Topological K-theory Research Articles

An index theorem of Callias type for pseudodifferential operators

AbstractWe prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + iΦ, where D is elliptic pseudodifferential with Hermitian symbols, and Φ is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give family versions of the previously known index formulas.

A Bott inverted model for equivariant unitary topological $K$-theory

Let $\mathsf A$ be an abelian compact Lie group. We show the Bott inverted suspension spectrum of the projective space that classifies $\mathsf A$-equivariant line bundles is a model for $\mathsf A$-equivariant unitary topological $K$-theory.

An involution on theK–theory of bimonoidal categories with anti-involution

We construct a combinatorially defined involution on the algebraic $K$-theory of the ring spectrum associated to a bimonoidal category with anti-involution. Particular examples of such are braided bimonoidal categories. We investigate examples such as algebraic K-theory of connective complex and real topological K-theory and Waldhausen's K-theory of spaces of the form BBG, for abelian groups G. We show that the involution agrees with the classical one for a bimonoidal category associated to a ring and prove that it is not trivial in the above mentioned examples.

Morita equivalences and KK-theory for Banach algebras

AbstractVincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences.

The equivariant index theorem in entire cyclic cohomology

AbstractLet G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifold M, with compact quotient G\M. There is an assembly map which associates to any G-equivariant K-homology class on M, an element of the topological K-theory of a suitable Banach completion of the convolution algebra of continuous compactly supported functions on G. The aim of this paper is to calculate the composition of the assembly map with the Chern character in entire cyclic homology . We prove an index theorem reducing this computation to a cup-product in bivariant entire cyclic cohomology. As a consequence we obtain an explicit localization formula which includes, as particular cases, the equivariant Atiyah-Segal-Singer index theorem when G is compact, and the Connes-Moscovici index theorem for G-coverings when G is discrete. The proof is based on the bivariant Chern character introduced in previous papers.

Strongly torsion generated groups from K-theory of real C*-algebras

AbstractWe pursue the program initiated in [7], which consists of an attempt by means of K-theory to construct a strongly torsion generated group with prescribed center and integral homology in dimensions two and higher. Using algebraic and topological K-theory for real C*-algebras, we realize such a construction up to homological dimension five. We also explore the limits of the K-theoretic approach.

On module bundles and topological K-theory

We define and study the group K(X) of a topological space X as the Grothendieck group of the category of suitable module bundles over X instead of the Grothendieck group of the category of vector bundles over X and prove some of its properties. Keywords Topological K-Theory, Module bundles, Waelbroeck algebra Mathematics Subject Classification (2000) 19 Lxx, 19 Axx, (Secondary) 46 Hxx

Comparison between algebraic and topological K-theory of locally convex algebras

This paper is concerned with the algebraic K-theory of locally convex C -algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological K-theory agree on the completed projective tensor product L ⊗ ˆ J , and that the obstruction for the comparison map K ( L ⊗ ˆ J ) → K top ( L ⊗ ˆ J ) to be an isomorphism is (absolute) algebraic cyclic homology. We prove the existence of an exact sequence (Theorem 6.2.1) ▪ We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Fréchet algebra whose topology is generated by a countable family of sub-multiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family (Theorem 8.3.3). This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485–496]. We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. Every Fréchet ideal is sub-harmonic, but not conversely; for example the Schatten ideal L p is sub-harmonic for all p > 0 but is Fréchet only if p ⩾ 1 . We prove a variant of the exact sequence above which essentially says that if A is a C -algebra and J is sub-harmonic, then the obstruction for the periodicity of K ∗ ( A ⊗ C J ) is again cyclic homology (Theorem 7.1.1). This generalizes to all algebras a result of Wodzicki for H-unital algebras announced in [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485–496]. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize in Theorem 6.1.6), and the excision theorem for infinitesimal K-theory, due to the first author.

Diagonal Subschemes and Vector Bundles

We study when a smooth variety X, embedded diagonally in its Cartesian square, is the zero scheme of a section of a vector bundle of rank dim(X) on X × X. We call this the diagonal property (D). It was known that it holds for all flag manifolds SLn/P . We consider mainly the cases of proper smooth varieties, and the analogous problems for smooth manifolds (“the topological case”). Our main new observation in the case of proper varieties is a relation between (D) and cohomologically trivial line bundles on X, obtained by a variation of Serre’s classic argument relating rank 2 vector bundles and codimension 2 subschemes, combined with Serre duality. Based on this, we have several detailed results on surfaces, and some results in higher dimensions. For smooth affine varieties, we observe that for an affine algebraic group over an algebraically closed field, the diagonal is in fact a complete intersection; thus (D) holds, using the trivial bundle. We conjecture the existence of smooth affine complex varieties for which (D) fails; this leads to an interesting question on projective modules. The arguments in the topological case have a different flavour, with arguments from homotopy theory, topological K-theory, index theory etc. There are 3 variants of the diagonal problem, depending on the type of vector bundle we want (arbitrary, oriented or complex). We obtain a homotopy theoretic reformulation of the diagonal property as an extension problem for a certain homotopy class of maps. We also have detailed results in several cases: spheres, odd dimensional complex projective quadric hypersurfaces, and manifolds of even dimension ≤ 6 with an almost complex structure.

The localization sequence for the algebraic K-theory of topological K-theory

We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra $K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})$ for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the K-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective $A_\infty$ ring spectrum R with the Quillen K-theory of the abelian category of finitely generated $\pi_{0}R$-modules.

Algebraic topology of Calabi–Yau threefolds in toric varieties

We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological.

Rational computations of the topological K-theory of classifying spaces of discrete groups

We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.

Induction Theorems and Isomorphism Conjectures for K- and L-Theory

The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C -algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with Z as kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups.

D -brane charges in Gepner models

We construct Gepner models in terms of coset conformal field theories and compute their twisted equivariant K-theories. These classify the D-brane charges on the associated geometric backgrounds and therefore agree with the topological K-theories. We show this agreement for various cases, in particular, the Fermat quintic.

Some Fréchet Algebras for which the Chern Character is an Isomorphism

Using similarities between topological K-theory and periodic cyclic ho- mology we show that, after tensoring with C, for certain Frechet algebras the Chern character provides an isomorphism between these functors. This is applied to prove that the Hecke algebra and the Schwartz algebra of a reductive p-adic group have isomorphic periodic cyclic homology. Later an appendix was added, to deal with infinite direct products of algebras. Mathematics Subject Classification (2000) 19D55, 19L10, 22E50, 46L80

Fluxes, brane charges and Chern morphisms of hyperbolic geometry

The purpose of this paper is to provide the reader with a collection of results which can be found in the mathematical literature and to apply them to hyperbolic spaces that may have a role in physical theories. Specifically, we apply K-theory methods for the calculation of brane charges and RR-fields on hyperbolic spaces (and orbifolds thereof). It is known that by tensoring K-groups with the rationals, K-theory can be mapped to rational cohomology by means of the Chern character isomorphisms. The Chern character allows one to relate the analytic Dirac index to a topological index, which can be expressed in terms of cohomological characteristic classes. We obtain explicit formulae for Chern character, spectral invariants, and the index of a twisted Dirac operator associated with real hyperbolic spaces. Some notes for a bivariant version of topological K-theory (KK-theory) with its connection to the index of the twisted Dirac operator and twisted cohomology of hyperbolic spaces are given. Finally, we concentrate on lower K-groups useful for description of torsion charges.

Stabilization of the Witt group

In this Note, using an idea due to Thomason, we define a “homology theory” on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmer's higher Witt groups. For more general rings, this homology is isomorphic to the KT-theory of Hornbostel, inspired by the work of Williams. For real or complex C ∗ -algebras, we recover – up to 2 torsion – topological K-theory. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

The operator dbar in holomorphic $K$-theory

The holomorphic (or semi-topological) K-theory of a smooth projective variety sits between the algebraic K-theory of the variety and the topological K-theory of the underlying topological space. We describe how to define a family of dbar operators on holomorphic K-theory in a manner analogous to Atiyah's construction of a family of elliptic operators in topological K-theory. In the process, we prove a result akin to Bott periodicity for holomorphic mapping spaces. These results first appeared in the author's Stanford University Ph.D. thesis under the direction of Ralph Cohen.

On [formula omitted]-algebras and K-theory for infinite-dimensional Fredholm manifolds

Let M be a smooth Fredholm manifold modeled on a separable infinite-dimensional Euclidean space E with Riemannian metric g. Given an augmented Fredholm filtration F of M by finite-dimensional submanifolds { M n } n = k ∞ , we associate to the triple ( M , g , F ) a non-commutative direct limit C ∗ -algebra A ( M , g , F ) = lim → A ( M n ) that can play the role of the algebra of functions vanishing at infinity on the non-locally compact space M. The C ∗ -algebra A ( E ) , as constructed by Higson–Kasparov–Trout for their Bott periodicity theorem, is isomorphic to our construction when M = E . If M has an oriented Spin q -structure ( 1 ⩽ q ⩽ ∞ ) , then the K-theory of this C ∗ -algebra is the same (with dimension shift) as the topological K-theory of M defined by Mukherjea. Furthermore, there is a Poincaré duality isomorphism of this K-theory of M with the compactly supported K-homology of M, just as in the finite-dimensional spin setting.

On the [formula omitted]-theory of complete regular local [formula omitted]-algebras

Let A be a noetherian F p -algebra that is finitely generated as a module over the subring A p of pth powers. We give an explicit formula for the de Rham–Witt complex of the power series ring A 〚 t 〛 in terms of that of the ring A. We use this formula to show that, for every complete regular local F p -algebra whose residue field is a finite extension of the subfield of pth powers, the canonical map from the algebraic K-theory with Z / p v -coefficients to the topological K-theory with Z / p v -coefficients is an isomorphism.