Waldhausen’s Nil groups and continuously controlled K-theory

Munkholm, Hans; Prassidis, Stratos

Fundamenta Mathematicae, Tome 159 (1999), p. 217-224 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $Γ = Γ_{1} *_{G} Γ_{2}$ be the pushout of two groups $Γ_{i}$ , i = 1,2, over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces $B Γ_{1} \leftarrow B G \leftarrow B Γ_{2}$ . Denote by ξ the diagram $I \frac{p}{\leftarrow} H \frac{1}{\to} X = H$ , where p is the natural map onto the unit interval. We show that the $N i l^{\sim}$ groups which occur in Waldhausen’s description of $K_{*} (ℤ Γ)$ coincide with the continuously controlled groups $_{*}^{c c} (ξ)$ , defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups $_{*}^{c c} (ξ^{+})$ which are known to form a homology theory in the variable ξ, with the “homology part” in Waldhausen’s description of $K_{* - 1} (ℤ Γ)$ . A similar result is also obtained for HNN extensions.

Publié le : 1999-01-01

Zbl 0938.19002

EUDML-ID : urn:eudml:doc:212401

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     author = {Hans Munkholm and Stratos Prassidis},
     title = {Waldhausen's Nil groups and continuously controlled K-theory},
     journal = {Fundamenta Mathematicae},
     volume = {159},
     year = {1999},
     pages = {217-224},
     zbl = {0938.19002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv161i1p217bwm}
}

Munkholm, Hans; Prassidis, Stratos. Waldhausen’s Nil groups and continuously controlled K-theory. Fundamenta Mathematicae, Tome 159 (1999) pp. 217-224. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv161i1p217bwm/

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