arXiv subjects

Oscar Harr

Publications and source records attributed to Oscar Harr.

5 recordsLinked to original sources

arXiv2026

Whitehead torsion and the kernel of assembly

For a topological space that is homeomorphic to a finite simplicial complex, we prove that the Bartels--Nikolaus assembly functor has a fully faithful right adjoint. Using this, we define for each such topological space $X$ a {\em Whitehead category}, whose K-theory is canonically identified with the Whitehead spectrum of $X$; and for a homotopy equivalence between two such spaces, we define an object in the Whitehead category of $X$ called the {\em torsion cosheaf} of the map, whose K-theory class recovers the classical Whitehead torsion.

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arXiv2023

The derived category of a locally compact space is rarely smooth

We show that the derived category of a locally compact Hausdorff space $X$ is smooth in the sense of non-commutative geometry if and only if $X$ is discrete and finite.

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arXiv2023

Compact sheaves on a locally compact space

We describe the compact objects in the $\infty$-category of $\mathcal C$-valued sheaves $\text{Shv} (X,\mathcal C)$ on a hypercomplete locally compact Hausdorff space $X$, for $\mathcal C$ a compactly generated stable $\infty$-category. When $X$ is a non-compact connected manifold and $\mathcal C$ is the unbounded derived category of a ring, our result recovers a result of Neeman. Furthermore, for $X$ as above and $\mathcal C$ a nontrivial compactly generated stable $\infty$-category, we show that $\text{Shv} (X,\mathcal C)$ is compactly generated if and only if $X$ is totally disconnected.

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arXiv2022

Twisted homology stability of O_n for valuation rings

In this article, we extend an argument of Vogtmann in order to show homology stability of the Euclidean orthogonal group $O_n(A)$ when $A$ is a valuation ring subject to arithmetic conditions on either its residue or its quotient field. In particular, it is shown that if $A$ is a henselian valuation ring, then the groups $O_n(A)$ exhibit homology stability if the residue field of $A$ has finite Pythagoras number. Our results include those of Vogtmann, and hold with various twisted coefficients. Using these results, we give analogues for fields $F\neq\mathbb R$ of some computations that appear in the study of scissor congruences.

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arXiv2022

Disordered arcs and Harer stability

We give a new proof of homological stability with the best known isomorphism range for mapping class groups of surfaces with respect to genus. The proof uses the framework of Randal-Williams-Wahl and Krannich applied to disk stabilization in the category of bidecorated surfaces, using the Euler characteristic instead of the genus as a grading. The monoidal category of bidecorated surfaces does not admit a braiding, distinguishing it from previously known settings for homological stability. Nevertheless, we find that it admits a suitable Yang-Baxter element, which we show is sufficient structure for homological stability arguments.

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