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Six-dimensional gauge theories and (twisted) generalized cohomologyAug 22 2019We consider the global aspects of the 6-dimensional $\mathcal{N}=(1, 0)$ theory arising from the coupling of the vector multiplet to the tensor multiplet. We show that the Yang-Mills field and its dual, when both are abelianized, combine to define a class ... More
C_2-equivariant stable homotopy from real motivic stable homotopyAug 22 2019We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant ... More
Colored five-vertex models and Lascoux polynomials and atomsAug 20 2019We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai [arXiv:1305.3030] and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson ... More
The Euler Characteristic Of A Transitive Lie AlgebroidAug 19 2019We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid $A$ over a compact manifold $M$ ... More
Abelian quandles and quandles with abelian structure groupAug 19 2019Sets with a self-distributive operation (in the sense of $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c)$), in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study ... More
Relative topological surgery exact sequence and additivity of relative higher rho invariantsAug 18 2019In this paper, we define the relative higher $\rho$ invariant for orientation preserving homotopy equivalence between manifolds with boundary in $K$-theory of relative obstruction algebra, i.e relative analytic structure group. We also show that the map ... More
On the Chow group of the self-product of a CM elliptic curve defined over a number fieldAug 18 2019In this note we study the cokernel of the restriction map from the Chow group of codimension 2 cycles on the spread of the self product of a CM-elliptic curve over the ring of integers of a number field to the codimension 2 cycles on the self of product ... More
On functional equations for Nielsen polylogarithmsAug 13 2019We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also ... More
A remark on algebraic cycles on cubic fourfoldsAug 13 2019In this short note we try to generalize the Clemens-Griffiths criterion of non-rationality for smooth cubic threefolds to the case of smooth cubic fourfolds.
Reduction of matrices over simple Ore domainsAug 13 2019We study the theory of diagonal reductions of matrices over simple Ore domains of finite stable range. We cover the cases of 2-simple rings of stable range 1, Ore domains and certain cases of Bezout domains.
Positive scalar curvature on stratified spaces, I: the simply connected caseAug 12 2019Let $M_\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $\beta M$ the singular locus and by $L$ the associated link. In this paper we study the problem of when such a space can be endowed with a wedge metric of positive ... More
The unit map of the algebraic special linear cobordism spectrumAug 11 2019In joint work with Elmanto, Hoyois, Khan and Sosnilo, we computed infinite $\mathbb{P}^1$-loop spaces of motivic Thom spectra, using the technique of framed correspondences. This result allows us to express non-negative $\mathbb{G}_m$-homotopy groups ... More
Groups with Spanier-Whitehead dualityAug 10 2019We introduce the notion of Spanier-Whitehead K-duality for a discrete group G, defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the ... More
First Betti numbers of orbits of Morse functions on surfacesAug 08 2019In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. ... More
Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairsAug 08 2019We develop a theory of \emph{modulus sheaves with transfers}, which generalizes Voevodsky's theory of sheaves with transfers. This repairs part of the flaw in a previous preprint by three of the authors (arXiv:1511.07124 [math.AG]).
On the cap product in Hochschild theoryAug 06 2019In this paper we give an axiomatic characterization of the cap product in the Hochschild theory of associative unital algebras which are projective over a commutative unital ring. We also give an interpretation of the cap product with coefficients in ... More
Modules over algebraic cobordismAug 06 2019We prove that the $\infty$-category of $\mathrm{MGL}$-modules over any scheme is equivalent to the $\infty$-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbb{P}^1$-loop spaces, we deduce ... More
Controlled Analytic Properties and the Quantitative Baum-Connes ConjectureAug 06 2019We show that the classical Baum-Connes assembly map is quantitatively an isomorphism for a class of lacunary hyperbolic groups, and we explain how to see that this class contains many examples of groups that contain graph sequences of large girth inside ... More
Locally finitely presented and coherent heartsAug 01 2019Given a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in a Grothendieck category $\mathcal{G}$, we study when the heart $\mathcal{H}_{\mathbf t}$ of the associated Happel-Reiten-Smalo $t$-structure in the derived category ${\mathbf D}(\mathcal{G})$ ... More
Linear homology in a nutshellJul 31 2019In 1985 Bayer and Billera defined a flag vector $f(X)$ for every convex polytope $X$, and proved some fundamental properties. The flag vectors $f(X)$ span a graded ring $\mathcal{R}=\bigoplus_{d\geq0}\mathcal{R}_d$. Here $\mathcal{R}_d$ is the span of ... More
An elementary study on realizable changes of homology groups of Reeb spaces of fold maps by fundamental surgery operationsJul 26 2019In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating their source ... More
Crystal structures for canonical Grothendieck functionsJul 26 2019We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. ... More
Enriched set-valued P-partitions and shifted stable Grothendieck polynomialsJul 24 2019We introduce an enriched analogue of Lam and Pylyavskyy's theory of set-valued $P$-partitions. An an application, we construct a $K$-theoretic version of Stembridge's Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this ... More
The $p$-completed cyclotomic trace in degree $2$Jul 24 2019We prove that for a quasi-regular semiperfectoid $\mathbb{Z}_p^{\rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;\mathbb{Z}_p)$ of $R$ to the topological cyclic homology ... More
The Riemann-Roch Theorem on higher dimensional complex noncommutative toriJul 24 2019We prove analogues of the Riemann-Roch Theorem and the Hodge Theorem for noncommutative tori (of any dimension) equipped with complex structures, and discuss implications for the question of how to distinguish "noncommutative abelian varieties" from "non-algebraic" ... More
Equivariant Dimensions of Graph C*-algebrasJul 23 2019We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^*$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For $C^*$-algebras of finite acyclic graphs and finite cycles, ... More
On weakly negative subcategories, weight structures, and (weakly) approximable triangulated categoriesJul 22 2019We prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated $C$ that is compactly generated by a single object $G$ is weakly approximable if $C(G,G[i])=0$ for $i>1$ (we say that $G$ is ... More
Notes on the filtration of the K-theory for abelian p-groupsJul 22 2019In this paper, I correct errors in my paper (Kodai Math. J. (2015)) about gamma filtrations for classifying spaces for abelian p-groups which are not elementary. We also extend Chetard's results for such 2-groups to p-groups for odd prime.
Galois theory and the categorical Peiffer commutatorJul 22 2019We show that the Peiffer commutator previously defined by Cigoli, Mantovani and Metere can be used to characterize central extensions of precrossed modules with respect to the subcategory of crossed modules in any semi-abelian category satisfying an additional ... More
Smooth Connes--Thom isomorphism, cyclic homology, and equivariant quantisationJul 21 2019Using a smooth version of the Connes--Thom isomorphism in Grensing's bivariant K-theory for locally convex algebras, we prove an equivariant version of the Connes--Thom isomorphism in periodic cyclic homology. As an application, we prove that periodic ... More
The Tamarkin--Tsygan calculus of an algebra a la StasheffJul 21 2019We show that the Tamarkin--Tsygan calculus of an associative algebra can be computed using a cofibrant replacement of it, by giving explicit formulas for the action of the 2-colored operad of such calculi on Hochschild (co)chains in terms of the chosen ... More
Up-to-homotopy algebras with strict units. (Extended abstract version.)Jul 19 2019We prove the existence of minimal models a la Sullivan for operads with nontrivial arity zero. So up-to-homotopy algebras with strict units are just operad algebras over these minimal models. As an application, we give another proof of the formality of ... More
A Quillen model structure on the category of Kontsevich-Soibelman weakly unital dg categoriesJul 18 2019In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital ... More
Characteristic classes of bundles of K3 manifolds and the Nielsen realization problemJul 17 2019Let $K$ be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller--Morita--Mumford classes for smooth bundles with fiber $K$ are non-zero. As a consequence, we fill a gap in a paper of the first author, and prove ... More
Exact sequence between real and complex bivariant K theories and application to the Z2 pairingJul 16 2019We give some formulas for the ZZ pairing in KO theory using a long exact sequence for bivariant K theory which links real and complex theories. This is discussed under the framework of real structures given by antilinear operators verifying some symmetries. ... More
A KK-theoretic perspective on deformed Dirac operatorsJul 14 2019We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of the form $\mathsf{D}+\mathrm{i}\mathsf{c}(X)$, where $\mathsf{c}(X)$ is a Clifford multiplication operator by an orbital vector field with respect to the action ... More
Two examples of vanishing and squeezing in $K_1$Jul 13 2019Aug 01 2019Controlled topology is one of the main tools for proving the isomorphism conjecture concerning the algebraic $K$-theory of group rings. In this article we dive into this machinery in two examples: when the group is infinite cyclic and when it is the infinite ... More
Two examples of vanishing and squeezing in $K_1$Jul 13 2019Controlled topology is one of the main tools for proving the isomorphism conjecture in algebraic $K$-theory. In this article we make an exposition of this machinery in two examples: the infinite cyclic group $\langle t \rangle$ with the trivial subgroup ... More
The $B_\infty$-structure on the derived endomorphism algebra of the unit in a monoidal categoryJul 13 2019Consider a monoidal category which is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a $B_{\infty}$-algebra which is $A_{\infty}$-quasi-isomorphic to the derived endomorphism algebra ... More
Quadratic Algebras arising from Hopf operads generated by a single elementJul 12 2019The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated ... More
The KO-valued spectral flow for skew-adjoint Fredholm operatorsJul 11 2019In this article we give a comprehensive treatment of a `Clifford module flow' along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO${}_{*}(\mathbb{R})$ via the Clifford index of Atiyah-Bott-Shapiro. We develop ... More
(Co)homology of Crossed Products by Weak Hopf AlgebrasJul 09 2019We obtain a mixed complex simpler than the canonical one the computes the type cyclic homologies of a crossed product with invertible cocycle $A\times_{\rho}^f H$, of a weak module algebra $A$ by a weak Hopf algebra $H$. This complex is provided with ... More
Homotopy equivalence in unbounded KK-theoryJul 09 2019We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $\overline{U\!K\!K}(A,B)$ ... More
Homotopy equivalence in unbounded KK-theoryJul 09 2019Jul 23 2019We propose a new notion of unbounded $K\!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $\sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $\overline{U\!K\!K}(A,B)$ ... More
Word operads and admissible orderingsJul 09 2019We use Giraudo's construction of combinatorial operads from monoids to offer a conceptual explanation of the origins of Hoffbeck's path sequences of shuffle trees, and use it to define new monomial orders of shuffle trees. One such order is utilised to ... More
Commutative Lie algebras and commutative cohomology in characteristic $2$Jul 08 2019We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
From weight structures to (orthogonal) $t$-structures and backJul 08 2019A $t$-structure $t=(C_{t\le 0},C_{t\ge 0})$ on a triangulated category $C$ is right adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$; then $t$ can be uniquely recovered from $w$ and vice versa. We prove that if $C$ ... More
Bökstedt periodicity and quotients of DVRsJul 08 2019In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for B\"okstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the ... More
Cyclic homology for bornological coarse spacesJul 05 2019We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse ... More
L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfacesJul 02 2019We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) ... More
Distinguishing open symplectic mapping tori via their wrapped Fukaya categoriesJul 02 2019In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_\phi$ associated to a Weinstein domain $M$, and ... More
Framed motivic $Γ$-spacesJun 30 2019We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines $\Gamma$-spaces and framed correspondences into ... More
Determinant map for the prestack of Tate objectsJun 30 2019We construct a map from the prestack of Tate objects over a field $k$ of characteristic $0$ to the stack of $\mathbb{G}_{\rm m}$-gerbes. The result is obtained by combining the determinant map from the stack of perfect complexes as proposed by Sch\"urg-To\"en-Vezzosi ... More
The third homology of $\mathrm{SL}_2(\mathbb{Q})$Jun 27 2019We calculate the third homology of $\mathrm{SL}_2(\mathbb{Q})$ with half-integral coefficients. Corresponding to each prime $p$ there is an operator on this group with square the identity. The kernel of the (split surjective) homomorphism to the indecomposable ... More
On the classification of group actions on C*-algebras up to equivariant KK-equivalenceJun 26 2019We study the classification of group actions on C*-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-algebra. We show that a conjecture of Izumi is equivalent ... More
On the classification of group actions on C*-algebras up to equivariant KK-equivalenceJun 26 2019Jul 04 2019We study the classification of group actions on C*-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-algebra. We show that a conjecture of Izumi is equivalent ... More
Algebraic cycles on hyperplane sections of hypersurfaces in $\mathbb P^n$ for $n=5,6$Jun 25 2019Let $X$ be a cubic hypersurface in $\mathbb P^6$ or a hypersurface of degree greater than equal to $7$ in $\mathbb P^5$. In this note we try to understand, for a very general hyperplane section of $X$, the non-injectivity locus of the corresponding push-forward ... More
Spectral algebras and non-commutative Hodge-to-de Rham degenerationJun 22 2019We revisit the non-commutative Hodge-to-de Rham Degeneration Theorem of the first author, and present its proof in a somewhat streamlined and improved form that explicitly uses spectral algebraic geometry. We also try to explain why topology is essential ... More
Tate-Shafarevich group and Selmer group constructions for Chow group of an abelian varietyJun 19 2019In this note we are going to define the notion of Tate-Shafarevich group and Selmer group of the Chow group of an abelian variety defined over a number field. We prove the weak Mordell-Weil theorem for the group of degree zero cycles on the abelian variety ... More
Representability of Chow groups of codimension three cyclesJun 19 2019In this note we are going to prove that if we have a fibration of smooth projective varieties $X\to S$ over a surface $S$ such that $X$ is of dimension four and that the geometric generic fiber has finite dimensional motive and the first \'etale cohomology ... More
Equivariant higher twisted K-theory of SU(n) for exponential functor twistsJun 19 2019We prove that each exponential functor on the category of finite-dimensional complex inner product spaces and isomorphisms gives rise to an equivariant higher (ie. non-classical) twist of $K$-theory over $G=SU(n)$. This twist is represented by a Fell ... More
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
Enough vector bundles on orbispacesJun 13 2019We show that every coarsely finite-dimensional orbispace with isotropy groups of bounded order has enough (finite-dimensional) vector bundles. It follows that the K-theory of finite-dimensional vector bundles on compact orbispaces is well behaved. Global ... 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 2019Jun 20 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
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
Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold mapsJun 03 2019Jun 19 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
K-theory formulas for orthogonal and symplectic orbit closuresJun 03 2019Jul 23 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
Homotopy of the space of initial values satisfying the dominant energy condition strictlyMay 31 2019The dominant energy condition imposes a restriction on initial value pairs found on a spacelike hypersurface of a Lorentzian manifold. In this article, we study the space of initial values that satisfy this condition strictly. To this aim, we introduce ... 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
On CW-complexes over groups with periodic cohomologyMay 28 2019Jul 18 2019If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Wall's D2 problem ... 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 20 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