120 research outputs found
Smash products for secondary homotopy groups
We construct a smash product operation on secondary homotopy groups yielding
the structure of a lax symmetric monoidal functor. Applications on cup-one
products, Toda brackets and Whitehead products are considered. In particular we
prove a formula for the crossed effect of the cup-one product operation on
unstable homotopy groups of spheres which was claimed by
Barratt-Jones-Mahowald.Comment: We give a clearer description of the tensor product of symmetric
sequences of quadratic pair module
The characteristic cohomology class of a triangulated category
This is the final version of a series of papers uploaded in May 25, 2005. We
have splitted the long last paper of the previous version in two parts to make
it easier to understand. The results are essentially the same, although the
presentation has changed substantially. The first three papers have not
changed.
This is a collection of five papers on the foundation of triangulated
categories in the context of groupoid-enriched categories, termed track
categories, and characteristic cohomology classes. As a main result it is shown
that given an additive category A with a translation functor t: A --> A and a
class V in translation cohomology H^3(A,t) then two simple properties of V
imply that (A,t) is a triangulated category. The cohomology class V yields an
equivalence class (B,[s]) where B is a track category with homotopy category A
and [s] is the homotopy class of a pseudofunctor s: B --> B inducing t. The two
properties of V correspond to natural axioms on B and s which again imply that
(A,t) is a triangulated category.
The five papers of this volume depend on each other by cross references, but
each paper can be read independently of the others so that the reader is free
to choose one of the papers to start. Each paper has its own abstract,
introduction and literature.Comment: 166 pages, some diagrams do not appear correctly in the DVI fil
Toda brackets and cup-one squares for ring spectra
In this paper we prove the laws of Toda brackets on the homotopy groups of a
connective ring spectrum and the laws of the cup-one square in the homotopy
groups of a commutative connective ring spectrum.Comment: 22 page
Secondary homotopy groups
Secondary homotopy groups supplement the structure of classical homotopy
groups. They yield a track functor on the track category of pointed spaces
compatible with fiber sequences, suspensions and loop spaces. They also yield
algebraic models of homotopy types with homotopy groups concentrated in two
consecutive dimensions.Comment: We added further commets and references to make the paper more easily
readabl
Virtually abelian K\"ahler and projective groups
We characterise the virtually abelian groups which are fundamental groups of
compact K\"ahler manifolds and of smooth projective varieties. We show that a
virtually abelian group is K\"ahler if and only if it is projective. In
particular, this allows to describe the K\"ahler condition for such groups in
terms of integral symplectic representations
Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology
In this paper we determine the representation type of some algebras of
infinite matrices continuously controlled at infinity by a compact metrizable
space. We explicitly classify their finitely presented modules in the finite
and tame cases. The algebra of row-column-finite (or locally finite) matrices
over an arbitrary field is one of the algebras considered in this paper, its
representation type is shown to be finite.Comment: 33 page
Global Dimension of Polynomial Rings in Partially Commuting Variables
For any free partially commutative monoid , we compute the global
dimension of the category of -objects in an Abelian category with exact
coproducts. As a corollary, we generalize Hilbert's Syzygy Theorem to
polynomial rings in partially commuting variables.Comment: 11 pages, 2 figure
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