282 papers
2604.03887Cohomology of special unitary groups and congruence subgroups
We prove a homotopy invariance result for the first cohomology group of the special unitary group $\mathrm{SU}_3(F[t])$ with coefficients in irreducible representations of $\mathrm{PGL}_2(F)$. The main theorem establishes that this cohomology is naturally isomorphic to the corresponding cohomology of $\mathrm{PGL}_2(F)$.
2604.03887Apr 2026
View2604.01863Chromatic Noshift
The chromatic redshift philosophy, introduced by Ausoni and Rognes, suggests that algebraic $K$-theory raises chromatic height by $1$. We show that the analogue of this philosophy fails in the case of rigid symmetric monoidal stable $\infty$-categories. More precisely, we construct examples of rigid $T(n)$-local categories $C$ where a refinement $\mathrm{Dim}$ of the dimension morphism induces an equivalence $$K(C)\to \mathrm{End}(\mathbf{1}_C)^{BS^1}$$ and for which $K(C)$ therefore vanishes $T(n+1)$-locally. In fact, we prove that this equivalence always holds for $\aleph_1$-Nullstellensatzian rigid $T(n)$-local categories in the sense of Burklund, Schlank and Yuan. We study more in depth the rational version of these results to find a rigid rational additive $1$-category witnessing the failure of redshift at height $0$. Finally, we use our methods to prove and generalize a conjecture of Levy about categorification of ordinary rings.
2604.01863Apr 2026
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Anick Resolution for Lawvere Theories from Algebraic Discrete Morse Theory
Inspired by Brown's collapsing method (or discrete Morse theory) to obtain a free resolution of $\bbZ$ over the monoid ring $\bbZ M$, we apply algebraic discrete Morse theory to compute the homology groups of Lawvere theories, which is defined as Tor of a certain module. We reinterpret known partial free resolutions arising from complete term rewriting systems in terms of collapsing of the normalized bar resolution. This perspective yields homological inequalities that bound the number of equational axioms in presentations and recovers classical results, such as lower bounds for group axiomatizations. Our main contribution is to extend these resolutions to higher dimensions.
2603.28382Mar 2026
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Bordisms and unbounded $KK$-theory
This monograph studies $KK$-theory in its unbounded model. The central object is the $KK$-bordism group obtained by imposing the $KK$-bordism relation on unbounded $KK$-cycles. In the paradigm of noncommutative geometry, an unbounded $KK$-cycle is a noncommutative geometry in its own right and our approach allow for the study of mildly noncommutative geometries (orbifolds, foliations et cetera) as if they were closed manifolds. The techniques we introduce enable us to directly import manifold techniques and arguments into the important yet technical field of unbounded $KK$-theory. Recent decades has seen a tremendous progress in the study of the unbounded model for $KK$ as well as secondary invariants, the first motivated by refining computational tools in Kasparov's $KK$-theory and the second by applications to geometry and topology. The aim of this work is to provide a common framework for these two areas: equipping unbounded $KK$-cycles with a geometrically motivated relation recovering Kasparov's $KK$-theory that is computationally tractable for working with secondary invariants.
2603.26450Mar 2026
View2603.26222On the K-theory of algebraic Cuntz-Pimsner rings
We establish a long exact sequence for the homotopy K-theory groups of the algebraic Cuntz-Pimsner rings introduced by Carlsen and Ortega [CO11] by adapting Pimsner's original proof [Pim97] to Cuntz's formalism.
2603.26222Mar 2026
View2603.23191Toeplitz Operators on Contact Manifolds and Equivariant K-homology
We present an equivariant generalization of Boutet de Monvel's index theorem for Toeplitz operators on contact manifolds. We prove that the Dirac operator and the Szegö projection determine the same class in equivariant $K$-homology, generalizing a theorem of Baum-Douglas-Taylor. We do not assume that the contact manifold is the boundary of a strictly pseudoconvex domain. The proof proceeds by a deformation linking the principal symbols of the classical and Heisenberg pseudodifferential calculi. At the level of symbols, the projection defining the Dirac class deforms to the principal Heisenberg symbol of the Szegö projection. This deformation implies equality of the corresponding classes in K-homology. This, in turn, gives an equivariant generalization of Boutet de Monvel's index formula for Toeplitz operators.
2603.23191Mar 2026
View2603.23157The universal property of graded $KK^G$-theory
A universal category-theoretical characterization of groupoid equivariant $KK^G$-theory for ${\mathbb{Z}}_2$-graded $C^*$-algebras is established, by observing the ``$KK$-axiom'' that for each $[s,{\cal E} \oplus B, \mathbb{F}] \in KK^G(A,B)$, the `corner-embedding' $*$-homomorphism ${\bf j}: B \rightarrow {\sf cl} \big({\cal K}_B({\cal E} \oplus B) + s(A) + \mathbb{F} \cdot s(A) \big)$ is invertible in $KK^G$. This $KK$-axiom and homotopy-invariance characterize graded $KK^G$-theory universally and completely, thus directly extending the well-known characterization of $KK$-theory for ungraded $C^*$-algebras via stability, homotopy invariance and splitexactness by Higson.
2603.23157Mar 2026
View2603.18288On the K-theory of matroids with Tutte coverings
The aim of this work is to explicitly compute the K-theory of the category of matroids with respect to the covering family of Tutte coverings. In particular, we show that this is equivalent to the K-theory spectrum of the category of graphic matroids on looped forests, with the covering family generated by isomorphisms. Further, we show that this yields an equivalence of $C_2$-spectra.
2603.18288Mar 2026
View2603.15873Equivariant localizing motives and multiplicative norms on algebraic K-theory
We construct multiplicative norms on equivariant nonconnective algebraic $K$-theory for finite groups $G$. We also construct a genuine equivariant version of THH equipped with a Dennis trace map from K-theory compatible with the multiplicative norms. To do so, we follow the general strategy of Blumberg-Gepner-Tabuada in the nonequivariant case by generalizing their category of localizing motives to the genuine equivariant context, building upon the theory of perfect $G$-stable categories of the first-named author. Crucially, we proceed using the recent perspective on noncommutative motives by the second-named author with Sosnilo and Winges which allows us to deal with non-exact functors on this category of motives. Together with an isotropy separation argument for equivariant cubes, we prove our main theorem that norms of stable categories preserve equivariant motivic equivalences. As an immediate consequence, we obtain a unique equivariant multiplicative refinement of nonconnective algebraic $K$-theory. From these constructions and results, we draw several applications, namely: (1) that the endofunctor of (equivariant) tensor powers on ordinary perfect stable categories preserve motivic equivalences; (2) that the multiplicative norms also preserve the additive motivic equivalences, thus yielding a motivic refinement of a result of Elmanto-Haugseng and Cnossen-Haugseng-Lenz-Linskens that connective algebraic K-theory admits multiplicative norms; (3) we construct a genuine equivariant version of topological Hochschild homology equipped with a Dennis trace map that is compatible with multiplicative norms; and (4) we prove that every genuine $G$-spectrum is the K-theory of a perfect $G$-stable category.
2603.15873Mar 2026
View2603.13692A general Mayer-Vietoris sequence in algebraic $K$-theory
This paper investigates the Mayer-Vietoris sequence for the Milnor square. While such sequences often involve elusive intermediate terms, we provide an explicit characterization of the key group $X$ in a new, more general variant of the sequence. By identifying $X$ as a categorical pullback, we provide a full, constructive proof of the modified Mayer-Vietoris sequence. Furthermore, we show that $X$ fits into a structural exact sequence involving the relative $K$-groups $K_{*}(A, B, I)$. Finally, we provide a homotopy-theoretic description of $X$ as the homotopy group of a suitable fiber, clarifying its structure, kernel , and image.
2603.13692Mar 2026
View2603.10501Quantum cellular automata are a coarse homology theory
We show that quantum cellular automata naturally form the degree-zero part of a coarse homology theory. The recent result of Ji and Yang that the space of QCA forms an Omega-spectrum in the sense of algebraic topology is a direct consequence of the formal properties of coarse homology theories.
2603.10501Mar 2026
View2603.08653Theorem of the heart for Weibel's homotopy $K$-theory
In this paper we prove the theorem of the heart for Weibel's homotopy $K$-theory $KH.$ Namely, if $\mathcal{C}$ is a small stable $\infty$-category with a bounded $t$-structure, then the realization functor $D^b(\mathcal{C}^{\heartsuit})\to \mathcal{C}$ induces an equivalence of spectra $KH(\mathcal{C}^{\heartsuit})\xrightarrow{\sim}KH(\mathcal{C}).$ In a certain sense this result is dual to the Dundas-Goodwillie-McCarthy theorem. We deduce the dévissage theorem for $KH$ of abelian categories, also on the level of spectra (in all degrees). More generally, we prove these results for dualizable categories with nice $t$-structures and for the so-called coherently assembled abelian categories. The proof is heavily based on another new result, which is a much stronger version of Barwick's theorem of the heart. Its special case states the following: if $\mathcal{C}$ is a small stable category with a bounded $t$-structure, such that for some $n\geq 1$ the realization functor induces isomorphisms on $\operatorname{Ext}^{\leq n}$ between the objects of $\mathcal{C}^{\heartsuit},$ then the map $K_j(\mathcal{C}^{\heartsuit})\to K_j(\mathcal{C})$ is an isomorphism for $j\geq -n-1,$ and a monomorphism for $j = -n-2.$ Moreover, we prove that these estimates are sharp, even for dg categories over a field. In particular, the naive $K$-theoretic theorem of the heart fails for $K_{-3}.$
2603.08653Mar 2026
View2603.08344A remark on the invariance of $K$-theory under duality
In this short remark, we explain that two examples of invariance under duality for a localizing invariant $F$ hold purely formally when $F$ is $K$-theory, whereas the general statement for arbitrary localizing invariants does not reduce to a formal statement. We record a counterexample to the claim that the universal localizing invariant is invariant under the operation of taking opposite categories, originally due to Tabuada.
2603.08344Mar 2026
View2603.04834The Hochschild cohomlogy ring of a self-injective Nakayama algebra is a Batalin-Vilkoviskys algebra
Lambre, Zhou and Zimmermann showed that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra. They asked whether the semisimplicity condition is necessary. In this paper, we show that for a self-injective Nakayama algebra, the Hochschild cohomology ring is always a Batalin-Vilkovisky algebra. In course of proofs, we correct some inaccuracies in the literature, hoping not to introduce new errors.
2603.04834Mar 2026
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Uniformly elliptic boundary value problems
We study boundary conditions for elliptic operators on non-compact manifolds with boundary via uniform K-homology, a version of K-homology sensitive to the large-scale geometry of the manifold. To that end, we develop the theory of relative uniform K-homology. We show that boundary conditions for uniformly elliptic differential operators define classes in the relative and non-relative uniform K-homology of the manifold, depending on the assumed regularity of the boundary condition. Moreover, we define and study a relative index map on relative uniform K-homology that combines uniform coarse information on the interior with secondary information on the boundary. As an application, we compute that on a spin manifold with product structure and uniformly positive scalar curvature on the boundary the image of the relative uniform K-homology class of the Dirac operator under this relative index map is closely connected to a uniform version of the higher $ρ$-invariant of the boundary. In particular, a delocalized APS-index theorem of Piazza and Schick is proved in the uniform setting.
2602.22748Feb 2026
View2602.20961A K-theoretic note on the spectral localiser
We review the construction of the spectral localiser (due to Loring and Schulz-Baldes) from a K-theoretic perspective. We first give a K-theoretic argument providing a spectral flow expression for the even or odd index pairing in terms of the "infinite volume" spectral localiser. Our approach towards this first step is more direct, treats the even and odd cases on an equal footing, and has the advantage that the construction of the spectral localiser becomes immediately apparent from the computation of the index pairing via a Kasparov product. In a second step of "spectral truncation", we then describe how this spectral flow expression can be computed in terms of the signature of the "finite volume" spectral localiser. Throughout, we do not require invertibility of the operator representing the K-homology class, and the even index pairing then obtains an additional contribution coming from the Fredholm index.
2602.20961Feb 2026
View2602.19787A going-down principle for {é}tale groupoids and the Baum-Connes conjecture
We study a going-down principle for {é}tale groupoids and its applications, extending the earlier results for locally compact groups by Chabert, Echterhoff and Oyono-Oyono, and for ample groupoids by B{ö}nicke and by B{ö}nicke-Dell'Aiera. The proof in the general {é}tale groupoid setting is based on a more detailed study of groupoid simplicial complexes. We also study a bicategorical functoriality involving the induction functors from {é}tale groupoid correspondences, which was introduced by Miller. This yields a bicategorical interpretation of the induction-restriction adjunction. As an application of the going-down principle, we provide a proof of the split injectivity of Baum-Connes assembly map for {é}tale groupoids that are strongly amenable at infinity, recovering a result obtained by B{ö}nicke and Proietti via a categorical approach. The going-down principle is also applied on the proof of continuity of topological K-theory of {é}tale groupoids and the study of scope of validity of K{ü}nneth formulas.
2602.19787Feb 2026
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Algebraic $K$-theory of stably compact spaces
We compute the value of finitary localizing invariants, including algebraic $K$-theory, on categories of sheaves over stably locally compact spaces $X$. Our formula simultaneously generalizes the cases of locally compact Hausdorff and coherent (spectral) spaces and recovers several smaller $K$-theory calculations as special instances.
2602.18245Feb 2026
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Syntomic cohomology of truncated Brown--Peterson spectra
We compute the $\mathrm{MU}$-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n})$, of all $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of the truncated Brown--Peterson spectrum $\mathrm{BP}\langle n\rangle$. As qualitative consequences, we resolve the Lichtenbaum--Quillen, telescope, and redshift questions for the algebraic K-theories of all $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP} \langle n\rangle$. This extends work of the Hahn and Wilson. We also explicitly compute the algebraic K-theory of arbitrary $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 2\rangle$ at all primes $p\ge 5$ extending previous work of the author, Ausoni, Culver, Höning, and Rognes.
2602.143801Feb 2026
View2602.14112One Decomposition of $K_2$ for Certain Quotients over $\mathbb{Z}[G]$ with $G$ a Finite Abelian $p$-Group
This paper investigates the structure of $K_2$-groups for certain quotient rings of the integral group ring $\mathbb{Z}[G]$. Let $G$ be a finite abelian $p$-group with $p$-rank $r > 1$, let $Γ$ be the maximal $\mathbb{Z}$-order of $\mathbb{Q}[G]$, and let $\widetilde{G}$ denote the sum of all elements of $G$ in the group ring. By employing the framework of Kähler differentials, we first determine that the relative $K$-group $K_2(\mathbb{F}_p[G], (\widetilde{G}))$ is an elementary abelian $p$-group of rank $r$. Building upon this result, we establish an explicit isomorphism: $$ K_2(\mathbb{Z}[G]/(|G|Γ\cap p\mathbb{Z}[G])) \cong K_2(\mathbb{Z}[G]/|G|Γ) \oplus K_2(\mathbb{F}_p[G], (\widetilde{G})). $$
2602.14112Feb 2026
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