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Let A be a local domain of characteristic 2 such that its residue field has more than 64 elements. Then we find an exact relation between the third integral homology of the group SL 2 (A) and Hutchinson's refined Bloch group RB (A). [ABSTRACT FROM AUTHOR]

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line S[t] as a functor defined on the 1-category of cyclotomic spectra with values in the1-category of spectra with Frobenius lifts, refining a result of Blumberg-Mandell. We define the notion of an integral topological Cartier module using Barwick's formalism of spectral Mackey functors on orbital1-categories, extending the work of Antieau-Nikolaus in the p-typical setting. As an application, we show that TR evaluated on a connective E1-ring admits a description in terms of the spectrum of curves on algebraic K-theory, generalizing the work of Hesselholt and Betley-Schlichtkrull. [ABSTRACT FROM AUTHOR]

In this paper, we review some new results we recently obtained about the infrared physics of 3d 풩=2 SQCD with a unitary gauge group, in particular in the presence of a nonzero Fayet–Iliopoulos parameter and with generic values of the Chern–Simons levels. We review the 3d GLSM (also known as 3d A-model) approach to the computation of the 3d 풩=2 twisted chiral ring of half-BPS lines. For particular values of the Chern–Simons levels, this twisted chiral ring has a neat interpretation in terms of the quantum K-theory (QK) of the Grassmannian manifold. We propose a new set of line defects of the 3d gauge theory, dubbed Grothendieck lines, which represent equivariant Schubert classes in the QK ring. In particular, we show that double Grothendieck polynomials, which represent the equivariant Chern characters of the Schubert classes, arise physically as Witten indices of certain quiver supersymmetric quantum mechanics. We also explain two distinct ways how to compute K-theoretic enumerative invariants using the 3d GLSM approach. [ABSTRACT FROM AUTHOR]

A family of quasi-1D Schrödinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibits changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed C∗-algebra. The quotient of this algebra by the ideal of compact operators is studied, and an index theorem is deduced from these investigations. This result corresponds to a topological version of Levinson’s theorem in the presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the last two sections of the paper, the K-theory of the main C∗-algebra and the dependence on an external parameter are carefully analyzed. In particular, a surface of resonances is exhibited, probably for the first time. The contents of these two sections are of independent interest, and the main result does not depend on them. [ABSTRACT FROM AUTHOR]

We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory. [ABSTRACT FROM AUTHOR]

We give applications of the higher Lefschetz theorems for foliations of Benameur and Heitsch (J. Funct. Anal. 259:131–173, 2010), primarily involving Haefliger cohomology. These results show that the transverse structures of foliations carry important topological and geometric information. This is in the spirit of the passage from the Atiyah–Singer index theorem for a single compact manifold to their families index theorem, involving a compact fiber bundle over a compact base. For foliations, Haefliger cohomology plays the role that the cohomology of the base space plays in the families index theorem. We obtain highly useful numerical invariants by paring with closed holonomy invariant currents. In particular, we prove that the non-triviality of the higher A ^ class of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions. We then construct a large collection of examples for which no such actions exist. Finally, we relate our results to Connes’ spectral triples, and prove useful integrality results. [ABSTRACT FROM AUTHOR]

We prove a purity property in telescopically localized algebraic K-theory of ring spectra: For n\geq 1, the T(n)-localization of K(R) only depends on the T(0)\oplus \dots \oplus T(n)-localization of R. This complements a classical result of Waldhausen in rational K-theory. Combining our result with work of Clausen–Mathew–Naumann–Noel, one finds that L_{T(n)}K(R) in fact only depends on the T(n-1)\oplus T(n)-localization of R, again for n \geq 1. As consequences, we deduce several vanishing results for telescopically localized K-theory, as well as an equivalence between K(R) and TC(\tau _{\geq 0} R) after T(n)-localization for n\geq 2. [ABSTRACT FROM AUTHOR]

In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that T(k)-local TR vanishes on connective L_n^{p,f}-acyclic \mathbb {E}_1-rings for every 1 \leq k \leq n and deduce consequences for connective Morava K-theory and the Thom spectra y(n). The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive \infty-categories which was recently established by Córdova Fedeli [ Topological Hochschild homology of adic rings , Ph.D. thesis, University of Copenhagen, 2023]. [ABSTRACT FROM AUTHOR]

We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable \infty-category of non-\mathbb {A}^1-invariant motivic spectra, which turns out to be equivalent to the \infty-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this \infty-category satisfies \mathbb {P}^1-homotopy invariance and weighted \mathbb {A}^1-homotopy invariance, which we use in place of \mathbb {A}^1-homotopy invariance to obtain analogues of several key results from \mathbb {A}^1-homotopy theory. These allow us in particular to define a universal oriented motivic \mathbb {E}_\infty-ring spectrum \mathrm {MGL}. We then prove that the algebraic K-theory of a qcqs derived scheme X can be recovered from its \mathrm {MGL}-cohomology via a Conner–Floyd isomorphism \[ \mathrm {MGL}^{**}(X)\otimes _{\mathrm {L}{}}\mathbb {Z}[\beta ^{\pm 1}]\simeq \mathrm {K}{}^{**}(X), \] where \mathrm {L}{} is the Lazard ring and \mathrm {K}{}^{p,q}(X)=\mathrm {K}{}_{2q-p}(X). Finally, we prove a Snaith theorem for the periodized version of \mathrm {MGL}. [ABSTRACT FROM AUTHOR]

Let X be a discrete metric space with bounded geometry. In this paper, we show that if X admits an "A-by-CE" coarse fibration, then the canonical quotient map λ : C max ∗ (X) → C ∗ (X) from the maximal Roe algebra to the Roe algebra of X , and the canonical quotient map λ : C u , max ∗ (X) → C u ∗ (X) from the maximal uniform Roe algebra to the uniform Roe algebra of X , induce isomorphisms on K -theory. A typical example of such a space arises from a sequence of group extensions { 1 → N n → G n → Q n → 1 } such that the sequence { N n } has Yu's property A, and the sequence { Q n } admits a coarse embedding into Hilbert space. This extends an early result of Špakula and Willett [Maximal and reduced Roe algebras of coarsely embeddable spaces, J. Reine Angew. Math. 678 (2013) 35–68] to the case of metric spaces which may not admit a coarse embedding into Hilbert space. Moreover, it implies that the maximal coarse Baum–Connes conjecture holds for a large class of metric spaces which may not admit a fibered coarse embedding into Hilbert space. [ABSTRACT FROM AUTHOR]

We compare two approaches which use K-theory for C*-algebras to classify symmetry protected topological phases of quantum systems described in the one particle approximation. In the approach by Kellendonk, which is more abstract and more general, the algebra remains unspecified and the symmetries are defined using gradings and real structures. In the approach by Alldridge et al., the algebra is physically motivated and the symmetries implemented by generators which commute with the Hamiltonian. Both approaches use van Daele's version of K-theory. We show that the second approach is a special case of the first one. We highlight the role played by two of the symmetries: charge conservation and spin rotation symmetry. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new ‘Thom class’ formulation of topological T-duality for principal torus bundles. This definition is equivalent to the established one of Bunke, Rumpf, and Schick but has the virtue of removing the global assumptions on the H-flux required in the old definition. With the new definition, we provide easier and more transparent proofs of the classification of T-duals and generalize the local formulation of T-duality for circle bundles by Bunke, Schick, and Spitzweck to the torus case. [ABSTRACT FROM AUTHOR]

We realize the number of bound states of a Schrödinger operator on ℝn as an index pairing in all dimensions. Expanding on ideas of Guillopé and others, we use high-energy corrections to find representatives of the K-theory class of the scattering operator. These representatives allow us to compute the number of bound states using an integral formula involving heat kernel coefficients. [ABSTRACT FROM AUTHOR]

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with F p -coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and ⊗ has no zero-divisors. It provides a large supply of new points for the Balmer spectrum of the Voevodsky motivic category. We also prove the Morava K-theory version of the above result, which permits to construct plenty of new points for the Balmer spectrum of the Morel-Voevodsky A 1 -stable homotopy category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem. [ABSTRACT FROM AUTHOR]

Inspired by a remarkable work of Félix, Halperin, and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for torsion rank of homotopy groups, one using ordinary homology and one using K -theory. We use these to obtain explicit quantitative asymptotic lower bounds on the torsion rank of the homotopy groups for many interesting spaces after suspension, including Moore spaces, Eilenberg–MacLane spaces, complex projective spaces, complex Grassmannians, Milnor hypersurfaces, and unitary groups. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the equivariant and ordinary Grothendieck ring and the equivariant and ordinary topological K$K$‐ring of flag Bott manifolds of the general Lie type. This will generalize the results on the equivariant and ordinary cohomology of flag Bott manifolds of the general Lie type due to Kaji, Kuroki, Lee, and Suh. [ABSTRACT FROM AUTHOR]

In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in [3] that R ζ = c ζ 2 where R ζ is their map on K 3 defined using the cyclic quantum dilogarithm and c ζ is the Chern class map on K 3. [ABSTRACT FROM AUTHOR]

Let K be a finite extension of the p -adic numbers Q p with ring of integers O K and residue field κ. Let X a regular scheme, proper, flat, and geometrically irreducible over O K of dimension d , and X K its generic fiber. We show, under some assumptions on X , that there is a reciprocity isomorphism of locally compact groups H a r 2 d − 1 (X K , Z (d)) ≃ π 1 a b (X K) W from the cohomology theory defined in [10] to an integral model π 1 a b (X K) W of the abelianized fundamental group π 1 a b (X K). After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups. The key ingredient is the duality result in [10]. [ABSTRACT FROM AUTHOR]

We introduce and investigate some examples of C ∗ -algebras which are related to multiplication maps in the ring of p-adic integers. We find ideals within these algebras and use the corresponding short exact sequences to compute the K-theory. [ABSTRACT FROM AUTHOR]

Using the root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of T(2)_*\mathrm {K}(ku) for p>3. Through this, we also produce a new algebraic K-theory computation; namely we obtain T(2)_*\mathrm {K}(ku/p), where ku/p is the 2-periodic Morava K-theory spectrum of height 1. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and let ρ : ℤ / p → GL n ⁡ (ℤ) be an action of ℤ / p on a lattice and let Γ := ℤ n ⋊ ρ ℤ / p be the corresponding semidirect product. The torus bundle M := T ρ n × ℤ / p S ℓ over the lens space S ℓ / ℤ / p has fundamental group Γ. When ℤ / p fixes only the origin of ℤ n , Davis and Lück (2021) compute the L-groups L m 〈 j 〉 ⁢ (ℤ ⁢ [ Γ ]) and the structure set 풮 geo , s ⁢ (M) . In this paper, we extend these computations to all actions of ℤ / p on ℤ n . In particular, we compute L m 〈 j 〉 ⁢ (ℤ ⁢ [ Γ ]) and 풮 geo , s ⁢ (M) in a case where E ¯ ⁢ Γ has a non-discrete singular set. [ABSTRACT FROM AUTHOR]

Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group C*-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style. [ABSTRACT FROM AUTHOR]

We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real K$K$‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of TMF$\mathrm{TMF}$ is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of TMF$\mathrm{TMF}$ and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group. [ABSTRACT FROM AUTHOR]

In the early 1940s, P. A. Smith showed that if a finite p-group G acts on a finite dimensional complex X that is mod p acyclic, then its space of fixed points, XG, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a G-space X is a equivariant homotopy retract of the p-localization of a based finite G-C.W. complex, given H < G and n, what is the smallest r such that if XH is acyclic in the (n+r)th Morava K-theory, then XG must be acyclic in the nth Morava K-theory? Barthel et. al. then answered this when G is abelian, by finding general lower and upper bounds for these "blue shift" numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equiv- alent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod p homology, and thus applies to all finite dimensional G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for K(n)*-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava K-theories of all real Grassmanians. [ABSTRACT FROM AUTHOR]

In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted K-theory of connected, simply-connected and simple compact Lie groups G, in both equivariant and nonequivariant settings. [ABSTRACT FROM AUTHOR]

We construct a natural generalization of the Grothendieck group K0 to the case of possibly unpointed categories admitting pushouts by using the concept of heaps recently introduced by Brezinzki. In case of a monoidal category, the defined K0 is shown to be a truss. It is shown that the construction generalizes the classical K0 of an abelian category as the group retract along the isomorphism class of the zero object. We finish by applying this construction to construct the integers with addition and multiplication as the decategorification of finite sets and show that in this K0(Top̲) one can identify a CW-complex with the iterated product of its cells. [ABSTRACT FROM AUTHOR]

An error in the original paper is identified and corrected. The |$\textrm {C}^{\ast }$| -algebras with approximately inner flip, which satisfy the UCT, are identified (and turn out to be fewer than what is claimed in the original paper). The action of the flip map on K-theory turns out to be more subtle, involving a minus sign in certain components. To this end, we introduce new geometric resolutions for |$\textrm {C}^{\ast }$| -algebras, which do not involve index shifts in K-theory and thus allow for a more explicit description of the quotient map in the Künneth formula for tensor products. [ABSTRACT FROM AUTHOR]

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve |$E$| in |$\mathbb P^{2}$| and blow up nine general points on |$E$|⁠. Then the complement |$M$| of the strict transform of |$E$| in the blow-up has countably many algebraic structures. Moreover, each algebraic structure comes from an embedding of |$M$| into a blow-up of |$\mathbb P^{2}$| in nine points lying on an elliptic curve |$F\not \simeq E$|⁠. We classify algebraic structures on |$M$| using a Hopf transform : a way of constructing a new surface by cutting out an elliptic curve and pasting a different one. Next, we introduce the notion of an analytic K-theory of varieties. Manipulations with the example above lead us to prove that classes of all elliptic curves in this K-theory coincide. To put in another way, all motivic measures on complex algebraic varieties that take equal values on biholomorphic varieties do not distinguish elliptic curves. [ABSTRACT FROM AUTHOR]

We derive cancellation-free Chevalley-type multiplication formulas for the T -equivariant quantum K -theory ring of Grassmannians of type A and C , and also those of two-step flag manifolds of type A. They are obtained based on the uniform Chevalley formula in the T -equivariant quantum K -theory ring of arbitrary flag manifolds G / B , which was derived earlier in terms of the quantum alcove model, by the last three authors. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the topological K-ring in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K-ring, and hence the Grothendieck ring of algebraic vector bundles over flag Bott–Samelson varieties. [ABSTRACT FROM AUTHOR]

We consider the semi-direct products |$G={\mathbb{Z}}^{2}\rtimes GL_{2}({\mathbb{Z}}), {\mathbb{Z}}^{2}\rtimes SL_{2}({\mathbb{Z}})$|⁠ , and |${\mathbb{Z}}^{2}\rtimes \Gamma (2)$| (where |$\Gamma (2)$| is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum–Connes conjecture, namely the equivariant |$K$| -homology of the classifying space |$\underline{E}G$| for proper actions on the left-hand side, and the analytical K-theory of the reduced group |$C^{*}$| -algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for |$\underline{E}G$|⁠ , which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in |$G$|⁠ , leading to an extensive study of the wallpaper groups associated with finite subgroups. For the first and third groups, the computations in |$K_{0}$| provide explicit generators that are matched by the Baum–Connes assembly map. [ABSTRACT FROM AUTHOR]

We analyze topological invariants, in particular Z 2 invariants, which characterize time reversal invariant topological insulators, in the framework of index theory and K-theory. After giving a careful study of the underlying geometry and K-theory, we formalize topological invariants as elements of KR theory. To be precise, the strong topological invariants lie in the higher KR groups of spheres; K R ̃ − j − 1 ( S D + 1 , d ). Here j is a KR-cycle index, as well as an index counting off the Altland-Zirnbauer classification of Time Reversal Symmetry (TRS) and Particle Hole Symmetry (PHS)—as we show. In this setting, the computation of the invariants can be seen as the evaluation of the natural pairing between KR-cycles and KR-classes. This fits with topological and analytical index computations as well as with Poincaré Duality and the Baum–Connes isomorphism for free Abelian groups. We provide an introduction starting from the basic objects of real, complex and quaternionic structures which are the mathematical objects corresponding to TRS and PHS. We furthermore detail the relevant bundles and K-theories (Real and Quaternionic) that lead to the classification as well as the topological setting for the base spaces. [ABSTRACT FROM AUTHOR]

This is the second of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to C*-algebraic Morita equivalence, and the verification of the Connes–Kasparov conjecture in operator K-theory for these groups. In Part I we presented the Morita equivalence and the Connes–Kasparov morphism. In this part we shall compute the morphism using David Vogan's description of the tempered dual. In fact we shall go further by giving a complete representation-theoretic description and parametrization, in Vogan's terms, of the essential components of the tempered dual, which carry the K-theory of the tempered dual. [ABSTRACT FROM AUTHOR]

This is the first of two papers dedicated to the detailed determination of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and a new and very explicit proof of the Connes–Kasparov conjecture for these groups using representation theory. In this part we shall give details of the C*-algebraic Morita equivalence and then explain how the Connes–Kasparov morphism in operator K-theory may be computed using what we call the Matching Theorem, which is a purely representation-theoretic result. We shall prove our Matching Theorem in the sequel, and indeed go further by giving a simple, direct construction of the components of the tempered dual that have non-trivial K-theory using David Vogan's approach to the classification of the tempered dual. [ABSTRACT FROM AUTHOR]

In this paper, we exhibit two unital, separable, nuclear C∗${\rm C}^*$‐algebras of stable rank one and real rank zero with the same ordered scaled total K‐theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K‐theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of C∗${\rm C}^*$‐algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the C∗${\rm C}^*$‐algebras of stable rank one and real rank zero. [ABSTRACT FROM AUTHOR]

Placing a Dirac--Schrödinger operator along the orbit of a flow on a compact manifoldM defines an R-equivariant spectral triple over the algebra of smooth functions on M. We study some of the properties of these triples, with special attention to their zeta functions. These zeta functions are defined for Re(s) > 1 by Trace(fpH-8), where fp is the uniformly continuous function on the real line obtained by restricting the continuous or smooth function f on M to the orbit of a point p ∈ M, and H = -∂²/∂x² + x² is the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions are related to ergodic time averages in dynamics. In the case of the periodic flow on the circle, one obtains a spectral triple over the smooth irration torus A∞h ⊂ Ah already studied by Lesch and Moscovici. We strengthen a result of these authors, showing that the zeta function Trace(aH-8) extends meromorphically to C for any element a of the C*-algebra Ah. Another variant of our construction yields a spectral cycle for Ah ⊗ A1/h and a spectral triple over a suitable subalgebra with the meromorphic continuation property if h satisfies a Diophantine condition. The class of this cycle defines a fundamental class in the sense that it determines a KK-duality between Ah and A1/h. We employ the local index theorem of Connes and Moscovici in order to elaborate an index theorem of Connes for certain classes of differential operators on the line and compute the intersection form on K-theory induced by the fundamental class. [ABSTRACT FROM AUTHOR]

Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the K-theory of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group K_0 with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups. [ABSTRACT FROM AUTHOR]

Recently, Ludewig and Thiang introduced a notion of a uniformly localized Wannier basis with localization centers in an arbitrary uniformly discrete subset D in a complete Riemannian manifold X. They show that, under certain geometric conditions on X, the class of the orthogonal projection onto the span of such a Wannier basis in the K-theory of the Roe algebra C*(X) is trivial. In this paper, we clarify the geometric conditions on X, which guarantee triviality of the K-theory class of any Wannier projection. We show that this property is equivalent to triviality of the unit of the uniform Roe algebra of D in the K-theory of its Roe algebra, and provide a geometric criterion for that. As a consequence, we prove triviality of the K-theory class of any Wannier projection on a connected proper measure space X of bounded geometry with a uniformly discrete set of localization centers. [ABSTRACT FROM AUTHOR]

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds Ug,1n:=#g(Sn×Sn+1)∖int(D2n+1)$U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}(D^{2n+1})$, for large g$g$ and n$n$, up to degree n−3$n-3$. The answer is that it is a free graded commutative algebra on an appropriate set of Miller–Morita–Mumford classes. Our proof goes through the classical three‐step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, and (c) use pseudoisotopy theory and algebraic K$K$‐theory to get at actual diffeomorphism groups. [ABSTRACT FROM AUTHOR]

In this paper, for 1 < p < ∞, the authors show that the coarse ℓp-Novikov conjecture holds for metric spaces with bounded geometry which are coarsely embeddable into a Banach space with Kasparov-Yu's Property (H). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a strongly quasi-local version of the coarse Novikov conjecture, which states that a certain assembly map from the coarse Khomology of a metric space to the K-theory of its strongly quasi-local algebra is injective. We prove that the conjecture holds for metric spaces with bounded geometry which can be coarsely embedded into Banach spaces with Property (H), as introduced by Kasparov and Yu. We also generalize the notion of strong quasi-locality to proper metric spaces and provide a (strongly) quasi-local picture for K-homology. [ABSTRACT FROM AUTHOR]

We develop the motivic integration theory over formal Deligne-Mumford stacks over a power series ring of arbitrary characteristic. This is a generalization of the corresponding theory for tame and smooth Deligne-Mumford stacks constructed in earlier papers of the author. As an application, we obtain the wild motivic McKay correspondence for linear actions of arbitrary finite groups, which has been known only for cyclic groups of prime order. In particular, this implies the motivic version of Bhargava's mass formula as a special case. In fact, we prove a more general result, the invariance of stringy motives of (stacky) log pairs under crepant morphisms. [ABSTRACT FROM AUTHOR]

We give an elementary calculation of the internal hom from Kähler differentials to Kähler differentials, and from Milnor K-theory to Kähler differentials in the category of presheaves with transfers. This answers a question of Merici and Saito. [ABSTRACT FROM AUTHOR]

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the K -theory of the reduced C ∗ -algebra of the group. We show the power of this method giving explicit computations for the group SL 2 (ℤ [ i ]). In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions. [ABSTRACT FROM AUTHOR]

We initiate the study of K$K$‐theory Soergel bimodules, a global and K$K$‐theoretic version of Soergel bimodules. We show that morphisms of K$K$‐theory Soergel bimodules can be described geometrically in terms of equivariant K$K$‐theoretic correspondences between Bott–Samelson varieties. We thereby obtain a natural categorification of K$K$‐theory Soergel bimodules in terms of equivariant coherent sheaves. We introduce a formalism of stratified equivariant K$K$‐motives on varieties with an affine stratification, which is a K$K$‐theoretic analog of the equivariant derived category of Bernstein–Lunts. We show that Bruhat‐stratified torus‐equivariant K$K$‐motives on flag varieties can be described in terms of chain complexes of K$K$‐theory Soergel bimodules. Moreover, we propose conjectures regarding an equivariant/monodromic Koszul duality for flag varieties and the quantum K$K$‐theoretic Satake. [ABSTRACT FROM AUTHOR]

For a connected reductive group |$G$| and an affine smooth |$G$| -variety |$X$| over the complex numbers, the localization functor takes |$\mathfrak{g}$| -modules to |$D_{X}$| -modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson–Ginzburg, and show that the localizations of Harish-Chandra |$(\mathfrak{g}, K)$| -modules onto |$X = H \backslash G$| have regular holonomic cohomologies when |$H, K \subset G$| are both spherical reductive subgroups. The relative Lie algebra homologies and |$\operatorname{Ext}$| -branching spaces for |$(\mathfrak{g}, K)$| -modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler–Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included. [ABSTRACT FROM AUTHOR]

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C*-algebras that are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities for the |$K_0$| -group, the |$K_1$| -group, and the tracial state space of such an algebra. We show that these C*-algebras satisfy the Universal Coefficient Theorem, which is new even for the already known example of an exact C*-algebra nonisomorphic to its opposite algebra produced in an earlier work. [ABSTRACT FROM AUTHOR]

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$ - and $\mathrm {T}(n)$ -local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$ , then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$ , which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$ , we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$ -localized K-theory, showing that they coincide for any $p$ -invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$. [ABSTRACT FROM AUTHOR]

Let A be an associative ring and M a finitely generated projective A-module. We introduce a category RBS (M) and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories RBS (M) naturally arise as generalisations of the exit path ∞ -category of the reductive Borel–Serre compactification of a locally symmetric space, and one of our main techniques is to find purely categorical analogues of some familiar structures in these compactifications. [ABSTRACT FROM AUTHOR]