A spectral sequence for fusion systems

Ramos, Antonio Díaz

doi:10.2140/agt.2014.14.349

Cited by 6 publications

(8 citation statements)

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“…Theorem 2 Let G D .S; F; L/ be a p-local compact group, and let M be a finite Z .p/ -module with trivial S -action. Then the natural map A version of the spectral sequence of A Díaz [15] for p-local compact groups is also deduced easily. The notation .…”

Section: D20 55r35 55r40mentioning

confidence: 98%

“…This spectral sequence has a rather interesting feature: it uses only a strongly closed subgroup of a fusion system, instead of a normal fusion subsystem, and thus is much more versatile. There are many interesting corollaries in [15,Sections 5 and 6], which we suspect to admit generalizations to the compact case. We leave this out of the scope of this paper for the sake of brevity.…”

Section: D20 55r35 55r40mentioning

confidence: 99%

“…Z .p/ Mod; defined in terms of the double complexes A n;m i .X / above. In particular, the functor X n;m i defined above corresponds to the case k D 2 (for the sake of brevity, the reader is referred to [15,Section 4] for details). Furthermore, we have a commutative triangle of functors:…”

Section: ˝Bmmentioning

confidence: 99%

“…which satisfies condition ( ) in Lemma 4.1, since, for each P Ä S and each i 0, the group H n .P \ S i I H m .P \ S i \ RI M // is finite and the Mittag-Leffler condition [36, Definition 3.5.6] applies. SetH n .S=RI H m .RI M // F D .X n;m / Fin order to match the notation of[15]. Let G D .S; F; L/ be a p-local compact group, let R Ä S be a strongly F-closed subgroup, and let M be a finite Z .p/ -module with trivial S-action.…”

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confidence: 99%

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Finite approximations of p–local compact groups

Gonzalez

2016

Geom. Topol.

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Section: D20 55r35 55r40mentioning

confidence: 98%

Section: D20 55r35 55r40mentioning

confidence: 99%

Section: ˝Bmmentioning

confidence: 99%

mentioning

confidence: 99%

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Finite approximations of p–local compact groups

Gonzalez

2016

Geom. Topol.

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“…A version of the spectral sequence of A. Díaz [15] for p-local compact groups is also deduced easily. The notation (−) F below denotes stable elements in F , see Section 4 for an explicit definition.…”

mentioning

confidence: 91%

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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A p-nilpotency criterion for finite groups

Ramos

Viruel

2018

Acta Math. Hungar.

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A spectral sequence for fusion systems

Cited by 6 publications

References 16 publications

Finite approximations of p–local compact groups

Finite approximations of p–local compact groups

Finite approximations of $p$-local compact groups

A p-nilpotency criterion for finite groups

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