Search arXiv⌕ Search

arXiv · math/0404513

A Controlled Approach to the Isomorphism Conjecture

Abstract

We use a hocolim approach to the Isomorphism Conjecture in K-Theory to analyze the case of groups of the form $G\rtimes Z$ and $G_1*_{G}G_2$. As an important corollary we prove that the isomorphism conjecture in K-Theory holds for a finitely generated free group.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel Juan-Pineda, Stratos Prassidis. 2004-04-28. A Controlled Approach to the Isomorphism Conjecture. https://arxiv.org/abs/math/0404513

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

A secondary index map on relative K-homology

We introduce a secondary index map from relative K-homology to the K-theory of a relative structure algebra. We prove an algebraic Mayer-Vietoris sequence for the relative structure algebra, and prove compatibility of the secondary index map with the Mayer-Vietoris boundary map. As an application, we compute the secondary index of the Dirac operator on an even-dimensional spin manifold with collared metric having uniformly positive scalar curvature on the boundary. This computation leads to a refinement of the delocalized APS-index theorem.

math.KT↗

Higher Noncommutative Motives and Higher K-theory

We construct higher additive noncommutative motives of stable $(\infty,n)$-categories, extending the additive motives of Blumberg, Gepner, and Tabuada and the secondary theory of Mazel-Gee and Stern. We formulate a notion of exact sequences and establish the universal property of the resulting additive motives. We give two equivalent constructions: a localization of enriched presheaves and a weighted Waldhausen $S$-construction on complete exact flags. We then compute their Hom objects, obtaining ordinary K-theory at the first level and lower additive motives recursively at higher levels. Our results recover the classical and secondary theories, provide coherent multiplicative structures, and yield an endofunctor on the category of Stefanich rings. In short, higher motives are higher K-theory.

math.KT↗

Perfect Forms and Torsion Bounds in Algebraic $K$-Theory over Totally Real and CM Fields

We extend the method of Van Woerden \cite{woerden} to totally real and CM number fields to obtain an upper bound for the number of $\GLo-$equivalence classes of perfect forms. Moreover, by considering appropriate barycentric subdivisions of the Voronoi tessellation, we extend these estimates to the number of $\GLo$-orbits of $k$-dimensional faces of the Voronoi tessellation. As a consequence, we improve the asymptotic order of the bounds for the torsion part of $K_m(\rint)$ obtained by Soulé \cite{soule_hermitian_bounds} and Houriet \cite{Juliet_PHD}. We emphasise the improvement of the general bound for $F=\mathbb Q$, unchanged since Soulé's work, and the independence of the discriminant exponent from the field degree for totally real and CM fields.

math.KT↗