The bv algebra in string topology of classifying spaces

Kuribayashi, Katsuhiko; Menichi, Luc

doi:10.48550/arxiv.1610.03970

Cited by 1 publication

(1 citation statement)

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“…Our product on H * (LBG) should agree with the product constructed by Chataur and Menichi [CM12, Cor. 18] (with sign corrections by Kuribayashi and Menichi [KM16]), although we will not prove the comparison here. Indeed, Chataur and Menichi's product also arises as a composite of an induced map and an umkehr map as in (2.3), but with the umkehr map concat !…”

Section: Construction Of the Productsmentioning

confidence: 97%

String topology of finite groups of Lie type

Grodal,

Lahtinen

2020

Preprint

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We show that the mod cohomology of any finite group of Lie type in characteristic p = admits the structure of a module over the mod cohomology of the free loop space of the classifying space BG of the corresponding compact Lie group G, via ring and module structures constructed from string topology, à la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is nontrivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration.We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over Fq, with q congruent to 1 mod . We also show how to deal with twistings and get rid of the congruence condition by replacing BG by a certain -compact fixed point group depending on the order of q mod , without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings. Contents 1. Introduction 1 2. Construction of the products 7 3. Pairings on Serre spectral sequences 28 4. Proofs of Theorems A-C 32 5. On the existence of a fundamental class: Proof of Theorem D 35 6. On the existence of a fundamental class for general σ: Proof of Theorem E 40 Appendix A. Parametrized homotopy theory 45 Appendix B. Dualizability in a symmetric monoidal category 49 Appendix C. Z -root data and untwisting of finite groups of Lie type 50 References 56

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Section: Construction Of the Productsmentioning

confidence: 97%