A Banach Space Whose Operator Algebra has K0-Group Q

We construct a Banach space X whose algebra of bounded linearoperators has K0-group the additive group of rational numbers.We achieve this using a Gowers-Maurey space. 2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05,55Q40, 55U10     

Configurations, braids, and homotopy groups

The main results of this article are certain connections between braid groups and the homotopy groups of the -sphere. The connections are given in terms of Brunnian braids over the disk and over the -sphere. The techniques arise from the natural structure of simplicial and -structures on fundamental groups of configuration spaces.

     

Linear Representations of the Braid Groups of Some Manifolds

In this paper we prove that the braid group B_n(S²) of 2-sphere, mapping class group M(0,n) of the n-punctured 2-sphere and the braid group B₃(P²) of the projective plane are linear. Partially supported by the Russian Foundation for Basic Research (grant number 02-01-01118).Mathematics Subject Classifications (2000) 20F28, 20F36, 20G35.     

Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic.     

A Categorical Structure for the Virtual Braid Group

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding.     

On Vanishing and Cofiniteness of Generalized Local Cohomology Modules

Pure braid groups and pure mapping class groups of a punctured sphere have many features in common. In this article, we study the graded Lie algebra of the descending central series of the pure mapping class group of a 2-sphere. A simple presentation of this Lie algebra is obtained.     

Rational homotopy theory for non-simply connected spaces

We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.

     

Spaces of knots in the solid torus,knots in the thickened torus,and links in the 3-sphere

We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We similarly study spaces of unframed links in the 3-sphere, modulo rotations, and spaces of knots in the thickened torus. The subgroup of meridional rotations splits as a direct factor of the fundamental group of the space of any framed link except the unknot. Its generators can be viewed as generalizations of the Gramain loop in the space of long knots. Taking the quotient by certain such rotations relates the spaces we study. All of our results generalize previous work of Hatcher and Budney. We provide many examples and explicitly describe generators of fundamental groups.

     

The Elementary Geometric Structure of Compact Lie Groups

We give geometric proofs of some of the basic structure theoremsfor compact Lie groups. The goal is to take a fresh look atthese theorems, prove some that are difficult to find in theliterature, and illustrate an approach to the theorems thatcan be imitated in the homotopy theoretic setting of p-compactgroups. 1991 Mathematics Subject Classification 22E15, 55P35.