Homotopy equivalences between p-subgroup categories

Gelvin, Matthew; Møller, Jesper M.

doi:10.1016/j.jpaa.2014.10.002

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…Proof: For any finite poset, the existence of a unique minimal, right cofinal, full subposet is guaranteed by [22,Theorem 4.3]. For S, it is convenient identify this subposet by considering covers of categories [17, 2.5].…”

Section: Vanishing Results and The Proof Of Lemma 51mentioning

confidence: 99%

“…To see that S is minimal, we must show that I / ∈ S implies that the full subposet of elements strictly greater than I denoted I ↓↓ S has a contractible realization [22,Theorem 4.3]. Since I / ∈ S, the maximal elements of I ↓↓ S constitute a nonempty subset of the maximal elements of S and provide a cover of I ↓↓ S as above.…”

Section: Vanishing Results and The Proof Of Lemma 51mentioning

confidence: 99%

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Recognizing nullhomotopic maps into the classifying space of a Kac-Moody group

Foley

2015

Preprint

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This paper extends certain characterizations of nullhomotopic maps between p-compact groups to maps with target the p-completed classifying space of a connected Kac-Moody group and source the classifying space of either a p-compact group or a connected Kac-Moody group. A well known inductive principle for p-compact groups is applied to obtain general, mapping space level results. An arithmetic fiber square computation shows that a null map from the classifying space of a connected compact Lie group to the classifying space of a connected topological Kac-Moody group can be detected by restricting to the maximal torus. Null maps between the classifying spaces of connected topological Kac-Moody groups cannot, in general, be detected by restricting to the maximal torus due to the nonvanishing of an explicit abelian group of obstructions described here. Nevertheless, partial results are obtained via the application of algebraic discrete Morse theory to higher derived limit calculations which show that such detection is possible in many cases of interest.

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Section: Vanishing Results and The Proof Of Lemma 51mentioning

confidence: 99%

Section: Vanishing Results and The Proof Of Lemma 51mentioning

confidence: 99%

Recognizing nullhomotopic maps into the classifying space of a Kac-Moody group

Foley

2015

Preprint

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“…The omnibus Theorem 7.5 below is mainly a translation of "classical" homotopy equivalences [GS06, Thm. 1.1], using the elementary Lemma 7.1; see also [GM15] for F, and [GS06] for historical references. As usual let S p (G) and A p (G) denote non-trivial p-groups and non-trivial elementary abelian psubgroups V ∼ = (Z/p) r respectively.…”

Section: Appendix: Varying the Collection C Of Subgroupsmentioning

confidence: 99%

Endotrivial modules for finite groups via homotopy theory

Grodal

2016

Preprint

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Classifying endotrivial kG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G, has been a long-running quest, which by deep work of Dade, Alperin, Carlson, Thévenaz, and others, has been reduced to understanding the subgroup consisting of modular representations that split as the trivial module direct sum a projective module when restricted to a Sylow p-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivial p-subgroups with values in the units k × , viewed as a constant coefficient system. We then use homotopical techniques to give a number of formulas for this group in terms of the abelianization of normalizers and centralizers in G, in particular verifying the Carlson-Thévenaz conjecture-this reduces the calculation of this group to algorithmic calculations in local group theory rather than representation theory. We also provide strong restrictions on when such representations of dimension greater than one can occur, in terms of the p-subgroup complex and p-fusion systems. We immediately recover and extend a large number of computational results in the literature, and further illustrate the computational potential by calculating the group in other sample new cases, e.g., for the Monster at all primes.

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Recognizing nullhomotopic maps into the classifying space of a Kac–Moody group

Foley¹

2022

Math. Z.

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Homotopy equivalences between p-subgroup categories

Cited by 5 publications

References 12 publications

Recognizing nullhomotopic maps into the classifying space of a Kac-Moody group

Recognizing nullhomotopic maps into the classifying space of a Kac-Moody group

Endotrivial modules for finite groups via homotopy theory

Recognizing nullhomotopic maps into the classifying space of a Kac–Moody group

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