Elements of Small Orders in K2F Ⅱ

In "Elements of small orders in K2(F)" (Algebraic K-Theory, Lecture Notes in Math., 966, 1982, 1-6.), the author investigates elements of the form {a, Φn(a)} in the Milnor group K2F of a field F, where Φn(x) is the n-th cyclotomic polynomial. In this paper, these elements are generalized. Applying the explicit formulas of Rosset and Tate for the transfer homomorphism for K2, the author proves some new results on elements of small orders in K2F.     

A New Inequality for Weakly （K1, K2）-Quasiregular Mappings

We obtain a new inequality for weakly （K1,K2）-quasiregular mappings by using the McShane extension method. This inequality can be used to derive the self-improving regularity of （K1, K2）-Quasiregular Mappings.     

The decomposition of Kv into K2×K5''''s

The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a decomposition exists if and only if v ≡ 1 (mod 25).     

The Smallest Degree Sum That Yields Potentially Kr＋1 - K3-Graphic Sequences

Let a（Kr,＋1 - K3,n） be the smallest even integer such that each n-term graphic sequence п= （d1,d2,…dn） with term sum σ（п） = d1 ＋ d2 ＋…＋ dn 〉 σ（Kr＋1 -K3,n） has a realization containing Kr＋1 - K3 as a subgraph, where Kr＋1 -K3 is a graph obtained from a complete graph Kr＋1 by deleting three edges which form a triangle. In this paper, we determine the value σ（Kr＋1 - K3,n） for r ≥ 3 and n ≥ 3r＋ 5.     

Algebraic K-Theory and the Conjectural Leibniz K-Theory

The analogy between algebraic K-theory and cyclic homology is used to build a program aiming at understanding the algebraic K-theory of fields and the periodicity phenomena in algebraic K-theory. In particular, we conjecture the existence of a Leibniz K-theory which would play the role of Hochschild homology. We propose a motivated presentation for the Leibniz K ₂-group ofa field.     

On connected factors inK 1,3-free graphs

A graph is said to beK _1,3-free if it contains noK _1,3 as an induced subgraph. It is shown in this paper that every 2-connectedK _1,3-free graph contains a connected [2,3]-factor. We also obtain that every connectedK _1,3-free graph has a spanning tree with maximum degree at most 3. This research is partially supported by the National Natural Sciences Foundation of China and by the Natural Sciences Foundation of Shandong Province of China.     

On potentially K 5-H-graphic sequences

Let K _m -H be the graph obtained from K _m by removing the edges set E(H) of H where H is a subgraph of K _m . In this paper, we characterize the potentially K ₅-P ₄ and K ₅-Y ₄-graphic sequences where Y ₄ is a tree on 5 vertices and 3 leaves. Research was supported by NNSF of China (10271105) and by NSF of Fujian (Z0511034), Fujian Provincial Training Foundation for “Bai-Quan-Wan Talents Engineering”, Project of Fujian Education Department, Project of Zhangzhou Teachers College and by NSERC.     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).     

Relative MilnorK-theory

LetR be a commutative, semi-local ring,I ₁, ...,I _s ideals. In this paper, we define therelative Milnor K-groups of (R;I ₁, ...,I _s ),K _p ^M (R;I ₁, ...,I _s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic to the weightp portion of the relative QuillenK-groupK _p (R;I ₁, ...,I _s ), after inverting (p–1)!. We also define the relative group homology of GL _n (R;I ₁, ...,I _s ), and show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic toH _p (GL_p(R;I ₁, ...,I _s ))/Im(H _p (GL _p–1 (R;I ₁, ...,I _s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK _p ^M (F)/l ^v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:

Note on theK0 of rings with Zariskian filtration

For a noncommutative Zariski ringR with associated graded ringG(R) of finite global dimension there is an injectionK _o(R) K _o (G(R)). The result applies to certain rings of differential operators, in particular to the ring of germs of microlocal differential operators that also provides an example of a nonpositively filtered Zariski ring.Research fellow at UIA, supported by a grant from the Province of Antwerp.Supported by an NFWO grant.     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite coefficients.     

不含有5-圈和k4平面图的森林分解

本文研究了不含有5-圈和K₄的平面图的森林分解问题.利用权转移法,证明了任意不含有5-圈和K₄的平面图能分解成三个森林,且其中有一个森林的最大度不超过2,这一结果推广了文献[2,3]中的结论.     

Trade Spectrum of K 4 − e

A K₄ − e trade consists of two disjoint decompositions of some simple graph H into copies of K₄ − e. The number of vertices of H is referred to as the foundation of the trade, while the number of copies of K₄ − e in each of the decompositions is called the volume of the trade. We determine the values of v and s for which there exists a K₄ − e trade of volume s and foundation v.     

On K0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K ₀(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X ² is not a sufficient condition of K ₀(B(X)) = 0.     

On a spectral sequence for equivariant K-theory

We apply the “homotopy coniveau” machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin–Graham) and an Atiyah–Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch–Lichtenbaum and Friedlander–Suslin. The first-named author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and support of the NSF via grants DMS-0140445 and DMS-0457195.     

Connected domination critical graphs with respect to relative complements

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The minimum number of vertices in a connected dominating set of G is called the connected domination number of G, and is denoted by γ _c (G). Let G be a spanning subgraph of K _s,s and let H be the complement of G relative to K _s,s ; that is, K _s,s = G ⊕ H is a factorization of K _s,s . The graph G is k-γ _c -critical relative to K _s,s if γ _c (G) = k and γ _c (G + e) < k for each edge e ∈ E(H). First, we discuss some classes of graphs whether they are γ _c -critical relative to K _s,s . Then we study k-γ _c -critical graphs relative to K _s,s for small values of k. In particular, we characterize the 3-γ _c -critical and 4-γ _c -critical graphs.     

Semi-exactitude du bifoncteur de Kasparov équivariant

We define a notion of strong K-theoretic amenability for a locally compact group G. This notion coincides with the K-theoretic amenability of many groups. We prove that all results obtained concerning the behavior of KK(.,.) with respect to exact sequences are generalized to the case of KK _G (.,.) for G strongly K-amenable.     

Milnor K-Groups and Zero-Cycles on Products of Curves over p-Adic Fields

We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH ₀(X)/m for X a product of curves over a p-adic field.     

K2 of a Ring via the G-Construction

We interpret the Steinberg symbols x_i,j(a) as homotopies contracting the elementary matrices e_i,j(a), the latters being represented by certain arcs in a simplicial model of the K-theory. We further prove the Steinberg relations for these homotopies. This provides an explicit map from K₂ of a ring, defined classically as ker(St(R) → GL(R)), to π₂ of the G-construction assigned to R. Though the two groups are known to be isomorphic, a certain work is to be done to prove that this explicit map is an isomorphism. Mathematics Subject Classification 1991: Primary 19B99, 19D99; secondary 18E10, 18F25.     

ComplexK-theory ofBSL3(Z)

ComplexK-theory ofBSL₃(Z) andBSt₃(Z) are computed. We first study a Brown-Peterson (BP) version of Soulé's arguments. Then we consider complexK-theory by using a Conner-Floyd type theorem.