Groups of unstable Adams operations on p–local compact groups

Levi, Ran¹ ; Libman, Assaf

¹Institute of Mathematics, University of Aberdeen, Fraser Noble Building 138, Aberdeen, AB24 3UE, United Kingdom

Algebraic and Geometric Topology, Tome 17 (2017) no. 1, pp. 355-418

Résumé

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.

Détail

DOI : 10.2140/agt.2017.17.355

Classification : 55R35, 20D20
Keywords: $p$–local compact groups, unstable Adams operations

Bibliographie

[1] K K S Andersen, The normalizer splitting conjecture for p–compact groups, Fund. Math. 161 (1999) 1

[2] K K S Andersen, J Grodal, The classification of 2–compact groups, J. Amer. Math. Soc. 22 (2009) 387 | DOI

[3] 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 | DOI

[4] K K S Andersen, B Oliver, J Ventura, Reduced, tame and exotic fusion systems, Proc. Lond. Math. Soc. 105 (2012) 87 | DOI

[5] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972)

[6] 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 | DOI

[7] C Broto, R Levi, B Oliver, The rational cohomology of a p–local compact group, Proc. Amer. Math. Soc. 142 (2014) 1035 | DOI

[8] K S Brown, Cohomology of groups, 87, Springer (1982)

[9] P Gabriel, M Zisman, Calculus of fractions and homotopy theory, 35, Springer (1967)

[10] G Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994) 191 | DOI

[11] S Jackowski, J Mcclure, B Oliver, Homotopy classification of self-maps of BG via G–actions, I, Ann. of Math. 135 (1992) 183 | DOI

[12] F Junod, R Levi, A Libman, Unstable Adams operations on p–local compact groups, Algebr. Geom. Topol. 12 (2012) 49 | DOI

[13] 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 | DOI

[14] B Oliver, p–stubborn subgroups of classical compact Lie groups, J. Pure Appl. Algebra 92 (1994) 55 | DOI

[15] A M Robert, A course in p–adic analysis, 198, Springer (2000) | DOI

[16] E H Spanier, Algebraic topology, Springer (1966)

[17] R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91 | DOI