A refinement of a conjecture of Quillen

Rahm, Alexander; Wendt, Matthias

doi:10.1016/j.crma.2015.03.022

Cited by 3 publications

(7 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Wendt's and the author's contribution is that we can determine precisely the module structure above the virtual cohomological dimension; this has allowed us to relate the Quillen conjecture for SL 2 to statements about Steinberg homology. This, together with the results of [43], has allowed us to find a refined version of the Quillen conjecture, which keeps track of all the types of known counter-examples to the original Quillen conjecture: Conjecture 3 (Refined Quillen conjecture [36]). Let K be a number field.…”

Section: Theorem 1 ([35]mentioning

confidence: 98%

An introduction to torsion subcomplex reduction

Rahm¹

2021

Preprint

View full text Add to dashboard Cite

This survey paper introduces to a technique called Torsion Subcomplex Reduction (TSR) for computing torsion in the cohomology of discrete groups acting on suitable cell complexes. TSR enables one to skip machine computations on cell complexes, and to access directly the reduced torsion subcomplexes, which yields results on the cohomology of matrix groups in terms of formulas. TSR has already yielded general formulas for the cohomology of the tetrahedral Coxeter groups as well as, at odd torsion, of SL2 groups over arbitrary number rings. The latter formulas have allowed to refine the Quillen conjecture. Furthermore, progress has been made to adapt TSR to Bredon homology computations. In particular for the Bianchi groups, yielding their equivariant K-homology, and, by the Baum-Connes assembly map, the K-theory of their reduced C * -algebras. As a side application, TSR has allowed to provide dimension formulas for the Chen-Ruan orbifold cohomology of the complexified Bianchi orbifolds, and to prove Ruan's crepant resolution conjecture for all complexified Bianchi orbifolds.

show abstract

Section: Theorem 1 ([35]mentioning

confidence: 98%

An introduction to torsion subcomplex reduction

Rahm¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…A more recent positive result is the fact that the Quillen conjecture is true above the virtual cohomological dimension for groups SL 2 (O K,S ), cf. [RW14]. Based on arguments of Henn-Lannes-Schwartz, counterexamples to Quillen's conjecture have been established by Dwyer for GL n (Z[1/2]), n ≥ 32, Henn and Lannes for GL n (Z[1/2]), n ≥ 14, and by Anton for GL n (Z[ζ 3 , 1/3]), n ≥ 27.…”

Section: Quillen's Conjecture On Cohomology Of Arithmetic Groupsmentioning

confidence: 99%

“…Based on arguments of Henn-Lannes-Schwartz, counterexamples to Quillen's conjecture have been established by Dwyer for GL n (Z[1/2]), n ≥ 32, Henn and Lannes for GL n (Z[1/2]), n ≥ 14, and by Anton for GL n (Z[ζ 3 , 1/3]), n ≥ 27. For precise literature references, we refer to the discussion in [Knu01] or [RW14]. Note that all the counterexamples to the Quillen conjecture known so far are established by first showing that Quillen's conjecture implies detection of cohomology classes on the diagonal matrices, and then producing examples of this failure of detection.…”

Section: Quillen's Conjecture On Cohomology Of Arithmetic Groupsmentioning

confidence: 99%

“…In the present situation, Quillen's conjecture would predict that the cohomology H • (GL 3 (k[E]); F ℓ ) is a free module over the ring H • top (GL 3 (C); F ℓ ) ∼ = F ℓ [c 1 , c 2 , c 3 ] of universal Chern-classes, but the computations in the paper show that this is false in far greater generality than known previously. On the other hand, having complete computations of the cohomology H • (GL 3 (k[E]); F ℓ ) allows to discuss possible reformulations of the conjecture, resulting in a further refinement and correction of the formulation attempted in [RW14].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the cohomology of $GL_3$ of elliptic curves and Quillen's conjecture

Wendt¹

2016

Preprint

Self Cite

View full text Add to dashboard Cite

The paper provides a computation of the additive structure as well as a partial description of the Chern-class module structure of the cohomology of GL 3 over the function ring of an elliptic curve over a finite field. The computation is achieved by a detailed analysis of the isotropy spectral sequence for the action of GL 3 on the associated Bruhat-Tits building. This provides insights into the function field analogue of Quillen's conjecture on the structure of cohomology rings of arithmetic groups. The computations exhibit a lot of explicit classes which are torsion for the Chern-class ring. In some examples, even the torsion-free quotient of cohomology fails to be free. A possible variation of Quillen's conjecture is also discussed.

show abstract

“…This result sheds light on the Quillen conjecture, showing that the Quillen conjecture holds in a number of rank one cases; it also sheds light on the relation between Quillen's conjecture and detection questions in group cohomology, where detection refers to injectivity of the restriction map to diagonal matrices. See [RW15a] for a more detailed discussion of these issues. It is also worthwhile pointing out that the failure of detection as in the previous theorem, i.e., the non-injectivity of the restriction map…”

Section: Introductionmentioning

confidence: 99%

On Farrell–Tate cohomology of SL2 over S-integers

Rahm

Wendt

2018

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

In this paper, we provide number-theoretic formulas for Farrell-Tate cohomology for SL2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups.We expose three applications, to (I) detection questions for the Quillen conjecture, (II) the existence of transfers for the Friedlander-Milnor conjecture, (III) cohomology of SL2 over number fields.

show abstract

A refinement of a conjecture of Quillen

Cited by 3 publications

References 14 publications

An introduction to torsion subcomplex reduction

An introduction to torsion subcomplex reduction

On the cohomology of $GL_3$ of elliptic curves and Quillen's conjecture

On Farrell–Tate cohomology of SL2 over S-integers

Contact Info

Product

Resources

About