math.KT

0 Pith It

🧮 math.AG 2026-07-03

Gm tensor with mod-2 motivic cohomology is free on (2,1) generator

by Tom Bachmann, Robert Burklund +2 more

Motivic Hochschild homology of mod 2 motivic cohomology over algebraically closed fields

The computation over complex numbers yields a motivic analog of Bökstedt periodicity via tau comparisons and power operation relations.

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We compute the tensor of the multiplicative group scheme with the mod-$2$ motivic cohomology spectrum in normed motivic spectra over the complex numbers, and find that the resulting algebra is free on a generator in bidegree (2,1). This gives a motivic analog of B\"okstedt periodicity. The proof proceeds by comparing the tau-inverted and tau-reduced forms of the tensor. After inverting tau, the calculation reduces to classical B{\"o}kstedt periodicity via Betti realization. The reduction modulo tau is governed by a comparison between normed algebra structures and derived algebra structures on cellular modules over motivic cohomology mod tau. This comparison produces divided power operations and leads to mixed Cartan and Adem relations intertwining normed and topological power operations. A key input is a detailed analysis of motivic extended powers of spheres and their tau-torsion structure. In contrast with the corresponding simplicial-circle calculation due to Dundas-Hill-Ormsby-{\O}stv{\ae}r, the large families of tau-torsion classes disappear for the Gm-tensor, leaving a considerably more rigid algebraic structure.

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🧮 math.KT 2026-07-02

GL_n(Q) map equates cone complex to K-theory Gersten complex

by Peter Xu

A note on polyhedral cones and toric polylogarithms

The equivariant isomorphism connects sphere homology from simplicial cones to trace-fixed Milnor K-theory structures over the rationals.

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We extend some methods of our previous work on special elements in Milnor K-theory of algebraic tori, exhibiting in particular a $\mathrm{GL}_n(\mathbb{Q})$-equivariant isomorphism between a chain complex of simplicial cones, computing the homology of $S^{n-1}$, and the trace-fixed part of the weight-n Gersten complex for the Milnor K- theory of $\mathbb{G}_m^n$ over $\mathbb{Q}$. Via a relationship between graded pieces of algebras of cones and Steinberg modules, this refines a result of Charlton-Radchenko-Rudenko.

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🧮 math.AG 2026-07-02

Sheared Witt vectors provide decompletion of p-typical Witt vectors

by Bhargav Bhatt, Akhil Mathew +1 more

Sheared Witt Vectors

Exposition of the Drinfeld-Lau construction applies on rings whose reductions are perfect F_p-algebras.

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V. Drinfeld and E. Lau introduced a ``decompletion'' of the ring of $p$-typical Witt vectors, following earlier work of T. Zink. The goal of this paper is to offer an exposition of this construction, which we call the sheared Witt vectors, on the category of rings $R$ whose reduction is a perfect $\mathbb{F}_p$-algebra.

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🧮 math.GR 2026-07-02

Pro-2 Demushkin groups have A3-formal cochain algebras

by Ambrus Pál, Gereon Quick

A₃-formality for pro-2 Demushkin groups

Explicit computation of the obstruction class via their classification confirms the weak formality over F2.

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We study a weak form of formality for differential graded algebras, called $A_3$-formality, and show that the differential graded $\mathbb{F}_2$-algebras of continuous cochains of all pro-$2$ Demushkin groups are $A_3$-formal. We prove this result by an explicit computation of the Benson--Krause--Schwede canonical class using the classification of pro-$2$ Demushkin groups by Demushkin, Serre, and Labute. Compared to the case of odd primes, the new idea is to interpret the data of the canonical class as defining systems of higher Massey products.

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🧮 math.KT 2026-07-02

Explicit unitaries realize real K-theory generators on involuted spheres

by Jeffrey L. Boersema

The real K-theory of the sphere with an arbitrary involution

Matrices for dimensions up to 3 and a recipe for all higher dimensions turn abstract classes into concrete operators.

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We complete the investigation begun in a previous paper to find unitary representations of the non-trivial real $K$-theory elements for the sphere $S^d$ with an involution. Here we consider all involutions except the antipodal involutions. We write down explicit unitaries representing the generators in all cases for $d \leq 3$, and for $d > 0$ we describe a recipe for generating such unitaries.

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🧮 math.OA 2026-07-01

Real Robbin-Salamon theorem equates spectral flow to index

by Chris Bourne, Alan L. Carey +2 more

Analytic index theory and spectral flow in real Hilbert C^*-modules

Spectral flow of real Fredholm operators on Hilbert C*-modules equals a Fredholm index via Van Daele K-theory.

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We consider the analytic index and spectral flow of Fredholm operators on Hilbert $C^*$-modules. Our spaces and algebras are equipped with a real structure, so the analytic index and spectral flow takes value in the real $K$-theory group of a $\sigma$-unital $C^*$-algebra. We use Van Daele $K$-theory, which allows us to treat the eight real $K$-theory groups and the two complex groups on an equal footing. We provide a general definition of the analytic index for Clifford anti-linear and skew-adjoint Fredholm operators as well as self-adjoint and odd Fredholm operators. Our definition of spectral flow and its basic properties are valid for Wahl-continuous paths of Fredholm operators on a real Hilbert $C^*$-module. We also provide an analytic approach to the spectral flow as a decomposition into a finite sum of relative indices. Furthermore, we prove a real version of the Robbin-Salamon theorem, relating the spectral flow to a Fredholm index. Our description of the index and spectral flow relies on various isomorphisms between Kasparov's $KKR$-theory and Van Daele $K$-theory, which we systematically describe in the Appendix.

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🧮 math.AT 2026-06-30

Algebraic C_+ computes continuous homology of TP(MU)

by Sverre Lun{o}e-Nielsen, John Rognes

Continuous homology of topological periodic homology of complex cobordism

The construction supplies the E2-term of a multiplicative Adams-type spectral sequence to p-completed homotopy groups.

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We determine the continuous mod $p$ homology of the topological periodic homology $TP(MU)$ of the complex cobordism spectrum, as a graded algebra with Steenrod operations. The answer is given in terms of an explicit and purely algebraic construction $C_+$, analogous to Singer's construction $R_+$. Its $Ext$-algebra provides the $E_2$-term for a multiplicative Adams-type spectral sequence converging strongly to the homotopy of $p$-completed $TP(MU)$.

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🧮 math.DG 2026-06-30

Gromov's conjecture proved for scalar curvature in 3D

by Jinmin Wang, Zhizhang Xie +1 more

Gromov's dihedral rigidity conjecture in dimension three

Self-contained proof shows curvature and angle conditions force three-dimensional manifolds to be rigid.

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In this article, we present a self-contained proof of Gromov's dihedral rigidity conjecture on scalar curvature in the three-dimensional case. The proof avoids many of the technical complications that arise in higher dimensions, while still illustrating the essential ideas of the general approach developed in arXiv:2112.01510 (version 6) and arXiv:2203.09511. It is significantly shorter than the proof of the general case and is intended to be more accessible.

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🧮 math.DG 2026-06-29

Gromov conjecture confirmed under weakened group decay

by Qiaochu Ma, Guoliang Yu

Gromov's Conjecture on Positive Scalar Curvature and Simplicial Volume under a Fundamental Group Decay Property

Positive scalar curvature forces zero simplicial volume for manifolds whose fundamental groups meet a relaxed rapid decay condition.

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Gromov's simplicial volume is a fundamental invariant measuring the topological complexity of a manifold. A conjecture of Gromov predicts that every closed manifold admitting a metric of positive scalar curvature has vanishing simplicial volume. In this paper, we prove this conjecture under a natural weakening of the classical rapid decay (RD) property for the fundamental group.

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🧮 math.AT 2026-06-29

Generalized Euler characteristics interpreted via Morava E-theories

by Gijs Heuts, Irakli Patchkoria

Chromatic Euler characteristics and duality for infinite groups

For infinite groups with finite proper universal spaces, a new duality on equivariant spectra implies vanishing results for generalized Farr

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We study a family of generalizations of the notion of Euler characteristic of discrete groups (or of orbifolds, depending on one's perspective) indexed on the natural numbers. For $n=0$, this is the classical orbifold Euler characteristic as studied by Wall and Serre, whereas for $n \geq 1$ and finite groups, this is the chromatic cardinality as studied by Ben-Moshe--Carmeli--Schlank--Yanovski. For general $n$, we show that our generalized Euler characteristic admits a natural interpretation in terms of the Morava $E$-theories. Our work involves showing that the generalized cohomology of infinite groups $G$ with finite universal space for proper actions $\underline{E}G$ has a good theory of duality, as expressed by a new duality functor on the category of proper $G$-equivariant spectra. In particular, for such groups we prove the vanishing of Klein's generalized Farrell--Tate cohomology with $T(n)$-local coefficients. We compute our generalized orbifold Euler characteristics in a large number of examples. This includes many mapping class groups, where the classical calculation is a result of Harer--Zagier, and many arithmetic groups, whose classical orbifold Euler characteristics were computed by Harder.

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🧮 math.KT 2026-06-25

Matrix stability guarantees Morita invariance in K-theory

by Eugenia Ellis, Emanuel Rodríguez Cirone

Matrix stability and Morita invariance

Bivariant algebraic K-theory respects Morita equivalence for G-algebras and G-graded algebras, including crossed-product equivalences under

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Let $G$ be a group. We prove that matrix stability for either $G$-algebras or $G$-graded algebras guarantees Morita invariance. As a consequence, bivariant algebraic K-theory (either $G$-equivariant or $G$-graded) is Morita invariant. In particular, we show that if $G$ is a finite group acting freely on a finite simplicial set $X$, then $\ell^X\rtimes G$ and $\ell^{X/G}$ are kk-equivalent. Here, $\ell^Y$ denotes the $\ell$-algebra of piecewise polynomial functions on $Y$ with coefficients in the ground ring $\ell$.

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🧮 math.NT 2026-06-24

Fermat curves get explicit non-zero K_4 classes via trilogarithms

by François Brunault, David T.-B. G. Lilienfeldt +1 more

Elements in K₄ and regulator maps of Fermat curves

Uniform construction for every N yields regulators asymptotic to 3/2 zeta(3) N^2 and checks L-value predictions at s=3 for N=3,4,6.

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We construct explicit elements in the group $K_4^{(3)}$ of the Fermat curves $x^N+y^N=1$ for all $N\geq 3$. The construction, which is uniform in $N$, uses polylogarithmic complexes and a map of de Jeu to $K$-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to $\frac{3}{2}\zeta(3)N^2$ as $N\to +\infty$. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for $K_2$ groups of Fermat curves. Finally, we numerically verify some cases of Beilinson's conjectures on special values of $L$-functions at $s=3$ for $N\in \{ 3,4,6 \}$.

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🧮 math.OA 2026-06-24

Baum-Connes assembly map extended to inverse semigroup Fell bundles

by Diego Martínez

A Baum-Connes assembly map for essential semigroup crossed products

Functoriality of cross-sectional algebras allows the map for Cartan pairs and non-Hausdorff groupoids.

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We construct an equivariant E-theory and a Baum-Connes assembly map at the level of Fell bundles of inverse semigroups over separable C*-algebras. This generalizes previous work of several authors, and allows to discuss E-theoretic matters in the context of Cartan pairs; maximal and essential C*-algebras of non-Hausdorff groupoids; and Fell bundles over discrete groups and \'etale groupoids, among others. In order to do this we establish several functoriality properties for maximal, reduced and essential cross-sectional C*-algebras associated with a (saturated) Fell bundle of an inverse semigroup. This allows to discuss when these algebras give rise to short exact sequences, generalizing the classical case of discrete groups. We also introduce the adequate notion of ``proper'' Fell bundle, or ``proper'' action of an inverse semigroup, and prove a weak containment property for these. Using these functoriality properties and these proper actions we then introduce (maximal, reduced and/or essential) equivariant E-theory by means of adequately equivariant asymptotic morphisms, and construct a Baum-Connes assembly map that is both natural and reasonably well-behaved.

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🧮 math.KT 2026-06-23

Goncharov coalgebra gives weight-3 K-theory via polylogarithms

by Alexander Kupers, Daniil Rudenko +1 more

The Goncharov Lie coalgebra of a field

Symbolic descriptions for rational K4(3) and indecomposable K5(3) of fields extend known low-weight cases.

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This paper relates algebraic $K$-theory of fields to polylogarithms via general linear groups. We introduce the Goncharov Lie coalgebra, defined in terms of the $E_\infty$-homology of general linear groups. Using Steinberg modules, we find a presentation, compute its Lie cobracket, and construct motivic and Hodge realisations. Combining these results with the Rognes rank spectral sequence, we give symbolic descriptions of the rationalisation of the algebraic $K$-theory of fields beyond the cases studied by Matsumoto-Milnor and Bloch-Suslin: we express $K^{(3)}_4(F)$ and the indecomposable part of $K^{(3)}_5(F)$ in terms of Goncharov's polylogarithmic complex of weight 3.

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🧮 math.AT 2026-06-23

Short proof of Real Snaith equivalences via Wilson orientations

by Ryan Quinn, Qi Zhu

Structured Real Snaith Equivalences

Wilson space theory yields E6-complex orientations for even periodic spectra and recovers E2ρ-structure on Real BP

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We give a short proof of the Real Snaith equivalences and multiplicative refinements thereof. The key ingredient is control over structured Real orientations, which we manage through Wilson space theory. In particular, we develop a theory that produces $\mathbb{E}_6$-complex orientations of even periodic $\mathbb{E}_{\infty}$-ring spectra. This machinery can be used to recover an $\mathbb{E}_{2\rho}$-algebra structure on Real Brown-Peterson theory. We apply the Real Snaith theorems to compute $\mathrm{THR}(\mathrm{KU}_{\mathbb{R}})$ and $\mathrm{THR}(\mathrm{MUP}_{\mathbb{R}})$. This requires a norm inverted variant of the Real Snaith theorems, which we prove via the nilpotence theorem.

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🧮 math.KT 2026-06-22

Discretisation equates independent groupoids to discrete ones

by Xin Li, Alistair Miller

Discretisation and independent resolutions of ample groupoids

Independent resolutions then reduce homology and K-theory computations for ample groupoids to the discrete case.

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We develop a general framework for understanding and computing both the groupoid homology of an ample groupoid and the topological K-theory of its reduced C*-algebra, based on two main ideas: discretisation and independent resolutions. Discretisation shows that a special class of ample groupoids we term independent groupoids are homologically and K-theoretically equivalent to discrete groupoids. We introduce the notion of a resolution by independent groupoids and provide a recipe for building a controlled independent resolution of a given ample groupoid of interest, leading to a systematic way of studying its homology and K-theory. In order to illustrate our general ideas and methods, we work out several concrete examples and applications. Garside categories provide a wide range of examples, including higher rank graphs, self-similar groups and spherical Artin-Tits groups. We also present an application to the homology of Stein's groups.

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🧮 math.QA 2026-06-22

Calculus structure and flat connection for open-closed homotopy algebras

by Zekai Yu

Non-commutative calculus and Getzler-Gauss-Manin connections for Open-closed Homotopy Algebras

Hochschild invariants admit the structure; the Getzler-Gauss-Manin connection is flat up to chain homotopy on periodic cyclic chains.

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We establish the calculus structure on Hochschild invariants of open-closed homotopy algebras. We further define the Getzler-Gauss-Manin connection and show that it is flat up to chain homotopy on the open-closed periodic cyclic chain complex.

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🧮 math.KT 2026-06-22

K-theory of k[SL_2(F_q)] computed via cyclic assembly

by Isaac Moselle

The algebraic K-theory of k[operatorname{SL}₂(mathbb{F}_q)]

The higher groups reduce to the Sylow p-subgroup ring k[C_p^r] and are obtained from topological cyclic homology for any perfect field of ch

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We compute via trace methods the higher algebraic $K$-theory of the group ring $k[\operatorname{SL}_2(\mathbb{F}_q)]$, as well as the related groups $\operatorname{PSL}_2(\mathbb{F}_q)$, $\operatorname{PGL}_2(\mathbb{F}_q)$, and $\operatorname{GL}_2(\mathbb{F}_q)$, where $k$ is a perfect field of characteristic $p$ and $q=p^r$. At the core of the computation is the algebraic $K$-theory of the group ring of the Sylow $p$-subgroup, $k[C_p^r]$, which we determine via a theorem of L\"uck--Reich--Rognes--Varisco on cyclic assembly for topological cyclic homology. In the process, we reprove the cyclic assembly result in the language of Nikolaus--Scholze, analyse assembly for smaller families of subgroups, and develop further tools for computing topological cyclic homology of group rings.

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🧮 math.AG 2026-06-22

Cellular decomposition computes homology for stable curve moduli

by Jan Hennig

A cellular (co)homology computation for overline{M_(0,n)}

The stratification into trivial pieces recovers Chow groups and real cohomology as special cases over any base field.

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In this article we set up and showcase cellular computations for (co)homology with values in strictly $\mathbb{A}^1$-invariant sheaves. These computations encapsulate many classical invariants like Chow groups and singular cohomology of the real points. They also extend enumerative arguments from algebraically closed fields to more general fields. The spaces considered here have to admit a cellular structure. Instead of using the classical notion of cellularity, i.e. having a stratification by affine spaces, more general stratifications by cohomologically trivial spaces are used, following Morel--Sawant. Examples of cellular spaces include projective spaces and their products, but also spaces such as $\overline{M_{0,n}}$, the moduli space of stable genus $0$ curves with $n$ marked points. For these examples, we showcase the computations and show how to derive the classical results. Hopefully, the following text provides enough evidence to be convincing that such computations are doable and is encouraging to start computing the cohomology for more cellular spaces. This is part of the author's PhD thesis.

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🧮 math.RT 2026-06-18

Hopfological algebra gains a higher-categorical generalization

by Juan Omar Gómez, Gustavo Jasso +1 more

Hopfological algebra, revisited

Module categories in monoidal infinity-categories refine the foundations and extend the theory to new monoidal settings.

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We propose an $\infty$-categorical approach to Khovanov--Qi's Hopfological algebra that, in particular, refines several foundational aspects of the theory by recasting the previous constructions in terms of $\infty$-categories of modules in monoidal $\infty$-categories. This perspective leads to a more general variant of Hopfological algebra that takes place over an arbitrary rigidly-compactly generated symmetric monoidal stable $\infty$-category, which we also outline in the article. In the appendix, we compare the construction of Hopfological derived categories to that of Holm--J{\o}rgensen's $Q$-shaped derived categories.

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🧮 math.AT 2026-06-11

Only HP^n and CP^2 admit almost quaternionic structures among CROSSes

by Oliver Goertsches, Panagiotis Konstantis +2 more

Almost quaternionic structures on compact rank one symmetric spaces

The classification applies without assuming the structures are homogeneous.

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We prove that the only CROSSes that admit a (not necessarily homogeneous) almost quaternionic structure are ${\mathbb{H}} {\mathbb{P}}^n$ and ${\mathbb{C}} {\mathbb{P}}^2$.

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🧮 math.DG 2026-06-11

Index difference non-trivial for genus 3+ surfaces

by Samuel Lockman

A non-trivial index difference on surfaces of genus at least 3

The map from loops of Dirac-invertible metrics to KO^{-4} is shown non-zero on high-genus surfaces with bounding spin structures.

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For any closed surface of genus at least $3$, equipped with any bounding spin structure, we show that the index difference, viewed as a map from the fundamental group of the space of Dirac-invertible Riemannian metrics to $\KO^{-4}(*)$, is non-trivial. For products of two such surfaces, equipped with any spin structure, we prove that the corresponding space of Dirac-invertible Riemannian metrics is not contractible. We discuss the relationship of this result to the existence of metrics with harmonic spinors in dimension $4$.

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🧮 math.OA 2026-06-11

Schubert calculus blocks uniform property Γ in a nuclear C*-algebra

by Andrew S. Toms

Schubert Calculus and uniform property Gamma

A quadratic obstruction from degeneracy loci persists through inductive limits and prevents trace comparison of projections.

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We construct a simple, separable, unital, nuclear C$^*$-algebra without uniform property $\Gamma$. The construction is based on a new topological obstruction arising from the Thom-Porteous theory of degeneracy loci. Constructions of pathological nuclear C$^*$-algebras over the past 30 years have used Chern class calculations introduced by Villadsen to obstruct the existence of large trivial subbundles. Here, by contrast, we use determinantal Schur classes to force every bundle map between certain equal-rank vector bundles to vanish somewhere on the base space. A quadratic Schubert calculus computation shows that this obstruction can persist across an inductive system and ultimately obstructs the comparison of projections by traces in the uniform tracial completion. The relevant Thom-Porteous classes live in degree proportional to the square of the forced rank loss, which in turn forces dimension growth of the same order in the constituent homogeneous C$^*$-algebras of our example. This identifies a new geometric threshold in the structure theory of nuclear C$^*$-algebras, linking the presence or absence of uniform property $\Gamma$ to quadratic dimension growth.

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🧮 math.AT 2026-06-11

MML splits as MSL plus suspended MGL after retraction choice

by Ahina Nandy, Egor Zolotarev

On the metalinear algebraic cobordism spectrum

The decomposition gives explicit low-degree Milnor-Witt stems via K-theory spectra and identifies the geometric diagonal with Stong's spin c

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In this paper, we study the metalinear algebraic cobordism spectrum $\mathrm{MML}$ (also sometimes denoted $\mathrm{MSL}^c$), which is built from the structure groups of oriented vector bundles. We establish an interpolation between $\mathrm{MSL}$ and $\mathrm{MML}$ and deduce that the canonical morphism $\mathrm{MSL}\to \mathrm{MML}$ admits a retraction. We parametrize all such retractions in the category of $\mathrm{MSL}$-modules and, after fixing one of them, obtain an equivalence $\mathrm{MML}\cong\mathrm{MSL}\oplus \Sigma^{2,1}\mathrm{MGL}$. As an application of these results, we determine various invariants of the metalinear algebraic cobordism spectrum over a field (after inverting the exponential characteristic). More precisely, we determine the first few Milnor-Witt stems of $\mathrm{MML}$ in terms of the very effective algebraic and hermitian K-theory spectra, and the geometric diagonal of $\mathrm{MML}$ in terms of Stong's complex-spin cobordism ring. We also compute the slices and use them to describe the category of 2-inverted modules over the $\mathbb{E}_\infty$-ring spectrum $\mathrm{MML}$.

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🧮 math.RT 2026-06-11

Hochschild homology matches singular version for symmetric algebras

by Yu Wang, Xiaozhuan Liang

Singular Hochschild complex and Cartan matrix

For basic algebras the Cartan matrix is symmetric precisely when the dual mixed complex shifts by -1; counterexamples exist for general Frob

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If A is a symmetric algebra, then Hochschild homology of the dg enhancement of the singularity category of A agrees with singular Hochschild homology of A. For a basic finite dimensional k algebra A, the Cartan matrix of A is symmetric if and only if the k dual of the mixed complex of the dg enhancement of its singularity category is isomorphic to its shift by -1. We provide two counterexamples to show that neither result holds for general Frobenius algebras.

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🧮 math.AT 2026-06-10

Tensor product group completion gives rational K-theory

by Amartya Shekhar Dubey, Mattie Ji

Tensor Product K-theory is Rational Algebraic K-theory

For commutative rings, completing finitely generated free modules under tensor product produces the rationalized algebraic K-theory space up

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For a commutative ring $R$ with unity, its algebraic $K$-theory space $K(R)$ may be obtained by group-completing the symmetric monoidal category of finitely generated free $R$-modules under direct sum. A natural question is what happens when one group-completes with respect to the tensor product structure instead. In this note, we give a direct proof of the folklore theorem that the resulting group-completion is the rationalization of $K(R)$, up to $\pi_0$. We also discuss how a similar group-completion would give the $p$-perfection and, more generally, the localization of $K(R)$ at any non-trivial multiplicatively closed subset $S \subseteq \mathbb{Z}_{> 0}$. The localization statement can be recovered from a localization theorem of May. We give a plus-construction proof without using the full machinery of multiplicative infinite loop space theory.

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🧮 math.AT 2026-06-10

Sheaves are the unique six-functor formalism on LCH spaces

by Ulrich Bunke, Marco Volpe

A characterization of sheaves among six functor formalisms on LCH

A list of natural properties isolates sheaves, so every continuous formalism reduces to Shv with coefficients from its value at a point.

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Let $\mathcal{C}$ be any stable presentably symmetric monoidal $\infty$-category. In this paper, we characterize $\mathrm{Shv}(-,\mathcal{C})$ on locally compact Hausdorff spaces as the unique six functor formalism satisfying a list of very natural properties. As a consequence, we deduce that every continuous six functor formalism $D$ in the sense of Zhu is equivalent to $\mathrm{Shv}(-, D(\mathrm{pt}))$.

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🧮 math.CT 2026-06-09

Span functor from double ∞-categories admits squares right adjoint

by George Raptis, Wolfgang Steimle

The span-squares adjunction

The adjunction supplies new proofs that Q-, S-, cobordism and squares constructions give equivalent algebraic K-theory.

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We show a universal property of the span $\infty$-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double $\infty$-categories to $\infty$-categories, and show that this functor admits a right adjoint defined by the double $\infty$-categories of squares. Using this adjunction, we obtain new proofs of the equivalences between different models of algebraic $K$-theory, given by the $Q$-, the $S$-, the cobordism model, and the squares construction.

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🧮 math.GR 2026-06-08

Crossed modules define abelianization maps for Galois cohomology

by Mikhail Borovoi

Non-abelian hypercohomology of a group with coefficients in a crossed module, and Galois cohomology

A hypercohomology theory turns non-abelian Galois cohomology of reductive groups into abelian groups via new maps.

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We develop a hypercohomology theory of a group with coefficients in a crossed module, and apply it to define abelianization maps for Galois cohomology of reductive algebraic groups.

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🧮 math.AG 2026-06-08

K-theory of toric and flag varieties realized via virtual polytopes

by Leonid Monin, Evgeny Smirnov

Polyhedral models for K-theory of toric and flag varieties

Quotients of group algebras attached to linear families of polytopes recover the rings, their relations, and structure-sheaf classes.

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In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties. In this paper we extend Pukhlikov-Khovanskii type presentation to the case of K-theory of toric and flag varieties. First, we study the Frobenius algebras obtained as quotients of the group algebra of free abelian group (possibly of infinite rank). Then we apply this construction to define a K-ring associated to a linear family of (virtual) polytopes. We study in detail two examples of such families: the family of integer (virtual) polytopes with a fixed normal fan and the family of (virtual) Gelfand-Zetlin polytopes. We show that the K-theory of toric and flag varieties can be realized as K-rings of the above families and use this to get natural set of relations in the above K-rings. Further, we describe the classes of structure sheaves of toric orbit closures and Schubert varieties in type A flag varieties. Finally, we show that our results also hold true in T-equivariant setting.

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🧮 math.AT 2026-06-04

MSL slices match ANSS E2 page for MSU after e inversion

by Ahina Nandy, Oliver Röndigs +1 more

Slices of the special linear algebraic cobordism spectrum

The identification uses Novikov's explicit determination and yields the first three Milnor-Witt stems in terms of hermitian K-theory.

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Let $F$ be a field of exponential characteristic $e$. We compute the slices of $\mathbf{MSL}[e^{-1}]$, where $\mathbf{MSL}$ is the special linear algebraic cobordism spectrum defined by Panin and Walter. The answer is expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum, which was explicitly determined by Novikov. Its applicability is demonstrated by computations with the slice spectral sequence for $\mathbf{MSL}$, which determine the first few Milnor-Witt stems of its homotopy groups (up to the third) in terms of very effective hermitian $K$-theory. We also establish a decomposition of the rational special linear algebraic cobordism spectrum over an arbitrary qcqs scheme.

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🧮 math.OA 2026-06-03

Group homomorphisms induce maps on real K-theory spectral sequences

by Jeffrey L. Boersema, Sarah L. Browne +2 more

Functoriality of real crossed product K-theory spectral sequences with respect to group homomorphisms

For torsion-free groups satisfying Baum-Connes with amenable kernel, the induced K-map is approximated by the homology map.

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Spectral sequences are a key tool for computing the K-theory of a crossed product C$^*$-algebra. However, the impact of a group homomorphism $\Omega\colon G \to H$ on such a spectral sequence was unknown until quite recently, even when $G = \mathbb Z^\ell$, $H = \mathbb Z^{k}.$ Recent work [Mil25] of the fourth-named author in the complex case establishes that ABC spectral sequences are functorial with respect to group homomorphisms. In this paper, we obtain the analogous result for real K-theory and for united K-theory. Specifically, we first show that the ABC spectral sequence approximates KO$_*(G \ltimes_r A)$ with the group homology H$_p(G;KO_q(A))$ when $G$ is a torsion-free discrete group satisfying the Baum--Connes conjecture with coefficients in $A$. Then, for a homomorphism $\Omega \colon G \to H$ of such groups with amenable kernel, and a real $H$-C$^*$-algebra $A$, we show moreover that the map in K-theory induced by the $*$-homomorphism $G \ltimes_r A \to H \ltimes_r A$ is approximated by the natural map in group homology.

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🧮 math.AT 2026-05-28

Bordism model matches analytic K-homology

by Pierre Albin, Markus Banagl +1 more

Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps. Part 2

Equivalence proves analytic Gysin maps agree with topological versions on stratified spaces.

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In this Part 2 of our article we give a detailed discussion of the compatibility between the analytic Gysin maps we have defined in Part 1 and the topological Gysin maps defined by the second author. A significant role is played by a bordism-like description of K-homology due to Jakob which is closely related to the geometric K-homology theory of Baum and Douglas. We give a self-contained proof of the equivalence of the former with the analytic K-homology theory of Kasparov. As an intermediate step towards proving our main result we use Thom's transversality theorem to describe Gysin maps compatibly with Jakob's definition of K-homology.

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🧮 math.KT 2026-05-28

Bounded multi-complexes present higher K-groups

by Bernhard Köck

On presentations of K-groups by generators and relations

The combinatorial model works with size bounds on generators and supplies the matching relations for the same groups.

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In Grayson's combinatorial description of higher K-groups, the generators are bounded acyclic binary multi-complexes of arbitrary size. Generalising work by Kasprowski, Winges and the author, we show in this paper that multi-complexes of bounded size suffice and we provide the corresponding relations. Furthermore, we report on the progress in our attempt to algebraically prove the surjectivity of Quillen's d\'evissage isomorphism for K_1 and we give an elementary and fairly simple example in the codomain which appears to require a more sophisticated approach.

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🧮 math.AG 2026-05-27

Refined obstructions find more Hodge conjecture counterexamples

by Eoin Mackall

Refined index obstructions for Brauer classes on an abelian variety

New index obstructions for Brauer classes on abelian varieties prove stricter than de Jong-Perry versions and identify additional integral H

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We produce refined index obstructions, generalizing recently constructed index obstructions due to de Jong and Perry, for topologically trivial Brauer classes on smooth and projective complex varieties. We show that our refined obstructions are more stringent than previous obstructions and, as a consequence, we produce more counterexamples to the integral Hodge conjecture. Throughout this work, we focus on algorithmic aspects of these obstructions and we illustrate many of these aspects through the concrete examples of complex abelian varieties.

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🧮 math-ph 2026-05-26

su(n) forms have zero tree scatterings beyond the vertex

by Eugenia Boffo, Ján Pulmann +1 more

Field theory of mathfrak{su}(n): the absence of non-zero scatterings

Homological perturbation theory proves all other trivalent diagrams cancel for any n, while larger field spaces allow non-trivial higher pro

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We inspect $\mathfrak{su}(n)$ forms, providing greater detail for $n=2,3$, as a toy model for a field theory in finite dimensions and with gauge symmetries. Relying on homological perturbation theory, we show that there are no scattering amplitudes with trivalent tree-level diagrams, except for the interaction vertex, thus extending a known argument of Cattaneo--Mn\"{e}v to arbitrary $n$. In contrast to this, we show how to obtain non-trivial higher products when transferring to a larger space of fields.

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🧮 math.RT 2026-05-26

Deformation maps classify all complements in Lie-Yamaguti algebras

by Apurba Das

Factorizations, classifying complements problem and deformation maps for Lie-Yamaguti algebras

One strong complement determines every other via maps that also cover homomorphisms, derivations and Rota-Baxter operators.

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A Lie-Yamaguti algebra is a non-associative algebraic structure that generalizes both Lie algebras and Lie triple systems. We first consider the factorization problem for Lie-Yamaguti algebras that essentially related to the bicrossed product of Lie-Yamaguti algebras. Next, given an inclusion $\mathfrak{g} \subset E$ of Lie-Yamaguti algebras and a strong $\mathfrak{g}$-complement $\mathfrak{h}$, we describe and classify all $\mathfrak{g}$-complements in $E$. In particular, we show that any other $\mathfrak{g}$-complement in $E$ is isomorphic to $\mathfrak{h}$ by some deformation map $r: \mathfrak{h} \rightarrow \mathfrak{g}$. Despite this importance, it turns out that a deformation map generalizes homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators on Lie-Yamaguti algebras. We define the cohomology of a deformation map unifying the cohomologies of all the operators mentioned above. Finally, we provide a Maurer-Cartan characterization and construct the governing $L_\infty$-algebra of a deformation map $r$ that controls the linear deformations of $r$.

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✦ hep-th 2026-05-25

Deformed unstable K-theory quantizes Type IIA brane fluxes

by Pinak Banerjee, Hisham Sati +1 more

Flux Quantization of Type IIA in Unstable K-Theory

It handles nonlinear relations from NS-branes and lifts to M-theory quantization, unlike stable K-theory.

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The traditional conjecture that RR-flux is quantized in stable K-cohomology fails to account for the presence of NS-brane sources: These impose nonlinear relations -- reductions of the famous quadratic relation on M-brane flux -- that can only be captured by unstable nonabelian cohomology theories. Here we consider a deformation of unstable K-theory which properly quantizes the fluxes coupling to D0/D2/NS5-branes, find a twisted version that quantizes also the fluxes coupling to NS1/D4-branes, and show that this oxidizes to a proper electromagnetic quantization of M-brane fluxes.

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🧮 math.KT 2026-05-25

Singular vectors expand uniquely in even and odd Koschorke classes

by Kyouhei Horie

Odd Koschorke classes

Generalized classes indexed by partitions form a basis for Fredholm cohomology and yield the super-Virasoro expansion result.

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We introduce odd Koschorke classes in odd K-theory by using degeneracy loci of self-adjoint Fredholm operators. These classes are characteristic classes analogous to the even Koschorke classes in even K-theory. We study two aspects of these classes: their role as obstruction classes and their realization as characteristic classes with real coefficients for odd twisted K-theory. On the even side, we introduce generalized Koschorke classes indexed by arbitrary partitions via Cibotaru's notion of a quasi-manifold. These classes form a \(\mathbb{C}\)-basis of \(H^*(\mathrm{Fred}_0;\mathbb{C})\) and recover the usual Koschorke classes for rectangular partitions. Finally, by analogy with the correspondence between even Koschorke classes and singular vectors for a representation of the Virasoro algebra, we state a result for a representation of a super-Virasoro algebra: each singular vector has a unique finite expansion in terms of generalized Koschorke classes and generalized odd Koschorke classes.

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🧮 math.KT 2026-05-22 2 theorems

Secondary pairing detects missed K-homology classes

by Rufus Willett

A secondary pairing between K-theory and K-homology, relative eta invariants, and zeta maps

It pairs subgroups to Q/Z and links to eta invariants and classification sequences in good cases.

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The $K$-homology groups of a $C^*$-algebra are receptacles for information from topology, operator algebra theory, and representation theory. For applications, one often wants to know if two $K$-homology classes are the same: the simplest way to deduce this is typically via the `primary' pairing between $K$-homology and the dual theory ($K$-theory). However, this pairing will typically miss some information: for example, it cannot detect torsion elements of $K$-homology. In this paper, we introduce a `secondary' pairing between subgroups of $K$-homology and $K$-theory that takes values in $\mathbb{Q}/\mathbb{Z}$. In good cases we show that this pairing will detect all the classes in $K$-homology that are missed by the primary pairing. We then relate our secondary pairing to the relative eta invariants of Atiyah-Patodi-Singer, and to the Thomsen exact sequence and zeta maps from $C^*$-algebra classification theory.

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🧮 math.AT 2026-05-22 Recognition

Symmetric homology equals free E∞-algebra on E1-algebra

by Gabriel Angelini-Knoll, David Chan +3 more

Topological symmetric and braid homologies

Braid homology matches the free E2-algebra on an E1-algebra; the same tools give low-degree values and Thom-spectrum formulas.

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We identify topological symmetric homology as the free $\mathbb{E}_\infty$-algebra on an $\mathbb{E}_1$-algebra and topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra. In this way, topological symmetric homology and topological braid homology can be regarded as variants of $1$-dimensional representation homology. In order to identify topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra, we prove that the $\mathbb{E}_2$-monoidal envelope of the associative operad can be identified with the braided crossed simplicial group. Using this, we also compute the topological braid homology of grouplike $\mathbb{E}_1$-spaces. Further, we develop computational tools for topological symmetric and braid homologies. These tools allow us to perform low-degree computations of topological symmetric homology and prove that it is not Morita invariant. We also compute the topological $\Delta \mathbf{G}$-homology of Thom spectra in general and produce explicit formulas in the case of topological symmetric and braid homologies.

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🧮 math.AG 2026-05-22 2 theorems

Cohomology classifies bundles on real affine threefolds

by Samuel Lerbet

On the cohomological classification of vector bundles on smooth real affine surfaces and threefolds

Under assumptions on the real locus the classification matches the algebraically closed case and supplies the first non-stably-free module

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We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and in a coming joint paper with S. Banerjee and J. Fasel, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank $3$ bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine $\mathbb{R}$-algebra of dimension $3$ with trivial Chern classes which is not stably free.

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🧮 math.AG 2026-05-22 2 theorems

Unramified Milnor K-classes are n-divisible over separably closed fields

by Jean-Louis Colliot-Thélène, Stefan Schreieder

Divisibility phenomena in motivic Bloch--Ogus theory

The result holds for any n invertible in the field and generalizes to the Bloch-Ogus filtration on motivic cohomology.

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Let X be a smooth projective variety over a field k. For k separably closed, we prove that the subgroup of unramified classes in the Milnor K-group $K^M_i(k(X))$ of the function field of X is contained in the subgroup of n-divisible elements of $K^M_i(k(X))$ for any integer n invertible in k. This generalizes to a statement for unramified motivic cohomology of arbitrary bidegree. We further show that whenever k is finite or separably closed and l is a prime invertible in k, then all but the last step in the Bloch--Ogus filtration of the motivic cohomology of X are l-divisible up to torsion. Generalizations of this last result to arbitrary quasi-projective k-schemes are also proven.

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🧮 math.KT 2026-05-21 1 theorem

Abel summation yields non-trivial K-theory for unbounded complexes

by Thomas Huettemann, Dan Kucerovsky

The Abel Summation Method and Infinite Euler Characteristic

A new finiteness condition based on Abel sums makes algebraic K-theory of unbounded chain complexes non-trivial and containing an infinite-c

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We develop a finiteness notion for unbounded chain complexes over a commutative noetherian integral domain $R$ employing the Abel summation method. The algebraic K-theory of such complexes is defined, and shown to be non-trivial. We also exhibit a natural map from the (usual) algebraic K-theory of $R$ into the new K-theory and show that its image contains a canonical infinite cyclic subgroup.

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🧮 math.KT 2026-05-20 2 theorems

E-theory for spaces forms six-functor formalism equivalent to sheaves

by Ulrich Bunke

E-theory of X-C^(*)-algebras and functor formalisms

The equivalence unifies analytic K-theoretic invariants with geometric sheaf theory on locally compact Hausdorff spaces and cosheaves on fin

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We show that $E$-theory for locally compact Hausdorff spaces constitutes a six-functor formalism which is equivalent to the six-functor formalism of $\mathrm{E}$-valued sheaves. We furthermore show that the $E$-theory category for locales that can be written as unions of finite open sublocales is equivalent to the category of $\mathrm{E}$-valued cosheaves.

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🧮 math.AT 2026-05-13 2 theorems

Multivalued homology vanishes in positive degrees for compact Hausdorff spaces

by Alejandro O. Majadas-Moure

Singular multivalued homology

The groups H^M_n(X) are zero for all n greater than zero whenever X is compact and Hausdorff, extending the known one-dimensional case.

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Let $X$ be a compact, Hausdorff topological space. Then $H^M_n(X)=0$ for all $n>0$, where $H_n$ is the multivalued analogue of singular homology. The case $n=1$ is already known [8].

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🧮 math.CV 2026-05-13 3 theorems

Condensed sets recover Serre duality and Riemann-Roch for complex manifolds

by Dustin Clausen, Peter Scholze

Condensed Mathematics and Complex Geometry

Reproves finiteness of cohomology and GAGA by treating analytic spaces through condensed mathematics

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This is a slightly revised version of lectures notes for a course in Summer 2022 joint between Bonn and Copenhagen, intended as a stable citable version. The goal of this course is to make our general approach to analytic geometry via condensed mathematics more concrete by concentrating on the case of complex-analytic geometry. Instead of trying to develop new kinds of geometry, here we only try to redevelop the classical theory from a different point of view. More precisely, we reprove some important theorems for compact complex manifolds, including finiteness of coherent cohomology, Serre duality, GAGA and (Grothendieck--)Hirzebruch--Riemann--Roch.

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🧮 math.AG 2026-05-13

Motive decomposition determines full Chow-Witt ring of M-bar 1,2

by Nanjun Yang

The equivariant Milnor-Witt motive of overline{mathcal{M}}_(1,2)

The split of the equivariant Milnor-Witt motive for the genus-one two-pointed stable curve space yields the complete ring structure.

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We provide a decomposition of the equivariant Milnor-Witt motive for the moduli space of stable curves $\overline{\mathcal{M}}_{1,2}$. As a result, the equivariant Chow-Witt ring $\widetilde{CH}^*(\overline{\mathcal{M}}_{1,2})$ is fully determined.

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🧮 math.NT 2026-05-12 3 theorems

K2 element proven 2-indivisible for elliptic curves

by Neil Dummigan, Vasily Golyshev +2 more

The 2-part of the Bloch-Kato conjecture, and indivisibility results, for K₂ of some elliptic curves

The proof aids verification of the 2-part of the Bloch-Kato conjecture at s=2 for a family of curves.

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For certain integers $u$, we investigate the 2-part of the Bloch-Kato conjecture for $L(E_u,2)$, where $E_u: y^2=x(x+1)(x+u^2)$ is part of a (twisted) Legendre family that is 2-isogenous to a family studied by Boyd. For this, we first work out the corresponding 2-parts of the Tamagawa factors and Galois invariants. Then we give an explicit description of the 2-torsion in the Selmer group $H_f^1(\mathbb{Q},E_u[2^\infty](-1))$. We construct a specific element in the kernel of the tame symbol for $K_2$ on an integral model of $E_u$, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for $K_2$ on $E_u$. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map. Finally, in many cases we investigate numerically the validity of the 2-part of the Bloch-Kato conjecture.

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🧮 math.KT 2026-05-12 1 theorem

Equivariant Hochschild cohomology reduces to relative Ext

by Andrada Pojar, Constantin-Cosmin Todea

Equivariant Hochschild cohomology of group algebras and relative operatorname{Ext}

The isomorphism holds for any field k and supplies necessary conditions for non-vanishing when characteristic divides group order.

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For a finite group $\Gamma$, acting on a finite group $G,$ we find necessary conditions for which the first $\Gamma_0$-equivariant Hochschild cohomology of the group algebra $kG$ is non-trivial, where $k$ is a field of characteristic $p$ dividing the order of $G$ and $\Gamma_0$ is the stabilizer subgroup in $\Gamma$ of some element in $G.$ For any field $k$ we show that the $\Gamma$-equivariant Hochschild cohomology of $\Gamma$-algebras with coefficients in a $\Gamma$-equivariant bimodule (Jensen, 1996) is isomorphic with some $k\Gamma$-relative $\operatorname{Ext},$ in the context of relative homological algebra.

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🧮 math.NT 2026-05-11 2 theorems

Merkurjev construction yields non-R-trivial similitudes in degree 6

by M. Archita, Karim Johannes Becher

Non-R-trivial proper projective similitudes in type A₃equiv D₃

Algebras with orthogonal involution admit proper projective similitudes outside R-equivalence when the field has an anisotropic torsion 3-

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Over an arbitrary field of characteristic different from $2$ admitting an anisotropic torsion $3$-fold Pfister form, we apply a construction due to Merkurjev to produce an algebra with orthogonal involution of degree $6$ which admits proper projective similitudes that are not $R$-trivial. In particular, such examples exist over every finitely generated transcendental extension of a local or global number field, as well as over every finitely generated extension of transcendence degree $3$ of $\mathbb{R}$.

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🧮 math.AT 2026-05-11 1 theorem

Semisimplicial modules match chain-complex homotopy theories

by Atabey Kaygun

Combinatorial Models for Linear Homotopy Theories

Equivalences at both localization and Quillen levels hold over characteristic-zero fields, with a partial adjunction from semicubical models

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For a field $k$ of characteristic $0$, we compare $k$-linear chain complexes, semisimplicial vector spaces, augmented semisimplicial vector spaces, semicubical vector spaces, and arboreal vector spaces through small differential categorical algebras. We prove that semisimplicial modules and augmented semisimplicial modules are equivalent to appropriate chain-complex homotopy theories, both at the Gabriel--Zisman localization and the Quillen model-categorical level. The semicubical sign embedding gives a natural comparison from semicubical modules to augmented semisimplicial modules and induces a Quillen adjunction, but not a Quillen equivalence on the full semicubical category since there is an obstruction in augmented homology at degree $-1$.

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🧮 math.KT 2026-05-07

The paper shows that the algebraic K-theory groups of the thick subcategory generated by…

by Xiaojun Chen, Farkhod Eshmatov +1 more

Algebraic K-theory, cohomotopy K-groups, and Koszul duality

K_n(thick_A(k)) are identified as candidates for Loday's contravariant K-groups by combining Blumberg-Mandell Koszul duality equivalence…

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Let $A$ be an augmented differential graded algebra over a field $k$ of characteristic zero, and let $A^!=\mathbf{R}\mathrm{Hom}_A(k,k)$ be its Koszul dual algebra. Blumberg and Mandell showed that, under some finiteness conditions of $A$, the derived Koszul duality provides an equivalence between the $K$-theory $K(\mathrm{thick}_A(k))$ of the triangulated thick subcategory generated by $k$ and the $K$-theory $K(A^!)$ of the derived category of perfect $A^!$-modules. Combining this equivalence with the Jones-Goodwillie Chern character and the Jones-McCleary isomorphism, we obtain that the $K$-groups $K_n(\mathrm{thick}_A(k))$ are a concrete candidate for Loday's conjectural contravariant $K$-groups.

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🧮 math.KT 2026-05-06

Deformation groupoids yield functorial Gysin maps for Lie groupoids

by Paulo Carrillo Rouse, Quentin Karegar Baneh Kohal

Gysin maps and wrong way functoriality via geometric deformation groupoids

The maps compose correctly under groupoid morphisms, recover earlier constructions, and give new results for equivariant orbifold K-theory.

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In this article we study the normal bundle and the deformation to the normal cone functors to get deformation Lie groupoids that allow us to construct pushforward maps in any suitable (co)homology theory for Lie groupoids (not only K-theory) and in a natural and geometric way. The main theorems being the functoriality for these pushforward maps which recovers, unifies and generalizes many previous cases. The main new example we develop in this paper is the wrong way functoriality for equivariant (twisted) Orbifold K-theory with respect to a groupoid action.

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🧮 math.AT 2026-05-04

Stable categories reconstruct from hearts via two-step completion

by Thomas Nikolaus, Phil Pützstück

Unbounded Weight Structures: (Re)construction and Completion

A reconstruction theorem recovers any stable category with compatible weight and weak t-structures from its heart under left weight and

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We develop a theory of completeness for weight structures on stable categories, dual to the theory of complete t-structures. As in the bounded case, we show that complete weight structures are determined by their weight heart, giving rise to a universal construction $A \mapsto K(A)$ that assigns a complete weight category to an additive category and recovers classical examples such as homotopy categories of chain complexes. We also give a general construction of weight structures on presentable stable categories generated by a small set of objects, generalizing a result of Bondarko. This recovers the standard weight structure on spectra and an exotic one related to Anderson duality. We identify their completions with Bousfield--Kan completions arising in Adams-type spectral sequences. To treat naturally occurring examples - such as derived categories of abelian categories and module categories over ring spectra - which are often only partially weight complete, we introduce the notion of weak t-structures. Within this framework, we prove that any stable category equipped with compatible weight and weak t-structures, and satisfying left weight completeness and right t-completeness, can be reconstructed from its heart via a two-step completion process $A \mapsto \widehat{K}(A)$.

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🧮 math.AT 2026-05-04

Geodesic subspace posets realize as wedges of spheres

by Alexander Kupers, Ezekiel Lemann +3 more

Scissors automorphism groups II: Solomon-Tits theorems

A Solomon-Tits variant holds for collections generated by points or hyperplanes in Euclidean, hyperbolic, and spherical geometry.

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The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic subspaces of Euclidean, hyperbolic, or spherical geometry, assuming the collection is generated either by points or by hyperplanes. In the third paper of this series of papers, we will combine this with the homological stability theorems from the first paper to compute the homology of groups of scissors automorphisms in these geometries.

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🧮 math.AG 2026-05-01

Quillen-Lichtenbaum conjecture extends to noncommutative complex schemes

by Chunhui Wei

Noncommutative Quillen-Lichtenbaum Conjecture

Comparison maps between algebraic and topological K-groups become isomorphisms in ranges after refinement for finite type schemes.

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We establish isomorphism ranges for the comparison maps between algebraic and topological K-groups, extending classical Quillen-Lichtenbaum conjecture to separated complex schemes of finite type after refinement. Additionally, we generalizes the conjecture through the lens of noncommutative geometry.

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🧮 math.KT 2026-04-29

Equivariant coarse Baum-Connes equals groupoid version

by Liang Guo

A groupoid approach to the equivariant coarse Baum--Connes conjecture

For spaces with proper free isometric group actions, a coarse embedding into Hilbert space makes the assembly map injective.

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In this paper, we develop a groupoid approach to the equivariant coarse Baum--Connes conjecture. For a bounded geometry metric space $X$ equipped with a proper, free, and isometric action of a countable discrete group $\Gamma$, we introduce the equivariant coarse groupoid $G(X, \Gamma)$. We prove that the groupoid Baum--Connes conjecture for $G(X, \Gamma)$ with coefficients in $\ell^{\infty}(X,\mathcal{K})^\Gamma$ is equivalent to the equivariant coarse Baum--Connes conjecture for $(X, \Gamma)$ using a localization algebra description of equivariant $KK^\mathcal{G}$-theory for \'{e}tale groupoids. As applications of this framework, we prove that if the space $X$ admits a coarse embedding into Hilbert space (which is not required to be $\Gamma$-equivariant), then the equivariant coarse Novikov conjecture holds for $(X, \Gamma)$, i.e., the assembly map $\mu_{X,\Gamma}$ is an injection. We also obtain a new proof of the equivariant coarse Baum--Connes conjecture if $X$ admits an equivariant coarse embedding into Hilbert space.

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🧮 math.AT 2026-04-29

KR-theory yields immersions of C2-projective spaces

by Manyi Guo, Jackson Morris +2 more

Immersions of C₂-projective spaces via Kmathbb{R}-theory

The groups also give an equivariant James periodicity as an immediate corollary.

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We compute the Atiyah Real $K$-theory of $C_2$-equivariant projective spaces and construct immersions of such spaces into multiples of the regular representation. These computations are made tractable by the recent geometric filtration of equivariant projective spaces due to Bhattacharya-Waugh-Zeng-Zou, together with a variant of the localized slice spectral sequence introduced by Meier-Shi-Zeng. As an immediate corollary of these computations, we obtain an equivariant analogue of James periodicity.

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🧮 math.AG 2026-04-28

P1-unstable theory yields Gysin maps for motivic spectra

by Longke Tang

The P¹-motivic Gysin map

The construction for regular immersions supplies a uniform definition across many cohomology theories.

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We develop a $\mathbf{P}^1$-unstable non-$\mathbf{A}^1$-invariant theory of motivic spaces and spectra, and construct the Gysin map therein for regular immersions. This in particular gives the Gysin map in the Annala--Hoyois--Iwasa $\mathbf{P}^1$-motivic spectra, and thus gives a uniform construction for the Gysin maps of various cohomology theories.

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🧮 math-ph 2026-04-28

Pauli stabilizer codes match Clifford QCAs one dimension higher

by Bowen Yang, Matthew Yu

The Classification of Pauli Stabilizer Codes: A Lattice and Continuum Treatise

Bulk-boundary correspondence classifies lattice codes up to gapped interfaces using algebraic L-theory and reveals differences from framedTQ

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We classify mobile Pauli stabilizer codes up to gapped interfaces and coarse-graining using the framework of algebraic $\mathrm{L}$-theory. We compare this classification with that of framed TQFTs, theories that arise naturally in the continuum, highlighting a close structural relationship between the two. Our approach is formulated in the category of perfect chain complexes equipped with quadratic functor over the Laurent polynomial ring $R = \mathbb{Z}/p[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$, within which the collection of topological operators of Pauli stabilizer codes arise naturally as objects. In particular, we establish a bulk-boundary correspondence for lattice theories: the equivalence class of a Pauli stabilizer code up to gapped interface is described by a Clifford QCA in one dimension higher. This is done using the universal target category for stabilizer codes, which is the categorical spectrum whose existence and universal properties are introduced in this work. We conclude by highlighting subtle differences between the classification of Pauli stabilizer codes and TQFTs, leading to qualitative distinctions between lattice and continuum theories.

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🧮 math.AT 2026-04-28

A∞-algebras model rational fiberwise THH transfer

by Florian Naef, Robin Stoll

A rational model for the fiberwise THH transfer II: A_infty-algebras

The description generalizes Bouc and yields a rational Becker-Gottlieb model plus vanishing results for graph classes on manifold bundles.

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In Part I, we proved that a rational model for the fiberwise THH transfer of a map $f$ of fibrations over a base space is given by the Hochschild homology transfer of a cdga model of $f$. In this paper, we provide an explicit description of this Hochschild homology transfer in terms of $A_\infty$-algebras, generalizing work of Bouc. Using a result of Lind-Malkiewich, we deduce a rational model for the Becker-Gottlieb transfer. We furthermore use our results for the following applications to manifold topology. Firstly, we consider the rational characteristic classes constructed by Berglund for fibrations with fiber a Poincar\'e complex (which generalize classes found by Berglund-Madsen); they are defined via the Lie graph complex, and we prove that the classes corresponding to non-trivalent graphs with exactly one loop vanish when evaluated on fiber bundles with fiber a compact simply connected topological manifold. Secondly, we provide a rational model for the space of fiberwise THH-simple structures, which is a step towards obtaining rational models for the classifying spaces of diffeomorphisms and homeomorphisms of a compact simply connected manifold in the rational concordance stable range.

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🧮 math.RA 2026-04-28 2 theorems

Every lim¹ group equals the cokernel of a module to its completion

by Ioannis Emmanouil

The structure of lim¹-groups

A functorial filtration sequence realizes any derived limit as the obstruction to surjectivity onto the completion.

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If $(A_n)_n$ is a decreasing filtration of a module $A$ and $\widehat{A} = \lim_n A/A_n$, then $\lim^1_n A_n$ is identified with the cokernel of the canonical map $A \longrightarrow \widehat{A}$. In this note, we show that any $\lim^1$-group is canonically of that form: For any inverse sequence of modules $(X_n)_n$ there exists an inverse sequence $(A_n)_n$ as above and a morphism $(A_n)_n \longrightarrow (X_n)_n$, depending functorially on $(X_n)_n$, that induces an isomorphism on $\lim^1$. The proof is based on Quillen's small object argument, as formulated by Eklof and Trlifaj in their investigation of the existence of enough injective objects in certain cotorsion pairs, and also uses a construction by Salce that provides enough projective objects therein.

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🧮 math.FA 2026-04-27

Regularized winding number gives spectral flow for Id+Schatten unitaries

by A. Alexander, A. Carey +2 more

Analytic spectral flow formula for unitaries and Levinson's theorem

The integral formula counts bound states from the potential in Schrödinger systems and extends to open paths and Cayley-transformed operator

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We prove an integral formula for the spectral flow of differentiable loops of unitaries of the form ${\rm Id}+$Schatten. Our formula is in terms of a regularised winding number, expressed in terms of exact differential forms, and we show how the formula extends to non-closed paths. Applying these ideas to the scattering operator of Schr\"{o}dinger scattering systems yields explicit formulae for the number of bound states, possibly modified by the presence of resonances, of the system in terms of the potential. We finish by briefly considering the paths of unbounded operators obtained from unitary loops via the Cayley transform. These include cases of moving domain as well as paths with non-constant Hilbert space.

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🧮 math.AT 2026-04-27

Diagrams of Eilenberg-MacLane spaces are formal over Q

by Grigory Solomadin, Antoine Touzé

On formality of diagrams of Eilenberg-MacLane spaces

Spectral sequences for any such diagram over any small category collapse at page 2, but the result fails over rings without the rationals.

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In this paper, we establish formality (over $\mathbb{Q}$) for diagrams of Eilenberg-MacLane spaces of any height $n\geq 1$. This implies spectral sequence (over $\mathbb{Q}$) collapse at page $2$ for any diagram of EML spaces over any small category. We prove by functor calculus argument that formality does not hold over any fixed commutative ring $\mathbf{k}$ not containing $\mathbb{Q}$, where the category of diagrams is over the category generated by finite direct sums of a cyclic group.

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🧮 math.QA 2026-04-23

Tensor category cohomology conjecture reduces to Hochschild finiteness

by Petter Andreas Bergh

On the cohomology of finite tensor categories

The finite-generation conjecture for finite tensor categories is equivalent to the same property for Hochschild cohomology of endomorphismal

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It has been conjectured that finite tensor categories have finitely generated cohomology. We show that this is equivalent to finitely generated Hochschild cohomology for the endomorphism algebras of the projective generators.

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🧮 math.KT 2026-04-20

Witt ring of the real sphere is calculated

by Heng Xie

The Witt ring of the real sphere

Direct computation from standard definitions determines its full algebraic structure.

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We calculate the Witt ring of the real sphere.

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🧮 math.KT 2026-04-20

Two-symbol sums in char-2 K2 carry a K4 invariant

by Demba Barry, Adam Chapman +1 more

Sums of two symbols in K₂(F)/2K₂(F) in characteristic two

The invariant is zero precisely when the sum equals a single symbol modulo 4.

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In this paper, study sums $A=\{a,b\}_2+\{c,d\}_2$ of two symbols in $K_2(F)/2K_2(F)$ when $\operatorname{char}(F)=2$. We first prove a chain lemma that connects $A$ to $B=\{\alpha,\beta\}_2+\{\gamma,\delta\}_2$ by a finite sequence of small steps when $A \equiv B$. We use this lemma to prove that $\{a,b,c,d\}_2 \in K_4(F)/2K_4(F)$ is a well-defined invariant of $A$, and that this invariant is trivial if and only if $A$ is congruent to a single symbol in $K_2(F)/4K_2(F)$. We also bound the symbol length of $C$ in $K_2(F)/2^m K_2(F)$ from above when $C$ is the sum of up to four symbols in $K_2(F)/2^{m+1}K_2(F)$.

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🧮 math.LO 2026-04-20

Quotient encodings bound C*-algebra properties in the Borel hierarchy

by Tomasz Kania

Polish spaces for countable and separable structures through quotient encodings

Kernels form Polish spaces under Wijsman topology so nuclearity is Borel, AF-ness is Pi^0_3 and K-groups receive internal Borel codes.

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We develop a unified framework for locating natural properties of algebraic and analytic structures within the Borel hierarchy. Objects are presented as quotients of a universal generator and definability is read directly from the quotient data. For separable Banach-type structures (Banach algebras, $C^*$-algebras, Banach lattices, TROs) the kernel space is Polish under the Wijsman topology, and the quotient-norm functional $K\mapsto \|x+K\|$ is continuous, yielding a uniform definability scheme whose Borel ranks are bounded by quantifier alternation depth. For countable algebraic structures (groups, rings, lattices) we work on compact Polish spaces of congruences where atomic predicates are clopen. We obtain explicit Borel upper bounds: in the \emph{unital} $C^*$-algebra coding based on $C^*_{\max}(F_\infty)$, stable finiteness is closed, nuclearity is Borel, simplicity is~$G_\delta$, AF-ness lies in~$\Pi^0_3$, nuclear dimension~$\le n$ lies in~$\Pi^0_3$, and for fixed exact~$D$, $D$-absorption is analytic. For countable groups, soficity is~$G_\delta$; for abelian groups, slenderness is~$\Pi^0_3$. We give an internal Borel coding of the $K_0$-assignment in the quotient/Wijsman framework; for each fixed coordinate the corresponding section is $F_\sigma$, and suspension together with Bott periodicity yields Borel codings of all higher $K$-groups. We also show that several bounds are optimal ($\Sigma^0_2$- and $\Pi^0_2$-complete). To calibrate the method's reach, we exhibit a $\Pi^1_1$-complete property (separable dual in the commutative $C^*$-setting), provably outside the Borel hierarchy.

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🧮 math.OA 2026-04-17

This paper proves that the quantitative coarse Baum-Connes conjecture holds for the free…

by Jintao Deng, Ryo Toyota

The quantitative coarse Baum-Connes conjecture for free products

The quantitative coarse Baum-Connes conjecture holds for the free product G * H if it holds for the groups G and H.

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Let $G$ and $H$ be finitely generated groups. In this paper, we prove the quantitative coarse Baum--Connes conjecture for the free product $G* H$ under the assumption that the conjecture holds for both $G$ and $H$.

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🧮 math.AT 2026-04-17

Finite group actions preserve Witt pseudomanifolds and their L-classes

by Markus Banagl

Equivariant L-Classes of Atiyah-Singer-Zagier Type for Singular Spaces

Equivariant Atiyah-Singer-Zagier classes average to recover the Goresky-MacPherson L-class of the orbit space.

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If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds.

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🧮 math.KT 2026-04-16

Weighted limit defines Euler characteristic for unbounded complexes

by Thomas Huettemann, Dan Kucerovsky

An Euler Characteristic for Unbounded Chain Complexes

Inverse-length averages on finite truncations yield a homotopy invariant whose Grothendieck group is uncountable.

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We propose a definition of an Euler characteristic for unbounded chain complexes by taking the (usual) Euler characteristics of successively longer parts of the complex, weighted inversely proportional to the length, and passing to the limit. This amounts to taking the limit of the sequence of ranks of homology modules with alternating signs in the sense of the H\"older summation method. We establish the structure of a category with cofibrations and weak equivalences on unbounded complexes for which the infinite Euler characteristic is defined, and show that its Grothendieck group is unusually large (viz., uncountable).

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🧮 math.AG 2026-04-15

Polynomials represent suspended Hopf map over Z

by Jean Fasel, William Hornslien

Exotic Hopf maps, weight shifting and applications to vector bundles

The formulas produce an explicit rank-2 vector bundle on the Jouanolou device of P^3 over the integers.

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Using motivic homotopy theory we produce several explicit polynomial representatives of the suspension of the Hopf map defined over the integers. We derive from this computation an explicit rank 2 vector bundle on the Jouanolou device of the projective space of dimension 3 over the integers.

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🧮 math.AT 2026-04-14

Parametric T(n)-equivalences align two periodic localizations of spaces

by Shaul Barkan, Gijs Heuts +1 more

On periodic homotopy and homology equivalences of spaces

The condition yields exact comparisons for general spaces and an explicit L_n^f formula for infinite loop spaces whose spectra vanish at the

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There are at least two ways to approach the homotopy theory of spaces `at chromatic height $n$': one may localize with respect to $T(n)$-homology or with respect to $v_n$-periodic homotopy groups. It was already observed by Bousfield that these two options yield rather different results. We build on his work to prove precise comparison results between the two notions. A crucial concept is a more robust notion of $T(n)$-equivalence that we call `parametric $T(n)$-equivalence': this is a map of spaces that induces an equivalence on $\infty$-categories of local systems valued in $T(n)$-local spectra. Our results are sharpest in the case of infinite loop spaces, where amongst other things we prove a $T(n)$-local version of a result of Kuhn on the Morava $K$-theory of the Whitehead tower. As a corollary of our results we also produce a formula for the $L_n^f$-localization of an infinite loop space $\Omega^\infty E$ of a spectrum satisfying $L_{n-1}^f E \simeq 0$.

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🧮 math.KT 2026-04-13

K-theory of constant Tambara fields is torsion

by Noah Wisdom

The K-theory of finite Tambara fields: away from p

The groups are fully determined after inverting p and follow a simple pattern, with nontrivial p-power torsion in general.

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In previous work, the author and Chan computed the algebraic $K$-theory of the constant $C_2$-Tambara field with value the field with two elements, using a method which fails at odd primes. Herein we make progress towards the corresponding odd primary computations using a completely new idea. Particularly, we show that the $K$-theory groups of any constant $C_{p^n}$-Tambara field with value a characteristic $p$ finite field are torsion, and we completely determine these groups after inverting $p$. The away-from-$p$-torsion satisfies a simple pattern predicted by previous work, and a computer-aided computation shows that the $p$-power torsion is nontrivial in general.

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🧮 math.KT 2026-04-13

Relative Vorst theorem improves K1 stability bounds

by Sourjya Banerjee, Kuntal Chakraborty

Improved injective stability for relative K₁Sp-groups

A new relative version combined with Karoubi periodicity sharpens the thresholds for when relative linear and symplectic K1 groups stabilize

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We prove a relative version of Vorst's theorem concerning the equality of the group of all invertible matrices and the group of all elementary matrices over $R[X]$ with respect to an ideal $I\subset R$ such that $R/I$ is regular, where $R$ is a regular $k$-spot. We then introduce a relative version of the symplectic elementary Witt group and show that it fits into a relative version of the Karoubi periodicity sequence. Combining these results, we improve the existing injective stability bounds for relative linear and symplectic $\mathrm{K_1}$-groups of smooth affine algebras over various base fields.

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🧮 math.KT 2026-04-10 2 theorems

Bredon sheaf cohomology is fixed by descent conditions alone

by Guido Arnone, Devarshi Mukherjee +1 more

Bredon sheaf cohomology

For finite group actions on locally compact Hausdorff spaces, any dualizable stable functor obeying open descent and compact codescent must,

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For a finite group $G$, we compute the algebraic $K$-theory of the category of equivariant sheaves on a locally compact Hausdorff $G$-space, generalizing a result of Efimov, and determine the equivariant $E$-theory of the $C^*$-algebra of continuous functions. These invariants admit natural descriptions in terms of a new equivariant cohomology theory, which we call Bredon sheaf cohomology. This theory recovers classical Bredon cohomology for $G$-CW complexes and ordinary sheaf cohomology when $G$ is trivial. We establish its basic structural properties and prove a strong uniqueness theorem: any functor from the category of locally compact Hausdorff $G$-spaces to a dualizable stable category satisfying equivariant open descent and cofiltered compact codescent is equivalent to Bredon sheaf cohomology, generalizing a result of Clausen.

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🧮 math.NT 2026-04-09 2 theorems

Reduced unitary Whitehead group vanishes over p-adic curve fields

by Zitong Pei

Reduced Unitary Whitehead Groups over Function Fields of p-adic Curves

This holds for every period-2 central simple algebra equipped with an involution of the second kind when the residue characteristic is odd.

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Let $F_0$ be the function field of a curve over a $p$-adic field $K,$ and let $F$ be a quadratic extension over $F_0$. Let $A$ be a central simple algebra over $F$ of period $2,$ and let $\tau$ be a $F/F_0$-involution on $A$. We show the triviality of the reduced unitary Whitehead group $SK_1U( A, \tau)$ if $p\neq 2$.

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🧮 math.OA 2026-04-07 Recognition

Split exact sequences give KK-equivalences for amplified graph C*-algebras

by Jesse Reimann, Sophie Emma Zegers

On split exact sequences and KK-equivalences of amplified graph C*-algebras

The method produces explicit equivalences to C^N for quantum Grassmannians and equates classical and quantum projective lines.

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We give a general methodology for constructing split exact sequences of amplified graph C*-algebras with sinks. This in turn allows us to construct explicit KK-equivalences with $\mathbb{C}^N$ for a large class of C*-algebras, including the quantum Grassmannian $\mathrm{Gr}_q(2,4)$. We discuss compatibility with known (quantum) CW-constructions and give an explicit KK-equivalence between the classical and quantum projective spaces $\mathbb{C}P^1$ and $\mathbb{C}P_q^1$.

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🧮 math.KT 2026-04-06 2 theorems

SU₃(F[t]) first cohomology isomorphic to PGL₂(F)

by Claudio Bravo

Cohomology of special unitary groups and congruence subgroups

Homotopy invariance holds for these groups when coefficients are irreducible representations of PGL₂(F)

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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 $\mathrm{PGL}_2(F)$.

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