Squeezing and Higher Algebraic K-Theory

Bartels, Arthur

doi:10.1023/a:1024166521174

Cited by 70 publications

(70 citation statements)

References 22 publications

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“…This is almost [Bar03b,Corollary 4.3]. In [Bar03b], the spherical metric rather then the Euclidean metric is used, but this does not affect the argument. It is only important that the metric is the same on every simplex.…”

Section: A Vanishing Resultsmentioning

confidence: 99%

“…Later on, Higson was able to remove the finite classifying space assumption [Hig00]. In [Bar03b], the first author proved the algebraic K-and L-theory versions of Yu's work by developing a squeezing theorem for higher algebraic K-theory to use with the approach established in [Yu98]. For algebraic K-theory, this was also achieved in [CG04a].…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to extend the results from [Bar03b] by relaxing the finite BΓ assumption to allow for groups Γ with torsion. Specifically, we prove the following theorem.…”

Section: Introductionmentioning

confidence: 99%

“…This work was supported by the SFB 478 "Geometrische Strukturen in der Mathematik". considered in [Bar03b,CG04a], factors through the assembly map (0.1) via (0.3) H n (BΓ; K R ) ∼ = H Γ n (EΓ; K R ) → H Γ n (EΓ; K R ). In particular, the cokernel of (0.2) contains the cokernel of (0.3) in the situation of Theorem A.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the K-theory of groups with finite asymptotic dimension

Bartels

Rosenthal

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Section: A Vanishing Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to extend the results from [Bar03b] by relaxing the finite BΓ assumption to allow for groups Γ with torsion. Specifically, we prove the following theorem.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the K-theory of groups with finite asymptotic dimension

Bartels

Rosenthal

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…However, it is necessary for the homotopy-invariance arguments given in [3,5] and earlier papers to work. See [1,6] for further discussion of this point.…”

Section: Coarse Homology Theoriesmentioning

confidence: 99%

Addendum to “Coarse homology theories”

Mitchener

2003

Algebr. Geom. Topol.

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An etale approach to the Novikov conjecture

Dranishnikov

Ferry

Weinberger

2007

Comm Pure Appl Math

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Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups. §1 Introduction Ten years ago, the most popular approach to the Novikov conjecture went via compactifications. If a compact aspherical manifold, say, has a universal cover which suitably equivariantly compactifies, already Farrell and Hsiang [FH] In recent years other coarse methods have supplanted the compactification method (most notably the embedding method of [STY] in the C* algebra setting). The reason for this is that it seemed to have a better chance of applying generally, while compactifications effective for proving the Novikov conjecture seemed to require some special geometry for their construction. For a brief time, it seemed that this could conceivably not be the case. Higson introduced a general compactification of metric spaces (somewhat reminiscent of the Stone-Čech compactification) that automatically has half of the properties necessary for application to the Novikov conjecture. The missing property was acyclicity, which holds for the StoneČech Unfortunately, it was soon realized by and others (see [Kee, DF]) that the Higson compactification, even for manifolds as small as R, has nontrivial rational cohomology. Thus, it was felt that general compactifications were not suitable for the problem -one has to use geometric compactifications. However, Gromov has recently harpooned the embedding approach by constructing finitely generated groups which do not uniformly embed in Hilbert space [G2]. Moreover, a number of authors (e.g. [HLS] [O]) have shown that Gromovs groups can be used to construct counterexamples to general forms of the Baum-Connes conjecture.Moreover, for reasons that are not entirely clear, the embedding method has never been translated into pure topology; the results on the integral Novikov conjecture so obtained, from the L-theoretic viewpoint never account for the prime 2, and there do not seem to be many results in pure algebraic K-theory (or A-theory) provable by this method. This paper seeks to rehabilitate the Higson compactification approach. We will show that Theorem 1. If the Higson compactification of EΓ is mod 2 acyclic, and BΓ is finite type, then the Novikov conjecture for Γ holds at the prime 2.

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Squeezing and Higher Algebraic K-Theory

Abstract: Abstract. We prove that the Assembly map in algebraic Ktheory is split injective for groups of finite asymptotic dimension admitting a finite classifying space.

Cited by 70 publications

References 22 publications

On the K-theory of groups with finite asymptotic dimension

On the K-theory of groups with finite asymptotic dimension

Addendum to “Coarse homology theories”

An etale approach to the Novikov conjecture

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