Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.15551

K_1 of matroids

Thor Wittich
View original source ↗Read source document ↗Export metadata ↓

Abstract

Using the geometric realization spectral sequence and the Serre spectral sequence, we compute $K_1(\operatorname{Mat}_*)$, the first K-theory group of the proto-exact category of pointed matroids, answering a question of Jun, Sistko and Wright.

Explore related subjects

math.KTmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Thor Wittich. 2026-10-11. K_1 of matroids. https://arxiv.org/abs/2610.15551

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

KEEP EXPLORING

Related papers

arXiv2026

Generalized Thom Spectra, Conjugate Stacks, and truncated Brown-Peterson Spectra

We propose a of definition conjugate stack for a class of suitable $\mathbb{E}_3$-$\mathrm{MU}$-algebras that includes the truncated Brown-Peterson spectra $\mathrm{BP}\langle n \rangle$ for all heights $n$. Then we calculate part of $\mathrm{BP}\langle n \rangle^{\mathrm{conj}}$ as a quotient stack in terms of Witt vectors. To perform this computation we study generalized Thom spectra and their relative Topological Hochschild Homology. We exhibit the Eilenberg-Mac Lane spectrum $\mathrm{H}\mathbb{F}_2$ as a $\mathbb{E}_3$-$\mathrm{MU}$-Thom spectrum, the $p$-local Eilenberg-Mac Lane spectra $\mathrm{H}\mathbb{Z}/p^n$ (except for $\mathrm{H}\mathbb{Z}/4$) and $\mathrm{H}\mathbb{Z}_{(p)}$ as $\mathbb{E}_3$-$\mathrm{MU}_{(p)}$-Thom spectra, and the connective complex K-theory spectrum $ku$ as $\mathbb{E}_1$-$\mathrm{MU}$-Thom spectra. We accompany these ``go" results with ``no-go" results that show these ``go" results do not admit $\mathbb{E}_{\infty}$-refinements.

math.KT↗
arXiv2026

A higher categorical approach to graded bivariant $K$-theories for Real $C^*$-algebras

We present an $\infty$-categorical refinement of classical $KK$-theory and $E$-theory for graded, $G$-equivariant for a locally compact second countable group $G$, Real $C^*$-algebras. As a consequence, we obtain a couple of fiber sequences connecting the different $K$-theory spectra, and explicitly analyse the induced endomorphisms on $K_0$-groups. We also answer precisely some folklore questions in the area. Among other things, we show that graded $KK$- and $E$-categories are not a Dwyer-Kan localization of graded $C^*$-algebras.

math.KT↗
arXiv2026

Classifying quasi-representations of groups via their characters

For every countable discrete group, we construct a character for finite-dimensional unitary quasi-representations, with values in the inverse limit of the equivariant $K$-theory of its Rips complexes. The character is a complete invariant for quasi-representations up to stable equivalence for amenable groups and, more generally, for groups satisfying the Haagerup property and QR-quasidiagonality. For Haagerup groups satisfying quantitative QR-quasidiagonality, we also obtain a quantiative version of the theorems. These conclusions apply when the maximal group $C^*$-algebra is quasidiagonal and has the lifting property, including amenable groups, free groups, and closed orientable surface groups. For surface groups, rank and winding number give complete invariants for the classification. Finally, composing with the equivariant Chern character gives a delocalized cohomological obstruction to stable equivalence of a quasi-representation with genuine representations.

math.KT↗