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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]

The endomorphism ring of the projective plane over a field F of characteristic neither two nor three is slightly more complicated in the Morel–Voevodsky motivic stable homotopy category than in Voevodsky's derived category of motives. In particular, it is not commutative precisely if there exists a square in F which does not admit a sixth root. A byproduct of these computations is a proof of Suslin's conjecture on the Suslin–Hurewicz homomorphism from Quillen to Milnor K-theory in degree four, based on work of Asok et al. (Invent Math 219:39-73, 2020). [ABSTRACT FROM AUTHOR]

We study a symplectic variant of algebraic K-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of Q . We compute this action explicitly. The representations we see are extensions of Tate twists Z p (2 k - 1) by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties. [ABSTRACT FROM AUTHOR]

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in Pic (X) ⊗ C . We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples. [ABSTRACT FROM AUTHOR]

We construct a zig–zag from the once delooped space of pseudoisotopies of a closed 2n-disc to the once looped algebraic K-theory space of the integers and show that the maps involved are p-locally (2 n - 4) -connected for n > 3 and large primes p. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik–Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa's stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of BDiff ∂ (D 2 n + 1) in degrees up to 2 n - 5 . [ABSTRACT FROM AUTHOR]

We study the étale sheafification of algebraic K-theory, called étale K-theory. Our main results show that étale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories. Consequently, we show that étale K-theory has surprisingly well-behaved properties, integrally and without finiteness assumptions. A key theoretical ingredient is the distinction, which we investigate in detail, between sheaves and hypersheaves of spectra on étale sites. [ABSTRACT FROM AUTHOR]

In a 2006 article Schlichting conjectured that the negative K–theory of any abelian category must vanish. This conjecture was generalized in a 2019 article by Antieau, Gepner and Heller, who hypothesized that the negative K–theory of any category with a bounded t–structure must vanish. Both conjectures will be shown to be false. [ABSTRACT FROM AUTHOR]

In this paper we obtain a localization formula in differential K-theory for S 1 -actions. We establish a localization formula for equivariant η -invariants by combining this result with our extension of Goette's result on the comparison of two types of equivariant η -invariants. An important step in our approach is to construct a pre- λ -ring structure in differential K-theory. [ABSTRACT FROM AUTHOR]

Let p ∈ Z be an odd prime. We prove a spectral version of Tate–Poitou duality for the algebraic K-theory spectra of number rings with p inverted. This identifies the homotopy type of the fiber of the cyclotomic trace K (O F) p ∧ → T C (O F) p ∧ after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of Z . [ABSTRACT FROM AUTHOR]

:PR->PS ht exists and is exact. We define a category HT ht as follows. Which is exact in the abelian category HT ht and in which the left-hand term has size 1. We claim that the stronger statement that all submodules of a finitely generated projective left HT ht -module again are finitely generated projective holds. It is a sequence of HT ht -modules, by the argument in [[19], pp. 71-72]. [Extracted from the article]

We show that an old conjecture of A. A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree n to Milnor K-theory in degree n admits an interpretation in terms of unstable A 1 -homotopy sheaves of the general linear group. Using this identification, we establish Suslin's conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3. We do this by linking the relevant unstable A 1 -homotopy sheaf of the general linear group to the stable A 1 -homotopy of motivic spheres. [ABSTRACT FROM AUTHOR]

We show that the Levine–Weibel Chow group of 0-cycles CH d (A) of a reduced affine algebra A of dimension d ≥ 2 over an algebraically closed field is torsion-free. Among several applications, it implies an affirmative solution to an old conjecture of Murthy in classical K-theory. [ABSTRACT FROM AUTHOR]

We prove that Thompson's group V is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman–Thompson groups V n , r with the homology of the zeroth component of the infinite loop space of the mod n - 1 Moore spectrum. As V = V 2 , 1 , we can deduce that this group is acyclic. Our proof involves establishing homological stability with respect to r, as well as a computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of any type n. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real K-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $$2n \ge 6$$ , the natural KO-orientation from the infinite loop space of the Madsen-Tillmann-Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any 2 n-dimensional spin manifold. For manifolds of odd dimension $$2n+1 \ge 7$$ , we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov-Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah-Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen-Weiss theorem, proven by Galatius and the third named author. [ABSTRACT FROM AUTHOR]

The cohomology theory known as $$\mathrm{Tmf}$$ , for 'topological modular forms,' is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from $$\mathrm{Tmf}$$ with level structure to forms of $$K$$ -theory. In particular, this allows us to construct a connective spectrum $$\mathrm{tmf}_0(3)$$ consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-étale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces $$\mathrm{Tmf}$$ with level structure. [ABSTRACT FROM AUTHOR]

We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle on the closed fibre lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme. [ABSTRACT FROM AUTHOR]

In this article I study a number of topological and algebraic dimension type properties of simple C-algebras and their interplay. In particular, a simple C-algebra is defined to be (tracially) $(m,\bar{m})$-pure, if it has (strong tracial) m-comparison and is (tracially) $\bar{m}$-almost divisible. These notions are related to each other, and to nuclear dimension. The main result says that if a separable, simple, nonelementary, unital C-algebra with locally finite nuclear dimension is $(m,\bar{m})$-pure, then it absorbs the Jiang-Su algebra $\mathcal{Z}$ tensorially. It follows that a separable, simple, nonelementary, unital C-algebra with locally finite nuclear dimension is $\mathcal{Z}$-stable if and only if it has the Cuntz semigroup of a $\mathcal{Z}$-stable C-algebra. The result may be regarded as a version of Kirchberg's celebrated theorem that separable, simple, nuclear, purely infinite C-algebras absorb the Cuntz algebra $\mathcal{O}_{\infty}$ tensorially. As a corollary we obtain that finite nuclear dimension implies $\mathcal{Z}$-stability for separable, simple, nonelementary, unital C-algebras; this settles an important case of a conjecture by Toms and the author. The main result also has a number of consequences for Elliott's program to classify nuclear C-algebras by their K-theory data. In particular, it completes the classification of simple, unital, approximately homogeneous algebras with slow dimension growth by their Elliott invariants, a question left open in the Elliott-Gong-Li classification of simple AH algebras. Another consequence is that for simple, unital, approximately subhomogeneous algebras, slow dimension growth and $\mathcal {Z}$-stability are equivalent. In the case where projections separate traces, this completes the classification of simple, unital, approximately subhomogeneous algebras with slow dimension growth by their ordered K-groups. [ABSTRACT FROM AUTHOR]

Let A be an approximately subhomogeneous (ASH) C-algebra with slow dimension growth. We prove that if A is unital and simple, then the Cuntz semigroup of A agrees with that of its tensor product with the Jiang-Su algebra $\mathcal{Z}$. In tandem with a result of W. Winter, this yields the equivalence of $\mathcal{Z}$-stability and slow dimension growth for unital simple ASH algebras. This equivalence has several consequences, including the following classification theorem: unital ASH algebras which are simple, have slow dimension growth, and in which projections separate traces are determined up to isomorphism by their graded ordered K-theory, and none of the latter three conditions can be relaxed in general. [ABSTRACT FROM AUTHOR]

We use Lee's work on the Khovanov homology to define a knot invariant s. We show that s( K) is a concordance invariant and that it provides a lower bound for the smooth slice genus of K. As a corollary, we give a purely combinatorial proof of the Milnor conjecture. [ABSTRACT FROM AUTHOR]

The K-theory of a polynomial ring R[ t] contains the K-theory of R as a summand. For R commutative and containing ℚ, we describe K*( R[ t])/ K*( R) in terms of Hochschild homology and the cohomology of Kähler differentials for the cdh topology. We use this to address Bass’ question, whether K n( R)= K n( R[ t]) implies K n( R)= K n( R[ t1, t2]). The answer to this question is affirmative when R is essentially of finite type over the complex numbers, but negative in general. [ABSTRACT FROM AUTHOR]

Let p≥5 be a prime, let ku be the connective complex K-theory spectrum, and let K( ku) be the algebraic K-theory spectrum of ku. In this paper we study the p-primary homotopy type of the spectrum K( ku) by computing its mod ( p, v1) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over the polynomial algebra ${\mathbb{F}}_{p}[b]$, where b is a class of degree 2 p+2 defined as a “higher Bott element”. [ABSTRACT FROM AUTHOR]

We prove localization and Zariski-Mayer-Vietoris for higher Gro-thendieck-Witt groups, alias hermitian K-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove Additivity, Fibration and Approximation theorems for the hermitian K-theory of exact categories with weak equivalences and duality. [ABSTRACT FROM AUTHOR]

Quillen’s algebraic K-theory is reconstructed via Voevodsky’s algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P1-spectrum MGL of Voevodsky is considered as a commutative P1-ring spectrum. Setting $\mathrm{MGL}^i = \bigoplus_{p-2q =i}\mathrm{MGL}^{p,q}$ we regard the bigraded theory MGL p, q as just a graded theory. There is a unique ring morphism $\phi\colon\mathrm{MGL}^0(k)\to\mathbb{Z}$ which sends the class [ X]MGL of a smooth projective k-variety X to the Euler characteristic $\chi(X, \mathcal{O}_X)$ of the structure sheaf $\mathcal{O}_X$ . Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories on the category $\mathcal{S}m\mathcal{O}p/S$ in the sense of [6], where K*( X on Z) is Thomason–Trobaugh K-theory and K′ * is Quillen’s K′-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented in the sense of [6] and ϕ respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism [1]. [ABSTRACT FROM AUTHOR]

Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in S 3. We will prove this conjecture for null-homologous knots in arbitrary closed 3-manifolds. Namely, if K is a knot in a closed 3-manifold Y, Y- K is irreducible, and $\widehat{HFK}(Y,K)$ is monic, then K is fibred. The proof relies on previous works due to Gabai, Ozsváth–Szabó, Ghiggini and the author. A corollary is that if a knot in S 3 admits a lens space surgery, then the knot is fibred. [ABSTRACT FROM AUTHOR]

We construct two new G-equivariant rings: $\mathcal{K}(X,G)$ , called the stringy K-theory of the G-variety X, and $\mathcal{H}(X,G)$ , called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack $\mathcal{X}$ , we also construct a new ring $\mathsf{K}_{\mathrm{orb}}(\mathcal{X})$ called the full orbifold K-theory of $\mathcal{X}$ . We show that for a global quotient $\mathcal{X} = [X/G]$ , the ring of G-invariants $K_{\mathrm{orb}}(\mathcal{X})$ of $\mathcal{K}(X,G)$ is a subalgebra of $\mathsf{K}_{\mathrm{orb}}([X/G])$ and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading. We prove that there is a ring isomorphism $\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)$ , which we call the stringy Chern character. We also show that there is a ring homomorphism $\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$ , which we call the orbifold Chern character, which induces an isomorphism $Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})$ when restricted to the sub-algebra $K_{\mathrm{orb}}(\mathcal{X})$ . Here $H_{\mathrm{orb}}^\bullet(\mathcal{X})$ is the Chen–Ruan orbifold cohomology. We further show that $\mathcal{C}\mathbf{h}$ and $\mathfrak{C}\mathfrak{h}_\mathrm{orb}$ preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds. We further prove that $\mathcal{H}(X,G)$ is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring. We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class. [ABSTRACT FROM AUTHOR]

The original version of this article unfortunately contained several typesetting mistakes. [ABSTRACT FROM AUTHOR]

The article discusses the mathematical equations of the bi-relative algebraic K-theory and topological cyclic homology. It has been sought that the algebraic K-theory does not preserve fiber products of rings, thus defined to measure the deviation as proven by Cortiñas. Furthermore, a mathematical analysis in proving the analogous result for the K-theory having finite coefficients has been provided.

Let f: A→ B be a ring homomorphism of not necessarily unital rings and $I\triangleleft{A}$ an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups K *( A: I)→ K *( B: f( I)) to be an isomorphism; it is measured by the birelative groups K *( A, B: I). Similarly the groups HN *( A, B: I) measure the obstruction to excision in negative cyclic homology. We show that the rational Jones-Goodwillie Chern character induces an isomorphism [ABSTRACT FROM AUTHOR]

Presents a correction to the article "Higher algebraic K-theory for actions of diagonalizable groups" that was previously published in the April 7, 2005 issue.

Presents a way to compute the middle perversity intersection cohomology of the singular quotients. Use of the equivariant Morse theory for the computation of the cohomology ring of the orbifold quotient; Algorithm for the computation of the Betti numbers; Construction of a natural injection under an assumption, named the almost balanced condition.

The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the “Segre map”) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to algebraic K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semi-topological K-theory and an interpretation of the “topological filtration” on singular cohomology groups in K-theoretic terms. [ABSTRACT FROM AUTHOR]

We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type. [ABSTRACT FROM AUTHOR]

The article presents an addendum by professor A. Bartel for an article related to the integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension which published in the article K- theory of this journal "Inventiones Mathematicae," vol. 28. Professor Bartels informs that he has independently, and by different methods, proved the main result of this paper in his paper Squeezing and higher algebraic K-theory published in K-theory, volume 28.