scholarly journals Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups

2021 ◽  
pp. 1-32
Author(s):  
David Muñoz ◽  
Jorge Plazas ◽  
Mario Velásquez
Keyword(s):  
Spectral Sequence ◽  
Discrete Group ◽  
Hecke Operators ◽  
Classifying Space ◽  
Main Interest ◽  
Proper Actions

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.

scholarly journals Hecke Operations and the Adams E2-Term Based on Elliptic Cohomology

1999 ◽  
Vol 42 (2) ◽  
pp. 129-138 ◽  
Author(s):  
Andrew Baker
Keyword(s):  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Hecke Operators ◽  
Elliptic Cohomology

AbstractHecke operators are used to investigate part of the E2-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of Ext1 which combines use of classical Hecke operators and p-adic Hecke operators due to Serre.

Properties of Selick's filtration of the double suspension E2

2014 ◽  
Vol 06 (03) ◽  
pp. 421-440
Author(s):  
Jelena Grbić ◽  
Stephen Theriault ◽  
Hao Zhao
Keyword(s):  
Hopf Algebra ◽  
Spectral Sequence ◽  
Homotopy Groups ◽  
Classifying Space ◽  
Special Cases ◽  
Text Study ◽  
Classical Homotopy

To help study the double suspension [Formula: see text] when localised at a prime p, Selick filtered Ω2S2n+1 by H-spaces which geometrically realise a natural Hopf algebra filtration of H*(Ω2S2n+1;ℤ/p). Later, Gray showed that the fiber Wn of E2 has an integral classifying space BWn and there is a homotopy fibration [Formula: see text]. In this paper we correspondingly filter BWn in a manner compatible with Selick's filtration and the homotopy fibration [Formula: see text], study the multiplicative properties and homotopy exponents of the spaces in the filtrations, and use the filtrations to filter exponent information for the homotopy groups of S2n+1. Our results link three seemingly different in nature classical homotopy fibrations given by Toda, Selick and Gray and make them special cases of a systematic whole. In addition we construct a spectral sequence which converges to the homotopy groups of BWn.

scholarly journals The mod 2 homology of Sp (n) instantons and the classifying space of the gauge group

1997 ◽  
Vol 55 (3) ◽  
pp. 503-512 ◽  
Author(s):  
Younggi Choi
Keyword(s):  
Spectral Sequence ◽  
Gauge Group ◽  
Moduli Space ◽  
Classifying Space ◽  
Serre Spectral Sequence ◽  
Homology Operations

We study the mod 2 homology of the moduli space of instantons associated with the prinicpal Sp (n) bundle over the four-sphere and the classifying space of the gauge group using the Serre spectral sequence and the homology operations.

Frobenius Induction for Higher Whitehead Groups

1987 ◽  
Vol 39 (1) ◽  
pp. 222-238 ◽  
Author(s):  
Andrew J. Nicas
Keyword(s):  
Group Ring ◽  
Discrete Group ◽  
Rational Numbers ◽  
Base Point ◽  
Classifying Space ◽  
Induced Representations ◽  
Image Position ◽  
Whitehead Groups

The theory of induced representations has served as a powerful tool in the computations of algebraicK-theory andL-theory ([2], [7], [4, 5], [9], [10, 11, 12, 13], [14], [17], [18]). In this paper we show how to apply this theory to obtain induction theorems for the higher Whitehead groups of Waldhausen. The same technique applies to the analogs of Whitehead groups in unitaryK-theory and inL-theory.For any ringAwith unit, letK(A) be the spectrum of the algebraicK-theory ofA([8, p. 343]). Given a discrete group Γ and a subringRof the rational numbers, Loday defines a map of spectra:*where (BΓ) is the classifying space of Γ union with a disjoint base point andRΓ is the group-ring of Γ overR.

On theK-theory of the classifying space of a discrete group

1992 ◽  
Vol 292 (1) ◽  
pp. 319-327 ◽  
Author(s):  
Alejandro Adem
Keyword(s):  
Discrete Group ◽  
Classifying Space

scholarly journals The completion theorem in K-theory for proper actions of a discrete group

Topology ◽  
2001 ◽  
Vol 40 (3) ◽  
pp. 585-616 ◽  
Author(s):  
Wolfgang Lück ◽  
Bob Oliver
Keyword(s):  
Discrete Group ◽  
Proper Actions ◽  
K Theory ◽  
Completion Theorem

scholarly journals On modules over infinite group rings

2016 ◽  
Vol 26 (03) ◽  
pp. 451-466 ◽  
Author(s):  
Gunnar Carlsson ◽  
Boris Goldfarb
Keyword(s):  
Group Ring ◽  
Discrete Group ◽  
Infinite Group ◽  
Homological Dimension ◽  
Group Rings ◽  
Finite Model ◽  
Classifying Space ◽  
Projective Resolutions ◽  
Algebraic Formula ◽  
Important Object

Let [Formula: see text] be a commutative ring and [Formula: see text] be an infinite discrete group. The algebraic [Formula: see text]-theory of the group ring [Formula: see text] is an important object of computation in geometric topology and number theory. When the group ring is Noetherian, there is a companion [Formula: see text]-theory of [Formula: see text] which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of [Formula: see text]-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. It has some expected properties such as independence from the choice of a word metric. We prove that, whenever [Formula: see text] is a regular Noetherian ring of finite global homological dimension and [Formula: see text] has finite asymptotic dimension and a finite model for the classifying space [Formula: see text], the natural Cartan map from the [Formula: see text]-theory of [Formula: see text] to [Formula: see text]-theory is an equivalence. On the other hand, our [Formula: see text]-theory is indeed better suited for computation as we show in a separate paper. Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain modules over group rings.

scholarly journals INJECTIVE HULLS OF CERTAIN DISCRETE METRIC SPACES AND GROUPS

2013 ◽  
Vol 05 (03) ◽  
pp. 297-331 ◽  
Author(s):  
URS LANG
Keyword(s):  
Metric Spaces ◽  
Positive Curvature ◽  
Hyperbolic Group ◽  
Finite Graph ◽  
Barycentric Subdivision ◽  
Classifying Space ◽  
Polyhedral Complex ◽  
Locally Finite ◽  
Proper Actions ◽  
Locally Finite Graph

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the '60s Isbell showed that every metric space X has an injective hull E (X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E (X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in [Formula: see text], for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space [Formula: see text] for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense.

Sign in / Sign up

Export Citation Format

Share Document