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String topology of finite groups of Lie type

Jesper Grodal, Anssi Lahtinen

Abstract

We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ 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, a la Chas-Sullivan. If a certain class in the homology of the finite group of Lie type, arising from the fundamental class of $G$, is nontrivial, then this module structure is free of rank one, providing a highly structured isomorphism between the two cohomologies. We verify the nontriviality of the class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and avoid the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the 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.

String topology of finite groups of Lie type

Abstract

We show that the mod cohomology of any finite group of Lie type in characteristic different from admits the structure of a module over the mod cohomology of the free loop space of the classifying space of the corresponding compact Lie group , via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain class in the homology of the finite group of Lie type, arising from the fundamental class of , is nontrivial, then this module structure is free of rank one, providing a highly structured isomorphism between the two cohomologies. We verify the nontriviality of the class in a range of cases, including all simply connected untwisted classical groups over the field of elements, with congruent to 1 mod . We also show how to deal with twistings and avoid the congruence condition by replacing by a certain -compact fixed point group depending on the order of mod , without changing the finite group. With this modification, we know of no examples where the 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.

Paper Structure

This paper contains 15 sections, 22 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Construction of the string products: Proof of Theorem \ref{['stringtop-arXiv-v2:thm:mainresult']}, part 1
  4. The string pairing and its properties
  5. The string pairing $\circ \colon \mathbb{H}^\ast P(g,h) \otimes \mathbb{H}^\ast P(f,g) \to \mathbb{H}^\ast P(f,h)$
  6. The string product on $\mathbb{H}^*(LBG)$ and the string module structure on $\mathbb{H}^*(BG^{h\sigma})$
  7. Outline of the rest of Section \ref{['stringtop-arXiv-v2:sec:products']}
  8. The first construction of the string pairing
  9. Consequences of the first construction of the string pairing
  10. The second construction of the string pairing
  11. Outline of the construction
  12. A brief summary of umkehr-maps
  13. The category hF^ fop
  14. The category P_B enriched in (F^ fop)^ op
  15. The functor fH.

Key Result

Theorem 1

Let $\ell$ be a prime and let $BG$ be a semisimple $\ell$--compact group of dimension $d$ with a self-map $\sigma\colon BG \to BG$. The cohomology groups $H^\ast(BG^{h\sigma};{\mathbb {F}}_\ell)$ admit a $\mathbb{H}^\ast(LBG;{\mathbb {F}}_\ell)$--module structure via the above constructions, extending the natural $H^*(BG;{\mathbb {F}}_\ell)$--module structure on these cohomology groups (Definitio

Theorems & Definitions (55)

  • Theorem 1
  • Definition 1.1
  • Theorem 2
  • Proposition 1.2
  • Theorem 3
  • Remark 1.3
  • Theorem 1.4: Untwisting theorem
  • Definition 1.5: $[G^{h\bbtau'}]$--fundamental class
  • Corollary 1.6
  • Theorem 4
  • ...and 45 more