Unstable Adams operations acting on<i>p</i>–local compact groups and fixed points

Gonzalez, Alex

doi:10.2140/agt.2012.12.867

Cited by 5 publications

(10 citation statements)

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“…From the perspective of homotopical group theory, unstable Adams operations on p-compact groups yield examples of p-local finite groups after taking homotopy fixed points [8] (see [24] similar results for p-local compact groups). To the extent that homotopy Kac-Moody groups may generalize homotopy Lie groups (as proposed e.g., in Grodal's 2010 ICM address [26]), unstable Adams operations on Kac-Moody groups and their homotopy fixed points are interesting.…”

Section: Introductionmentioning

confidence: 86%

Discrete approximations for complex Kac–Moody groups

Foley

2015

Advances in Mathematics

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Abstract. We construct a map from the classifying space of a discrete KacMoody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac-Moody group and prove that it is a homology equivalence at primes q different from p. This generalises a classical result of Quillen-Friedlander-Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac-Moody groups compatible with the Frobenius homomorphism. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac-Moody groups.

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Section: Introductionmentioning

confidence: 86%

Discrete approximations for complex Kac–Moody groups

Foley

2015

Advances in Mathematics

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Section: Families Of Unstable Adams Operationsmentioning

confidence: 97%

“…Essentially, we follow the same lines as [23]. However, the introduction of telescopic transporter systems allows a great deal of simplification, and it is actually thanks to this that we are finally able to prove Proposition 3.17, basically the missing step in [23] in proving Theorem 1. We have opted for reproving here every property that we need from [23] for the sake of completeness as well as for correcting mistakes: while working on Proposition 3.17 below, the author realized that the statement of [23,Lemma 2.11] is false.…”

Section: Families Of Unstable Adams Operationsmentioning

confidence: 97%

“…We have opted for reproving here every property that we need from [23] for the sake of completeness as well as for correcting mistakes: while working on Proposition 3.17 below, the author realized that the statement of [23,Lemma 2.11] is false. Nevertheless, this affects neither the main results of [23] nor the results that we present in this paper, and this comment is just intended as a warning to the interested reader.…”

Section: Families Of Unstable Adams Operationsmentioning

confidence: 99%

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Finite approximations of p–local compact groups

Gonzalez

2016

Geom. Topol.

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“…Lemma 1. 23 Let G = (S, F, L) be a p-local compact group, and let P ≤ S. Then P is fully F -centralized if and only if P • is fully F -centralized. Furthermore, if this is the case then C G (P) = C G (P • ).…”

Section: Proofmentioning

confidence: 99%

Finite approximations of $p$-local compact groups

Gonzalez

2015

Preprint

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In this paper we show how every p-local compact group can be described as a telescope of p-local finite groups. As a consequence, we deduce several corollaries, such as a Stable Elements Theorem for the mod p cohomology of their classifying spaces, and a generalized Dwyer-Zabrodsky description of certain related mapping spaces. 55R35; 55R40, 20D20For a fixed prime p, the concept of p-local compact group was introduced in the last decade by C. Broto, R. Levi and B. Oliver in [7]. It provides a unifying, categorical language to study several classes of groups from a p-local point of view, including finite groups, compact Lie groups, p-compact groups and algebraic groups over algebraic closures of fields of characteristic other than p.In view of the many classes of groups and other mathematical objects that p-local compact groups model, it is no surprise that this theory is not very well understood yet. Nevertheless several structural results have already been conjectured since its introduction in [7], of which two are of special relevance in this work. The first conjecture asks about the existence of a version of the Stable Elements Theorem by H. Cartan and S. Eilenberg [10, Theorem 10.1] for p-local compact groups, while the second conjecture suggests the extension of the classic result of W. Dwyer and A. Zabrodsky [17, Theorem 1.1] on mapping spaces to p-local compact groups.A reader familiar with the theory of p-local finite groups will know that the proofs of these results in the finite case, as well as many others, depend ultimately on a certain biset, which is at the core this theory. And it is there where the main obstacle to study p-local compact groups resides, since there is no such biset in the context of p-local compact groups.An alternative to the biset is necessary, and evidence that such alternative may exist is found in a result by E. M. Friedlander and G. Mislin,[19,20], that relates compact Lie groups to towers of finite groups. In this paper we prove a version of the result of Friedlander and Mislin for p-local compact groups.Many statements for p-local finite groups extend to p-local compact groups as a consequence of this approximation result that replaces the use of the biset by arguments on Mapping spaces 54Bibliography 69Acknowledgements. The author is specially grateful to C. Broto, R. Levi and B. Oliver for numerous inspirational conversations and their support. The author would like to thank also N. Castellana, A. Libman and R. Stancu for their help in various stages of this work, and A. Chermak and R. Molinier for their support during the final stages of this work. Finally, the author is truly indebted to the referee, without whose assistance and thorough job the reader's experience with this paper would be greatly diminished. Background on p-local compact groupsLet p a prime, to remain fixed for the rest of the paper unless otherwise stated. In this section we review all the definitions and results about p-local compact groups that we Finally, we say that P is fully F -centralized, resp...

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Unstable Adams operations acting onp–local compact groups and fixed points

Cited by 5 publications

References 17 publications

Discrete approximations for complex Kac–Moody groups

Discrete approximations for complex Kac–Moody groups

Finite approximations of p–local compact groups

Finite approximations of $p$-local compact groups

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