., Volume 17, Number 1 (2017), 355-418.

Received: 20 December 2015
Revised: 3 June 2016
Accepted: 17 June 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R35: Classifying spaces of groups and $H$-spaces
Secondary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure

$p$–local compact groups unstable Adams operations


Levi, Ran; Libman, Assaf. Groups of unstable Adams operations on $p$–local compact groups. Algebr. Geom. Topol. 17 (2017), no. 1, 355--418. doi:10.2140/agt.2017.17.355.

Export citation


  • K,K,S Andersen, The normalizer splitting conjecture for $p$–compact groups, Fund. Math. 161 (1999) 1–16
  • K,K,S Andersen, J Grodal, The classification of $2$–compact groups, J. Amer. Math. Soc. 22 (2009) 387–436
  • K,K,S Andersen, J Grodal, J,M Møller, A Viruel, The classification of $p$–compact groups for $p$ odd, Ann. of Math. 167 (2008) 95–210
  • K,K,S Andersen, B Oliver, J Ventura, Reduced, tame and exotic fusion systems, Proc. Lond. Math. Soc. 105 (2012) 87–152
  • A,K Bousfield, D,M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer, Berlin (1972)
  • C Broto, R Levi, B Oliver, Discrete models for the $p$–local homotopy theory of compact Lie groups and $p$–compact groups, Geom. Topol. 11 (2007) 315–427
  • C Broto, R Levi, B Oliver, The rational cohomology of a $p$–local compact group, Proc. Amer. Math. Soc. 142 (2014) 1035–1043
  • K,S Brown, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • P Gabriel, M Zisman, Ergeb. Math. Grenzgeb. 35, Springer, Berlin (1967)
  • G Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994) 191–207
  • S Jackowski, J McClure, B Oliver, Homotopy classification of self-maps of $BG$ via $G$–actions, I, Ann. of Math. 135 (1992) 183–226
  • F Junod, R Levi, A Libman, Unstable Adams operations on $p$–local compact groups, Algebr. Geom. Topol. 12 (2012) 49–74
  • R Levi, A Libman, Existence and uniqueness of classifying spaces for fusion systems over discrete $p$–toral groups, J. Lond. Math. Soc. 91 (2015) 47–70
  • B Oliver, $p$–stubborn subgroups of classical compact Lie groups, J. Pure Appl. Algebra 92 (1994) 55–78
  • A,M Robert, A course in $p$–adic analysis, Graduate Texts in Mathematics 198, Springer, New York (2000)
  • E,H Spanier, Springer, New York (1966)
  • R,W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109