The localization sequence for the algebraic <Emphasis Type="Italic">K</Emphasis>-theory of topological <Emphasis Type="Italic">K</Emphasis>-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.     

On the algebraic <Emphasis Type="Italic">K</Emphasis>- and <Emphasis Type="Italic">L</Emphasis>-theory of word hyperbolic groups

In this paper, the assembly maps in algebraic K- and L-theory for the family of finite subgroups are proven to be split injections for word hyperbolic groups. This is done by analyzing the compactification of the Rips complex by the boundary of a word hyperbolic group.Mathematics Subject Classification (2000): 20F67, 18F25     

Nonstable <Emphasis Type="Italic">K</Emphasis>-theory for Graph Algebras

We compute the monoid V(L _K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L _K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L _K (E) and the lattice of order-ideals of V(L _K (E)). When K is the field of complex numbers, the algebra is a dense subalgebra of the graph C ^*-algebra C ^*(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property. The first author was partially supported by the DGI and European Regional Development Fund, jointly, through Project BFM2002-01390, the second and the third by the DGI and European Regional Development Fund, jointly, through Project MTM2004-00149 and by PAI III grant FQM-298 of the Junta de Andalucía. Also, the first and third authors are partially supported by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.     

Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Gorenstein projective modules

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ‘resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different algebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-nite Gorenstein algebras.     

Higher Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Ring Epimorphisms

Given a homological ring epimorphism from a ring R to another ring S, we show that if the left R-module S has a finite-type resolution, then the algebraic K-group K _n (R) of R splits as the direct sum of the algebraic K-group K _n (S) of S and the algebraic K-group K _n (R) of a Waldhausen category R determined by the ring epimorphism. This result is then applied to endomorphism rings, matrix subrings, rings with idempotent ideals, and universal localizations which appear often in representation theory and algebraic topology.     

Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.     

<Emphasis Type="Italic">K</Emphasis>-theory for certain extension algebras of purely infinite simple <Emphasis Type="Italic">C</Emphasis>*-algebras

We consider extension algebras of unital purely infinite simple C*-algebras by purely infinite simple stable C*-algebras. K-theory of such extension algebras is described.     

The nilpotence conjecture in <Emphasis Type="Italic">K</Emphasis>-theory of toric varieties

It is shown that all nontrivial elements in higher K-groups of toric varieties are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. To Richard G. SwanMathematics Subject Classification (2000) 14M25, 19D55, 19D25, 19E08     

Bounded homotopy theory and the <Emphasis Type="Italic">K</Emphasis>-theory of weighted complexes

Given a bounding class B, we construct a bounded refinement BK(−) of Quillen’s K-theory functor from rings to spaces. As defined, BK(−) is a functor from weighted rings to spaces, and is equipped with a comparison map BK → K induced by “forgetting control.” In contrast to the situation with B-bounded cohomology, there is a functorial splitting BK(−) ≅ K(−)×BK ^rel(−) where BK ^rel(−) is the homotopy fiber of the comparison map.     

<Emphasis Type="Italic">K</Emphasis>-homology of <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-categories and symmetric spectra representing <Emphasis Type="Italic">K</Emphasis>-homology

We define K-homology groups K _* () for small C ^* -categories in terms of Hilbert modules over the C ^* -category . We also define a functor A ^f from the category of small C ^* -categories into the category of C ^* -algebras and show that there is a natural isomorphism . In addition, we give an easy construction of a functor from the category of C ^* -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor comes with a natural isomorphism for C ^* -algebras A. It then follows that the composition A ^f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.     

Strict <Emphasis Type="Italic">K</Emphasis>-monotonicity and <Emphasis Type="Italic">K</Emphasis>-order continuity in symmetric spaces

This paper is devoted to strict K-monotonicity and K-order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on \([0,\infty )\) implies that the embedding \(E\hookrightarrow {L^1}[0,\infty )\) does not hold. We present a complete characterization of an equivalent condition to K-order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K-monotonicity and K-order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec–Klee property for global convergence in measure in a symmetric space E.     

Techniques,computations, and conjectures for semi-topological <Emphasis Type="Italic">K</Emphasis>-theory

We establish the existence of an Atiyah-Hirzebruch-like spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.Mathematics Subject Classification (2000): 19E20, 19E15, 14F43The first author was partially supported by the NSF and the NSAThe second author was supported by the Helen M. Galvin Fellowship of Northwestern UniversityThe third author was partially supported by the NSF and the NSA     

The connecting homomorphism for <Emphasis Type="Italic">K</Emphasis>-theory of generalized free products

We interpret the connecting homomorphism of the long exact sequence of algebraic K-groups associated to a category with cofibrations and two notions of weak equivalence. One notion of weak equivalence is finer than the other and certain technical conditions involving mapping cylinders must also be satisfied. Such situations arise when one considers certain free product diagrams in the category of rings, such as those that arise in applications of the Seifert-van Kampen theorem where all fundamental group homomorphisms are injective. The techniques follow Waldhausen’s approach to algebraic K-theory of categories with cofibrations and weak equivalences.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

Automorphisms of Coxeter groups of type <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A Coxeter system (W, S) is said to be of type K _n if the associated Coxeter graph Γ_S is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by Γ_S. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type K _n then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).     

Torsion in the full orbifold <Emphasis Type="Italic">K</Emphasis>-theory of abelian symplectic quotients

Let (M, ω, Φ) be a Hamiltonian T-space and let H í T{H\subseteq T} be a closed Lie subtorus. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion in the integral full orbifold K-theory of the orbifold symplectic quotient [M//H]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T-space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S ¹-quotients of GKM spaces, have integral full orbifold K-theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our results using low-rank coadjoint orbits of type A and B.     

Integrals of <Emphasis Type="Italic">K</Emphasis> and <Emphasis Type="Italic">E</Emphasis> from lattice sums

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.     

Hermitian <Emphasis Type="Italic">K</Emphasis>-theory and 2-regularity for totally real number fields

We completely determine the 2-primary torsion subgroups of the hermitian K-groups of rings of 2-integers in totally real 2-regular number fields. The result is almost periodic with period 8. Moreover, the 2-regular case is precisely the class of totally real number fields that have homotopy cartesian “Bökstedt square”, relating the K-theory of the 2-integers to that of the fields of real and complex numbers and finite fields. We also identify the homotopy fibers of the forgetful and hyperbolic maps relating hermitian and algebraic K-theory. The result is then exactly periodic of period 8 in the orthogonal case. In both the orthogonal and symplectic cases, we prove a 2-primary hermitian homotopy limit conjecture for these rings.     

On the sequential order continuity of the <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-space

As shown in [1], for each compact Hausdorff space K without isolated points, there exists a compact Hausdorff P′-space X but not an F-space such that C(K) is isometrically Riesz isomorphic to a Riesz subspace of C(X). The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.     

Induced representations,transferred Chern classes and Morava rings <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">s</Emphasis>)*(<Emphasis Type="Italic">BG</Emphasis>): Some calculations

The ring structure of Morava K-theory K(s)*(BG) for the 2-group no. 38 of order 32 from the Hall-Senior list is calculated. Previously it was known that K(s)*(BG) is evenly generated and for s = 2 is generated by Chern characteristic classes.