Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.2324

Construction of Fredholm representations and a modification of the Higson-Roe corona

Alexander S. Mishchenko·Nicolae Teleman
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula. Earlier a natural family of the Fredholm representations was constructed that lead to a symmetric vector bundle on completion of the fundamental group with a modification of the Higson-Roe corona when the completion is a closed manifold. Here we will discuss a homology version of symmetry in the case when completion with a modification of the Higson-Roe corona is a manifold with boundary. The results were developed during the visit of the first author in Ancona on March, 2007. The second version is supplemented by details of consideration the case of manifolds with boundary.

Explore related subjects

math.KT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexander S. Mishchenko, Nicolae Teleman. 2008-02-16. Construction of Fredholm representations and a modification of the Higson-Roe corona. https://arxiv.org/abs/0709.2324

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

KEEP EXPLORING

Related papers

arXiv2026

Finite-coefficient Gersten injectivity fails in ramified mixed characteristic

Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\in K_2(A;\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the integral Gersten conjecture but it rules out a naive reduction to finite coefficients.

math.KT↗
arXiv2026

$E$-theory is compactly assembled

We show that the equivariant $E$-theory category $\mathrm{E}_{\mathrm{sep}}^{G}$ for separable $C^{*}$-algebras is a compactly assembled stable $\infty$-category. We derive this result as a consequence of the shape theory for $C^{*}$-algebras developed by Blackadar and Dardarlat and a new construction of $\mathrm{E}_{\mathrm{sep}}^{G}$. As an application we investigate a topological enrichment of the homotopy category of a compactly assembled $\infty$-category in general and argue that the results of Carrión and Schafhauser on the enrichment of the classical $E$-theory category can be derived by specialization.

math.KT↗
arXiv2026

Scissors automorphism groups I: Homological stability and K-theory

In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. In the classical case of scissors congruence of polytopes, this leads to calculations and structural results on the homology of the scissors automorphism group. In two sequels, we show that more generally our approach leads to homology calculations for various families of groups appearing in group theory and dynamics, recovering results of Szymik-Wahl, Li, and Tanner.

math.KT↗