Unstable Adams operations acting on \(p\)-local compact groups and fixed points. (English) Zbl 1259.55005

González, Alex

Algebr. Geom. Topol. 12, No. 2, 867-908 (2012).

F. Junod, R. Levi and A. Libman defined unstable Adams operations on \(p\)-local compact groups in [Algebr. Geom. Topol. 12, No. 1, 49–74 (2012; Zbl 1258.55010)]. One typical application of Adams operations for compact Lie groups \(G\) has been to use fixed points to approximate \(BG\) by classifying spaces of finite groups, see E. M. Friedlander and G. Mislin [Invent. Math. 83, 425–436 (1986; Zbl 0566.55011)]. Recently, C. Broto and J. M. Møller have performed this for \(p\)-compact groups [Algebr. Geom. Topol. 7, 1809–1919 (2007; Zbl 1154.55009)] and the aim of the article under review is to study to which extent this approach can be applied to \(p\)-local compact groups.
It turns out that each Adams operation defines a whole family of them – by iteration – and these determine a so-called “strong fixed points” transporter system. It is however difficult to check whether or not they correspond to saturated fusion systems. This is achieved in the case of \(p\)-local compact groups of rank one and for those coming from the unitary compact Lie group \(U(n)\). Here, the \(p\)-local compact group is indeed approximated by \(p\)-local finite groups and such an approximation yields a stable element theorem for computing the mod \(p\) cohomology.

Reviewer: Jérôme Scherer (Lausanne)

MSC:

55R35	Classifying spaces of groups and \(H\)-spaces in algebraic topology
20D20	Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
55S25	\(K\)-theory operations and generalized cohomology operations in algebraic topology

Keywords:

Adams operation; \(p\)-local compact group; \(p\)-local finite group; stable element

Citations:

Zbl 0566.55011; Zbl 1154.55009; Zbl 1258.55010

References:

[1]	A Adem, R J Milgram, Cohomology of finite groups, Grundl. Math. Wissen. 309, Springer (2004) · Zbl 1061.20044
[2]	A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972) · Zbl 0259.55004
[3]	C Broto, N Castellana, J Grodal, R Levi, B Oliver, Subgroup families controlling \(p\)-local finite groups, Proc. London Math. Soc. 91 (2005) 325 · Zbl 1090.20026
[4]	C Broto, N Castellana, J Grodal, R Levi, B Oliver, Extensions of \(p\)-local finite groups, Trans. Amer. Math. Soc. 359 (2007) 3791 · Zbl 1145.55013
[5]	C Broto, R Levi, B Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003) 779 · Zbl 1033.55010
[6]	C Broto, R Levi, B Oliver, A geometric construction of saturated fusion systems (editors D Arlettaz, K Hess), Contemp. Math. 399, Amer. Math. Soc. (2006) 11 · Zbl 1100.55005
[7]	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 · Zbl 1135.55008
[8]	C Broto, J M Møller, Chevalley \(p\)-local finite groups, Algebr. Geom. Topol. 7 (2007) 1809 · Zbl 1154.55009
[9]	E M Friedlander, Unstable \(K\)-theories of the algebraic closure of a finite field, Comment. Math. Helv. 50 (1975) 145 · Zbl 0307.18005
[10]	E M Friedlander, Étale homotopy of simplicial schemes, Annals of Math. Studies 104, Princeton Univ. Press (1982) · Zbl 0538.55001
[11]	E M Friedlander, G Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 59 (1984) 347 · Zbl 0548.55016
[12]	E M Friedlander, G Mislin, Locally finite approximation of Lie groups, I, Invent. Math. 83 (1986) 425 · Zbl 0566.55011
[13]	A González, The structure of \(p\)-local compact groups, PhD thesis, Autónoma de Barcelona (2010)
[14]	F Junod, Unstable Adams operations on \(p\)-local compact groups, PhD thesis, University of Aberdeen (2009) · Zbl 1258.55010
[15]	F Junod, R Levi, A Libman, Unstable Adams operations on \(p\)-local compact groups, Alg. Geom. Topol. 12 (2012) 49 · Zbl 1258.55010
[16]	R Kessar, R Stancu, A reduction theorem for fusion systems of blocks, J. Algebra 319 (2008) 806 · Zbl 1193.20006
[17]	S Mac Lane, Homology, Classics in Math., Springer (1995) · Zbl 0818.18001
[18]	M Mimura, H Toda, Topology of Lie groups. I, II, Transl. of Math. Monogr. 91, Amer. Math. Soc. (1991) · Zbl 0757.57001
[19]	B Oliver, J Ventura, Extensions of linking systems with \(p\)-group kernel, Math. Ann. 338 (2007) 983 · Zbl 1134.55011
[20]	D Quillen, Higher algebraic \(K\)-theory. I (editor H Bass), Lecture Notes in Math. 341, Springer (1973) 85 · Zbl 0292.18004
[21]	L Ribes, P Zalesskii, Profinite groups, Ergeb. Math. Grenzgeb. 40, Springer (2000) · Zbl 0949.20017
[22]	C A Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge Univ. Press (1994) · Zbl 0797.18001