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Remarks on classical number theoretic aspects of Milnor-Witt K-theoryJun 18 2019We record a few observations on number theoretic aspects of Milnor-Witt K-theory, focusing on generalizing classical results on reciprocity laws, Hasse's norm theorem and K_2 of number fields and rings of integers.

Galois descent criteriaJun 14 2019This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the \'etale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois cohomological objects ... More

The homotopy invariance of dihedral homology of involutive $A_\infty$-algebras over ringsJun 14 2019The dihedral homology functor $HD:A_\infty^{{\rm inv}}(K)\to GrM(K)$ from the category $A_\infty^{{\rm inv}}(K)$ of involutive $A_\infty$-algebras over any commutative unital ring $K$ to the category $GrM(K)$ of graded $K$-modules is constructed. Further, ... More

An alternative construction of equivariant Tamagawa numbersJun 11 2019We propose a new formulation of the equivariant Tamagawa number conjecture (ETNC) for non-commutative coefficients. We remove Picard groupoids, determinant functors, virtual objects and relative K-groups. Our Tamagawa numbers lie in an idele group instead ... More

Three real Artin-Tate motivesJun 07 2019We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by the second-named ... More

K- and L-theory of graph products of groupsJun 05 2019We compute the group homology, the algebraic $K$- and $L$-groups, and the topological $K$-groups of right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products.

New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifoldsJun 04 2019Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ ... More

A dévissage theorem of non-connective $K$-theoryJun 04 2019The purpose of this article is to show a version of d\'evissage theorem of non-connective $K$-theory. Our theorem contains Quillen's d\'evissage theorem, Waldhausen's cell filtration theorem and theorem of heart as special cases. In this sense, we give ... More

The Farrell--Jones Conjecture for normally poly-free groupsJun 04 2019We prove the $K$- and $L$-theoretic Farrell--Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $A\rtimes \mathbb{Z}$ where $A$ ... More

Nilpotence in normed MGL-modulesJun 04 2019We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies $E \wedge HZ \simeq 0$, then also $E \wedge MGL \simeq 0$. In words, motivic homology detects vanishing of normed modules over the algebraic ... More

Transfer maps in generalized group homology via submanifoldsJun 04 2019Let $N \subset M$ be a submanifold embedding of spin manifolds of some codimension $k \geq 1$. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that $M$ does not admit a metric of positive scalar curvature if $k = 2$ ... More

Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold mapsJun 03 2019Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional version, started ... More

Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold mapsJun 03 2019Jun 15 2019Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional version, started ... More

K-theory formulas for orthogonal and symplectic orbit closuresJun 03 2019The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these orbits. Our polynomials ... More

A Universal HKR TheoremMay 31 2019In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a \emph{filtered circle} interpolating between the usual topological circle and a formal version of it. By mapping to schemes we ... More

Transverse Kronecker flows and Connes' duality for the irrational rotation algebraMay 31 2019In this article we explain how a suitably chosen non-compact transversal to the Kronecker foliation of the 2-torus can be used to invert Connes' Poincar\'e duality map for the irrational rotation algebra. This supplies a geometrically interesting cycle ... More

Matrix factorizations for self-orthogonal categories of modulesMay 31 2019For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in \mathsf{C}$, ... More

A Simplicial Construction for Noncommutative SettingsMay 30 2019In this paper we present a general construction that can be used to define the higher Hochschild homology for a noncommutative algebra. We also discuss other examples where this construction can be used.

Higher invariants in noncommutative geometryMay 29 2019We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.

A Lichnerowicz Vanishing Theorem for the Maximal Roe AlgebraMay 29 2019We show that if a discrete group acts properly and isometrically on a spin manifold of bounded geometry with a uniformly positive scalar curvature metric, then the maximal equivariant index of the Dirac operator vanishes in K-theory of the maximal equivariant ... More

On CW-complexes over groups with periodic cohomologyMay 28 2019If $G$ has $4$-periodic cohomology and at most two one-dimensional quaternionic representations, then $G$ has the D2 property if and only if $G$ has a balanced presentation. We use this to solve Wall's D2 problem for several infinite families of non-abelian ... More

Complex motivic $kq$-resolutionsMay 28 2019We analyze the $kq$-based motivic Adams spectral sequence over the complex numbers, where $kq$ is the very effective cover of Hermitian K-theory defined over $\mathbb{C}$ by Isaksen-Shkembi and over general base fields by Ananyevskiy-R{\"o}ndigs-{\O}stv{\ae}r. ... More

Higher rho numbers and the mapping of analytic surgery to homologyMay 28 2019Let $\Gamma$ be a finitely generated discrete group and let $\widetilde{M}$ be a Galois $\Gamma$-covering of a smooth compact manifold $M$. Let $u:X\to B\Gamma$ be the associated classifying map. Finally, let $\mathrm{S}_*^\Gamma (\widetilde{M})$ be the ... More

The Novikov ConjectureMay 27 2019We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .

Symmetric approximation sequences, derived equivalences and quotient algebras of locally $Φ$-Beilinson-Green algebrasMay 27 2019In this paper, we introduce a class of locally $\Phi$-Beilinson-Green algebras which is a way of obtaining algebras from C-constructions, where $\Phi$ is an infinite admissible set of the integral numbers, and show that symmetric approximation sequences ... More

The homotopy invariance of cyclic homology of $A_\infty$-algebras over ringsMay 27 2019In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of ... More

Note on linear relations in {é}tale $K$-theory of curvesMay 25 2019In this paper we investigate a local to global principle for {\'e}tale K-theory of curves. More precisely, we show that the result obtained by G.Banaszak and the author in \cite{bk13} describing the sufficient condition for the local to global principle ... More

Koszul duality in exact categoriesMay 24 2019In this paper we establish Koszul duality type results in the setting of chain complexes in exact categories. In particular we prove generalisations of Vallette's cooperadic Koszul duality theorem, and operadic Koszul duality along the lines of Lurie. ... More

The Baum-Connes conjecture: an extended surveyMay 24 2019We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.

Lie algebras of differential operators III: ClassificationMay 23 2019Jun 02 2019In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this ... More

Lie algebras of differential operators III: ClassificationMay 23 2019Jun 05 2019In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this ... More

Lie algebras of differential operators III: ClassificationMay 23 2019Jun 08 2019In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this ... More

Lie algebras of differential operators III: ClassificationMay 23 2019May 30 2019In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this ... More

Lie algebras of differential operators III: ClassificationMay 23 2019In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this ... More

Lie algebras of differential operators III: ClassificationMay 23 2019Jun 12 2019In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this ... More

On support varieties and tensor products for finite dimensional algebrasMay 22 2019It has been asked whether there is a version of the tensor product property for support varieties over finite dimensional algebras defined in terms of Hochschild cohomology. We show that in general no such version can exist. In particular, we show that ... More

Twisted differential KO-theoryMay 22 2019We provide a systematic approach to twisting differential KO-theory leading to a construction of the corresponding twisted differential Atiyah-Hirzebruch spectral sequence (AHSS). We relate and contrast the degree two and the degree one twists, whose ... More

Topological cyclic homologyMay 22 2019This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B\"okstedt periodicity ... More

Trees are 1-TransferMay 21 2019The K-theoretic Farrell-Jones isomorphism conjecture for a group ring $R[G]$ has been proved for several groups. The toolbox for proving the Farrell-Jones conjecture for a given group depends on some geometric properties of the group as it is the case ... More

Band width estimates via the Dirac operatorMay 21 2019Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $\sigma > 0$, the distance between ... More

Koszul calculus of preprojective algebrasMay 20 2019We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A 1 and A 2 , vanishes in any (co)homological degree p > 2. Moreover, the (higher) cohomological calculus is isomorphic as a bimodule to the (higher) homological ... More

Filtrations on homotopy invariant sheaves with transfersMay 19 2019We construct filtrations on homotopy invariant sheaves with transfers and show that under Ayoub's conjectures on $n$-motives, our filtration agrees with the one conjectured by Ayoub and Barbieri-Viale if the latter exists. Our construction is directly ... More

Strong Novikov conjecture for low degree cohomology and exotic group C*-algebrasMay 19 2019We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra even holds for the reduced group C*-algebra. To achieve this we provide a ... More

Grothendieck polynomials and the Boson-Fermion correspondenceMay 19 2019We present a new characterization of the Grothendieck polynomial and its dual by using the Boson-Fermion correspondence. As an application, we give a new alternative proof of some fundamental theorems about the Grothendieck polynomials such as determinantal ... More

Grothendieck polynomials and the Boson-Fermion correspondenceMay 19 2019Jun 05 2019In this paper we study algebraic and combinatorial properties of Grothendieck polynomials and their dual polynomials by means of the Boson-Fermion correspondence. We show that these symmetric functions can be expressed as a vacuum expectation value of ... More

The Hodge Chern character of holomorphic connections as a map of simplicial presheavesMay 19 2019We define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable non connection preserving isomorphisms of vector bundles with holomorphic connections an appropriate sequence of holomorphic forms. We apply this ... More

T-structures and twisted complexes on derived injectivesMay 17 2019May 21 2019In the paper "Deformation theory of abelian categories", the last two authors proved that an abelian category with enough injectives can be reconstructed as the category of finitely presented modules over the category of its injective objects. We show ... More

The circle action on topological Hochschild homology of complex cobordism and the Brown-Peterson spectrumMay 16 2019We specify exterior generators for $\pi_* THH(MU) = \pi_*(MU) \otimes E(\lambda'_n \mid n\ge1)$ and $\pi_* THH(BP) = \pi_*(BP) \otimes E(\lambda_n \mid n\ge1)$, and calculate the action of the $\sigma$-operator on these graded rings. In particular, $\sigma(\lambda'_n) ... More

Hyperdescent and etale K-theoryMay 16 2019We study the etale sheafification of algebraic K-theory, called etale K-theory. Our main results show that etale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories. Consequently, we ... More

Sign choices for orientifoldsMay 15 2019We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant $p$-gerbes with $p\geq-1$, which give rise to sign choices and are related by coboundary maps. We ... More

Stability of Loday constructionsMay 14 2019We study the question for which commutative ring spectra $A$ the tensor of a simplicial set $X$ with $A$, $X \otimes A$, is a stable invariant in the sense that it depends only on the homotopy type of $\Sigma X$. We prove several structural properties ... More

On the homology of the commutator subgroup of the pure braid groupMay 13 2019We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we determine the cohomological ... More

Standard Conjecture D for matrix factorizationsMay 12 2019We prove the non-commutative analogue of Grothendieck's Standard Conjecture D for the dg-category of matrix factorizations of an isolated hypersurface singularity in characteristic 0. Along the way, we show the Euler pairing for such dg-categories of ... More

Some remarks in $C^*$- and $K$-theoryMay 09 2019This note consists of three unrelated remarks. First, we demonstrate how roughly speaking $*$-homomorphisms between matrix stable $C^*$-algebras are exactly the uniformly continuous $*$-preserving group homomorphisms between their genral linear groups. ... More

Cohomological Methods in Intersection TheoryMay 09 2019These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to see how motives ... More

Rigidity in equivariant algebraic $K$-theoryMay 08 2019If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $n\ge 1$ is an integer coprime to $|G|$ and such that $n\cdot |G|\in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra \[ K^G(R)/n\stackrel{\simeq}{\longrightarrow} ... More

Smooth classifying spaces for differential $K$-theoryMay 08 2019We construct a version of differential $K$-theory based on smooth Banach manifold models for the homotopy types $B \mathrm U\times Z$ and $\mathrm U$ that appear in the topological $K$-theory spectrum. These manifolds carry natural differential forms ... More

The equivariant K- and KO-theory of certain classifying spaces via an equivariant Atiyah-Hirzebruch spectral sequenceMay 08 2019Computation of the K- and KO-theory for the classifying G-spaces for proper actions of certain infinite discrete groups G via a special version of the equivariant Atiyah- Hirzebruch spectral sequence.

Factorization of symplectic matrices into elementary factorsMay 07 2019We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic matrices with entries in a Banach algebra or in the ring of ... More

Factorization of symplectic matrices into elementary factorsMay 07 2019May 22 2019We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic matrices with entries in a Banach algebra or in the ring of ... More

On pure derived and pure singularity categoriesMay 07 2019Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient ... More

On morphisms killing weights and Hurewicz-type theoremsApr 29 2019We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most ... More

Deformations of Loday-type algebras and their morphismsApr 28 2019We study a formal deformation of multiplications in an operad. This closely resemble Gerstenhaber's deformation theory for associative algebras. However, this is applicable to various Loday-type algebras and to their twisted analogues. We explicitly describe ... More

Deformations of Loday-type algebras and their morphismsApr 28 2019May 03 2019We study a formal deformation of multiplications in an operad. This closely resemble Gerstenhaber's deformation theory for associative algebras. However, this is applicable to various Loday-type algebras and to their twisted analogues. We explicitly describe ... More

An equivariant Atiyah-Patodi-Singer index theorem for proper actionsApr 25 2019Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. Then an equivariant Dirac type operator on $M$ under a suitable boundary condition has an equivariant ... More

Elliptic classes of Schubert cells via Bott-Samelson resolutionApr 24 2019We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety $G/B$. For this first we need to twist the notion of elliptic characteristic class of Borisov-Libgober by a line bundle, and thus allow ... More

Elliptic classes of Schubert cells via Bott-Samelson resolutionApr 24 2019May 16 2019We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety $G/B$. For this first we need to twist the notion of elliptic characteristic class of Borisov-Libgober by a line bundle, and thus allow ... More

Hopfological algebra for infinite dimensional Hopf algebrasApr 23 2019May 22 2019We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector ... More

Hopfological algebra for infinite dimensional Hopf algebrasApr 23 2019We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector ... More

Hopfological algebra for infinite dimensional Hopf algebrasApr 23 2019May 17 2019We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector ... More

Hopfological algebra for infinite dimensional Hopf algebrasApr 23 2019Jun 04 2019We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector ... More

Isotropic motivesApr 19 2019In this article we introduce the local versions of the Voevodsky category of motives with Z/p-coefficients over a field k, parameterized by finitely-generated extensions of k. We introduce the, so-called, flexible fields, passage to which is conservative ... More

Tangent of K-theoryApr 17 2019We show that the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. More precisely, we prove that the tangent of K-theory, in terms of (abelian) deformation problems over k, is cyclic ... More

A Dolbeault-Hilbert complex for a variety with isolated singular pointsApr 16 2019Apr 23 2019Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with ... More

A Dolbeault-Hilbert complex for a variety with isolated singular pointsApr 16 2019Given a compact complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one ... More

On analytic Todd classes of singular varietiesApr 15 2019Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the fist part, assuming either $\mathrm{dim}(\mathrm{sing}(X))=0$ or $\mathrm{dim}(X)=2$, we ... More

On analytic Todd classes of singular varietiesApr 15 2019May 06 2019Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the fist part, assuming either $\mathrm{dim}(\mathrm{sing}(X))=0$ or $\mathrm{dim}(X)=2$, we ... More

Infinitesimal Bloch regulatorApr 14 2019In this paper, we continue our project of defining and studying the infinitesimal versions of the classical, real analytic, invariants of motives. Here, we construct an infinitesimal analog of Bloch's regulator. Let $X/k$ be a scheme of finite type over ... More

Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebrasApr 13 2019We prove that an isomorphism of graded Grothendieck groups $K^{gr}_0$ of two Leavitt path algebras induces an isomorphism of their algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their associated graph $C^*$-algebras. ... More

Coassembly is a homotopy limit mapApr 11 2019We prove a claim by Williams that the coassembly map is a homotopy limit map. As an application, we show that the homotopy limit map for the coarse version of equivariant $A$-theory agrees with the coassembly map for bivariant $A$-theory that appears ... More

Mock modularity and a secondary elliptic genusApr 11 2019The theory of Topological Modular Forms suggests the existence of deformation invariants for two-dimensional supersymmetric field theories that are more refined than the standard elliptic genus. In this note we give a physical definition of some of these ... More

A K-theoretic Selberg trace formulaApr 09 2019Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of operators on the Hilbert space L^2(G/H) associated to compactly supported smooth functions ... More

Categorified Chern character and cyclic cohomologyApr 08 2019We examine Hopf cyclic cohomology in the same context as the analysis of the geometry of loop spaces $LX$ in derived algebraic geometry and the resulting close relationship between $S^1$-equivariant quasi-coherent sheaves on $LX$ and $D_X$-modules. Furthermore, ... More

On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3Apr 07 2019Let $F$ be a field, $\ell$ a prime and $D$ a central division $F$-algebra of $\ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $\lambda$ such that the cohomology class $(D)\cup (\lambda)\in H^3(F,\,\mathbb{Q}_{\ell}/\Z_{\ell}(2))$ ... More

Lie, associative and commutative quasi-isomorphismApr 07 2019May 27 2019Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. We also show the Koszul dual statement that two dg Lie algebras ... More

Lie, associative and commutative quasi-isomorphismApr 07 2019Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. We also show the Koszul dual statement that two dg Lie algebras ... More

The first Hochschild (co)homology when adding arrows to a bound quiver algebraApr 07 2019We provide a formula for the change of the dimension of the first Hoch\-schild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology ... More

On the $p$-adic Beilinson conjecture and the equivariant Tamagawa number conjectureApr 05 2019Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin $L$-functions attached ... More

A Bloch-Ogus Theorem for henselian local rings in mixed characteristicApr 05 2019We show a conditional exactness statement for the Nisnevich Gersten complex associated to an $\mathbb{A}^1$-invariant cohomology theory with Nisnevich descent for smooth schemes over a Dedekind ring with only infinite residue fields. As an application ... More

A geometric approach to K-homology for Lie manifoldsApr 05 2019We prove that the computation of the Fredholm index for fully elliptic pseudodifferential operators on Lie manifolds can be reduced to the computation of the index of Dirac operators perturbed by smoothing operators. To this end we adapt to our framework ... More

Auslander-Reiten triangles and Grothendieck groups of triangulated categoriesApr 04 2019We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a cogenerator, then the category has only finitely many isomorphism classes of indecomposable objects ... More

Subtle characteristic classes for Spin-torsorsApr 03 2019Extending [10], we obtain an explicit description of the motivic cohomology with $\mathbb{Z}/2$-coefficients of the Nisnevich classifying space of the spin group $Spin_n$ associated to the standard split quadratic form in bidegrees $(*')[*]$ satisfying ... More

Homotopical and operator algebraic twisted K-theoryApr 03 2019Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K-theory, both in the real ... More

Multiplicative parametrized homotopy theory via symmetric spectra in retractive spacesApr 03 2019In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding infinity-categorical product and ... More

A subcomplex of Leibniz complexMar 30 2019Using the free graded Lie algebras we introduce a natural subcomlex of the Loday's complex of a Leibniz algebra. Our conjecture says for free Leibniz algebras, the complex is acyclic.

A subcomplex of Leibniz complexMar 30 2019Apr 06 2019Using the free graded Lie algebras we introduce a natural subcomlex of the Loday's complex of a Leibniz algebra. Our conjecture says, that for free Leibniz algebras, the complex is acyclic.

Massey products for graph homologyMar 28 2019This paper shows that the operad encoding modular operads is Koszul. Using this result we construct higher composition operations on (hairy) graph homology which characterize its rational homotopy type.

Cohomology and deformations of dendriform algebras, and $\mathrm{Dend}_\infty$-algebrasMar 28 2019A dendriform algebra is an associative algebra whose product splits into two binary operations and the associativity splits into three new identities. These algebras arise naturally from some combinatorial objects and through Rota-Baxter operators. In ... More

Diagonal reduction of matrices over Bezout rings of stable range 1 with the Kazimirsky conditionMar 24 2019We constuct the theory of diagonalizability for matrices over Bezout rings of stable range 1 with the Kazimirsky condition. It is shown that a ring of stable range 1 with the right (left) Kazimirsky condition is an elementary divisor ring if and only ... More

Ramond-Ramond fields and twisted differential K-theoryMar 21 2019We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, ... More