The Approximation Theorem and the K-Theory of Generalized Free Products

Schwanzl, Roland; Staffeldt, Ross

doi:10.2307/2155013

Cited by 2 publications

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“…Letting Δ op ≤2,inj be the subcategory of the simplicial category with objects [0], [1], and [2] and injective, orderpreserving morphisms, this amounts to a functor Y : Δ op ≤2,inj → Δ op − Sets together with a natural transformation to the restriction of N wS • C to Δ op ≤2,inj . Then the left Kan extension to Δ op produces a bisimplicial set X and map φ : X → N wS • C .…”

Section: Sample Computationsmentioning

confidence: 99%

“…In [2] the category C is a category M V of diagrams associated to the generalized free product called Mayer-Vietoris presentations. Recalling some of the definitions, an object of There are also mapping cylinders, and the hypotheses of the Fibration Theorem hold.…”

Section: Sample Computationsmentioning

confidence: 99%

“…Thus, we may also apply Theorem 4.1 to obtain information about the connecting homomorphism ∂ # : K 1 (M V ; w) → K 0 (M V w ; v). We postpone till later interpretation of the connecting homomorphism in terms of the identifications K n (M V ; w) ∼ = K n (R) and K n−1 (M V w ; v) ∼ = K n−1 (C) ⊕ Nil n−1 (C; A , B ) proved in [2].…”

Section: Sample Computationsmentioning

confidence: 99%

“…It is of great interest to provide tools to interpret the connecting homomorphism ∂ # : K n (R) → K n−1 (C) ⊕ Nil n−1 (C; A , B ). The paper [2] rederived the decomposition results of [1] using the systematic results and methods of [3] applied to a certain category with cofibrations and two notions of weak equivalence. The present paper exploits more deeply techniques of [3] in order to provide one tool for the study of the connecting homomorphism.…”

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confidence: 99%

“…Finally, we discuss briefly the motivating example of generalized free products. When we work with the category of Mayer-Vietoris presentations and the notions of fine and coarse equivalence are as given in [2], the result of Theorem 4.1 simplifies, yielding a fairly simple recipe for K 1 (R) → K 0 (C) ⊕ Nil 0 (C; A , B ) in terms of Mayer-Vietoris presentations. Because the Mayer-Vietoris presentations play a role, explicit computation of examples will depend on specifics of the general free product situation and are postponed to later work.…”

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confidence: 99%

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The connecting homomorphism for K-theory of generalized free products

Staffeldt

2010

Geom Dedicata

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We interpret the connecting homomorphism of the long exact sequence of algebraic K -groups associated to a category with cofibrations and two notions of weak equivalence. One notion of weak equivalence is finer than the other and certain technical conditions involving mapping cylinders must also be satisfied. Such situations arise when one considers certain free product diagrams in the category of rings, such as those that arise in applications of the Seifert-van Kampen theorem where all fundamental group homomorphisms are injective. The techniques follow Waldhausen's approach to algebraic K -theory of categories with cofibrations and weak equivalences.

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Section: Sample Computationsmentioning

confidence: 99%

Section: Sample Computationsmentioning

confidence: 99%

Section: Sample Computationsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The connecting homomorphism for K-theory of generalized free products

Staffeldt

2010

Geom Dedicata

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K-Theory and Geometric Topology

Rosenberg

2005

Handbook of K-Theory

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Historically, one of the earliest motivations for the development of K-theory was the need to put on a firm algebraic foundation a number of invariants or obstructions that appear in topology. The primary purpose of this chapter is to examine many of these K-theoretic invariants, not from a historical point of view, but rather a posteriori, now that K-theory is a mature subject. There are two reasons why this may be a useful exercise. First, it may help to show K-theorists brought up in the "algebraic school" how their subject is related to topology. And secondly, clarifying the relationship between Ktheory and topology may help topologists to extract from the wide body of K-theoretic literature the things they need to know to solve geometric problems. For purposes of this article, "geometric topology" will mean the study of the topology of manifolds and manifold-like spaces, of simplicial and CWcomplexes, and of automorphisms of such objects. As such, it is a vast subject, and so it will be impossible to survey everything that might relate this subject to K-theory. I instead hope to hit enough of the interesting areas to give the reader a bit of a feel for the subject, and the desire to go off and explore more of the literature. Unless stated otherwise, all topological spaces will be assumed to be Hausdorff and compactly generated. (A Hausdorff space X is compactly generated if a subset C is closed if and only if C ∩K is closed, or equivalently, compact, for all compact subsets K of X. Sometimes compactly generated spaces are called k-spaces. The k stands both for the German Kompakt and for Kelley, who pointed out the advantages of these spaces.) This eliminates certain pathologies that cause trouble for the foundations of homotopy theory. "Map" will always mean "continuous map." A map f : X → Y is called a weak equivalence if its image meets every path component of Y and if f * : π n (X, x) → π n (Y, f (x)) is an isomorphism for every x ∈ X.

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The Approximation Theorem and the K-Theory of Generalized Free Products

Cited by 2 publications

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The connecting homomorphism for K-theory of generalized free products

The connecting homomorphism for K-theory of generalized free products

K-Theory and Geometric Topology

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