Mathematics - K-Theory and Homology Publications (50)

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Mathematics - K-Theory and Homology Publications

We prove a {\Gamma}-equivariant version of the algebraic index theorem, where {\Gamma} is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Read More


We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any elliptic curve $A$ in any characteristic. Read More


We give an algorithm to compute the $K$-groups of the crossed product by the flip automorphism for a nuclear C$^*$-algebra satisfying the UCT. Read More


Let $R$ be a regular domain of dimension $d\geq 2$ which is essentially of finite type over an infinite perfect field $k$. We compare the Euler class group $E^d(R)$ with the van der Kallen group $Um_{d+1}(R)/E_{d+1}(R)$. In the case $2R=R$, we define a map from $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$ and study it in intricate details. Read More


This text is based on a talk by the first named author at the first congress of the SMF (Tours, 2016). We present Bloch's conductor formula, which is a conjectural formula describing the change of topology in a family of algebraic varieties when the parameter specialises to a critical value. The main objective of this paper is to describe a general approach to the resolution of Bloch's conjecture based on techniques from both non-commutative geometry and derived geometry. Read More


Squier introduced a homotopical method in order to describe all the relations amongst rewriting reductions of a confluent and terminating string rewriting system. From a string rewriting system he constructed a $2$-dimensional combinatorial complex whose $2$-cells are generated by relations induced by the rewriting rules. When the rewriting system is confluent and terminating, the homotopy of this complex can be characterized in term of confluence diagrams induced by the critical branchings of the rewriting system. Read More


We define and study the quantum equivariant $K$-theory of cotangent bundles over the Grassmannians. For every tautological bundle in the $K$-theory we define its one-parametric deformation, referred to as quantum tautological bundle. We prove that the spectrum of operators of quantum multiplication by these quantum classes is governed by the Bethe ansatz equations for the inhomogeneous $XXZ$ spin chain. Read More


We prove the Burghelea Conjecture for all groups with finite virtual cohomological dimension. In particular, we proved the Burghelea Conjecture in its original formulation Read More


Given a graph of C*-algebras, we prove a long exact sequence in KK-theory for both the maximal and the vertex-reduced fundamental C*-algebras in the presence of possibly non GNS-faithful conditional expectations. We deduce from it the KK-equivalence between the full fundamental C*-algebra and the vertex-reduced fundamental C*-algebra even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Pimsner, Germain and Thomsen. Read More


We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Read More


Let $\varphi$ be a rational map $\mathbb{P}^2 \dashrightarrow\mathbb{P}^2$ which preserves the rational volume form $\frac{\mathrm{d}x}{x}\wedge\frac{\mathrm{d}y}{y}$. Sergey Galkin conjectured that in this case $\varphi$ is necessarily birational. We show that such a map preserves the element $\{x,y\}$ of the second K-group $K_2(\mathbf{k}(x,y))$ up to multiplication by a constant, and restate this condition explicitly in terms of mutual intersections of the divisors of coordinates of $\varphi$ in a way suitable for computations. Read More


We calculate the bivariant local cyclic cohomology of the Banach convolution algebra of summable functions on a word-hyperbolic group. Our result implies that the Banach algebraic assembly map in local cyclic homology is an isomorphism for such a group. Read More


We construct connections on $S^1$-equivariant Hamiltonian Floer cohomology, which differentiate with respect to certain formal parameters. Read More


We discuss a conjecture saying that derived equivalence of smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection $X$ of three quadrics in ${\mathbf P}^5$ and the corresponding double cover $Y \to {\mathbf P}^2$ branched over a sextic curve. Read More


We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. We also explain how this makes available the entire range of tools that comes with a Quillen homology theory, such as long exact sequences (transitivity) and excision isomorphisms (flat base change). Read More


We introduce the notions of a $\mathbf{D}$-standard abelian category and a $\mathbf{K}$-standard additive category. We prove that for a finite dimensional algebra $A$, its module category is $\mathbf{D}$-standard if and only if any derived autoequivalence on $A$ is standard, that is, given by a two-sided tilting complex. We prove that if the subcategory of projective $A$-modules is $\mathbf{K}$-standard, then the module category is $\mathbf{D}$-standard. Read More


We develop a $K$-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author. Read More


Young tableaux carry an associative product, described by the Schensted algorithm. They thus form a monoid $\mathbf{Pl}$, called \emph{plactic}. It is central in numerous combinatorial and algebraic applications. Read More


We show that, for a simplicial complex, the supported cap product operation on Borel-Moore homology coincides with the supported cap product on simplicial homology. For this we introduce the supported cap product for locally finite singular homology, and compare the cap product on the three homology theories. Read More


Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients -- which usually is that of abelian groups -- is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory. Read More


We establish new structures on Grothendieck-Witt rings, including a GW(k)-module structure on the unit group GW(k)^x and a presentation of \ul{GW}^x as an infinite Gm-loop sheaf. Even though our constructions are motivated by speculations in stable A1-homotopy theory, our arguments are purely algebraic. Read More


In this short note, we simply collect some known results about representing algebraic cycles by various kind of "nice" (e.g. smooth, local complete intersection, products of local complete intersection) algebraic cycles, up to rational equivalence. Read More


We pursue the study of local index theory for operators of Fourier-integral type associated to non-proper and non-isometric actions of Lie groupoids, initiated in a previous work. We introduce the notion of geometric cocycles for Lie groupoids, which allow to represent fairly general cyclic cohomology classes of the convolution algebra of Lie groupoids localized at isotropic submanifolds. Then we compute the image of geometric cocycles localized at units under the excision map of the fundamental pseudodifferential extension. Read More


We calculate homotopy G-fixed points and homotopy G-orbits in various types of *-categories. To this end we construct good model category structures on the category of *-categories and C-linear versions. For our calculation we then use injective or projective model category structure on the category G-equivariant *-categories and explicite (co)fibrant resolutions. Read More


Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum $R$ whose graded homotopy ring $\pi_*R$ is concentrated in dimensions divisible by a natural number $N \geq 5$ and has homological dimension at most three, the homotopy category of $R$-modules is equivalent to the derived category of $\pi_*R$. The Johnson-Wilson spectrum $E(3)$ and the truncated Brown-Peterson spectrum $BP\langle 2 \rangle$ for any prime $p \geq 5$ are our main examples. Read More


We revisit the characterisation of modules over non-unital C*-algebras analogous to sections of vector bundles. Using ideas from Kajiwara, Pinzari and Watatani, we identify a new class of modules which closely mirror the commutative case. Read More


K-theory for $ \sigma$-C*-algebras (countable inverse limits of C*-algebras) has been investigated by N. C. Phillips [{\it K-Theory} {\bf 3} (1989), 441--478]. Read More


In this article, using the recent theory of noncommutative motives, we compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely "fixed-point data". As a consequence, we recover, in a unified and conceptual way, the original results of Vistoli concerning algebraic K-theory, of Baranovsky concerning cyclic homology, of the second author with Polishchuk concerning Hochschild homology, and of Baranovsky-Petrov and Caldararu-Arinkin (unpublished) concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned computation is new in the literature. As an application, we verify Grothendieck's standard conjectures of type C and D, as well as Voevodsky's smash-nilpotence conjecture, in the case of "low-dimensional" orbifolds. Read More


A signed version of Putnam homology for Smale spaces is introduced. Its definition, basic properties and associated Lefschetz theorem are outlined. In particular, zeta functions associated to an Axiom A diffeomorphism are compared. Read More


Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov's $KK$-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Read More


In this short note, making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite. Read More


We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$ local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum. Read More


Given a filtration of a commutative monoid $A$ in a symmetric monoidal stable model category $\mathcal{C}$, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of $A$, and whose output is the higher order topological Hochschild homology of $A$. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring $R$, we get an upper bound on the size of the $THH$-groups of $E_{\infty}$-ring spectra $A$ such that $\pi_*(A) \cong R$. Read More


This note was prepared after the workshop on cdh descent and algebraic $K$-theory in Hara-mura, Japan (1 - 5 Sept. 2016), to complement the author's talk on the history and applications of pro cdh descent. In the final section pro cdh descent is used to define $K$-groups of rigid analytic varieties. Read More


We give a proof of Bott periodicity for real graded $C^\ast$-algebras in terms of K- theory and E-theory. Guentner and Higson proved a similar result in the complex graded case but we extend this to cover all graded $C^\ast$-algebras. We obtain the 8-fold periodicity in E-theory by constructing two maps that are inverse to each other. Read More


Periodic Hamiltonians on a three dimensional lattice which have a spectral gap not only on the bulk but also on two edges at the common Fermi level are considered. By using K-theory applied for quarter-plane Toeplitz algebras, two topological invariants are defined for such gapped Hamiltonians. One is defined for the bulk and edges, and the other corresponds to wave functions localized near the corner. Read More


Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a surjective ring homomorphism from the Grothendieck ring of varieties over F. Read More


We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and G\'alvez-Carrillo, Kock and Tonks. Examples of relative $2$-Segal spaces include the categorified unoriented cyclic nerve, real pseudo-holomorphic polygons in almost complex manifolds and the $\mathcal{R}_{\bullet}$-construction from Grothendieck-Witt theory. We show that a relative $2$-Segal space defines a categorical representation of the Hall algebra associated to the base $2$-Segal space. Read More


We apply controlled (or quantitative) $K$-theory to prove that a certain $L_p$ assembly map is an isomorphism for $p\in(1,\infty)$ when a countable discrete group $\Gamma$ acts with finite dynamic asymptotic dimension on a compact Hausdorff space $X$. When $p=2$, this is a model for the Baum-Connes assembly map for $\Gamma$ with coefficients in $C(X)$, and was shown to be an isomorphism by Guentner, Willett, and Yu. As a consequence, we see that the $K$-theory of the $L_p$ reduced crossed product is independent of $p\in (1,\infty)$ under the assumption of finite dynamic asymptotic dimension. Read More


Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Herv\'e Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative $K$-theory for the class of algebras of bounded linear operators on subquotients (i.e. Read More


We prove that algebraic K-theory satisfies `pro-descent' for abstract blow-up squares of noetherian schemes. As an application we derive Weibel's conjecture on the vanishing of negative K-groups. Read More


For simplicial modules, Eilenberg-Zilber's classical theorem states the existence of a product $sh : M\otimes N\to M\times N$ (the shuffle) and a coproduct $AW : M\times N\to M\otimes N$ (the Alexander-Whitney map), which are quasi-inverse of eachother. A cyclic version of this theorem was established in 1987 by Hood and Jones: they proved that $sh$ and $AW$ admit "coextensions" $sh_\infty$ and $AW_\infty$, using an acyclic-model method. Besides, an explicit formula for $sh_\infty$ has been discovered by several authors. Read More


The bulk-edge correspondence for two-dimensional type A topological insulators and topological superconductors is proved by using the cobordism invariance of the index. The idea of G. M. Read More


In this paper, we construct Chern classes from the relative K-group of a modulus pair to the relative motivic cohomology defined by Binda-Saito using algebraic cycles with modulus. Read More


We compute the $L_\infty$-theoretic dimensional reduction of the F1/D$p$-brane super $L_\infty$-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient $L_\infty$-algebras are naturally related by an $L_\infty$-isomorphism which we find to act on the super $p$-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between $K^0$ and $K^1$, rationally. Moreover, we show that these $L_\infty$-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra". Read More


We discuss Base Change functoriality for mod p eigenforms for GL(2) over number fields. We carry out systematic computer experiments and collect data supporting its existence in cases of field extensions K/F where F is imaginary quadratic and K is CM quartic. Read More


In this paper we extend Badzioch's, Dorabiala's, and Williams' definition of cohomological higher smooth torsion to a twisted cohomological higher torsion invariant. Additionally, we show that this still satisfies geometric additivity and transfer, and will also satisfy additivity and transfer for coefficients. Read More


We show that for finite groups the Loday assembly map with coefficients in finite fields is in general not injective. Read More


We provide some new computations of Farrell--Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell-Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise. We provide an algorithm computing an efficient subdivision of a complex to achieve this rigidity property. Read More