691 research outputs found
The 1-type of a Waldhausen K-theory spectrum
We give a small functorial algebraic model for the 2-stage Postnikov section
of the K-theory spectrum of a Waldhausen category and use our presentation to
describe the multiplicative structure with respect to biexact functors.Comment: We include more technical details which were left to the reader in
the previous version
On the functoriality of cohomology of categories
In this paper we show that the Baues-Wirsching complex used to define
cohomology of categories is a 2-functor from a certain 2-category of natural
systems of abelian groups to the 2-category of chain complexes, chain
homomorphism and relative homotopy classes of chain homotopies. As a
consequence we derive (co)localization theorems for this cohomology.Comment: 15 page
Holonomy Groups of Complete Flat Pseudo-Riemannian Homogeneous Spaces
We show that a complete flat pseudo-Riemannian homogeneous manifold with
non-abelian linear holonomy is of dimension at least 14. Due to an example
constructed in a previous article by Oliver Baues and the author, this is a
sharp bound. Also, we give a structure theory for the fundamental groups of
complete flat pseudo-Riemannian manifolds in dimensions less than 7. Finally,
we observe that every finitely generated torsion-free 2-step nilpotent group
can be realized as the fundamental group of a complete flat pseudo-Riemannian
manifold with abelian linear holonomy.Comment: 16 page
Stems and Spectral Sequences
We introduce the category Pstem[n] of n-stems, with a functor P[n] from
spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of
a space. We show how to associate to each simplicial n-stem Q an
(n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem
of a simplicial space X, the truncated spectral sequence for Q is the
truncation of the usual homotopy spectral sequence of X. Similar results are
also proven for cosimplicial n-stems. They are helpful for computations, since
n-stems in low degrees have good algebraic models
Flat Pseudo-Riemannian Homogeneous Spaces with Non-Abelian Holonomy Group
We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian
fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian
manifolds are complete and have abelian linear holonomy group. To the contrary,
we show that there do exist non-compact and non-complete examples, where the
linear holonomy is non-abelian, starting in dimensions , which is the
lowest possible dimension. We also construct a complete flat pseudo-Riemannian
homogeneous manifold of dimension 14 with non-abelian linear holonomy.
Furthermore, we derive a criterion for the properness of the action of an
affine transformation group with transitive centralizer
Self-maps of the product of two spheres fixing the diagonal
AbstractWe compute the monoid of essential self-maps of SnĆSn fixing the diagonal. More generally, we consider products SĆS, where S is a suspension. Essential self-maps of SĆS demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions
Strongly minimal PD4-complexes
We consider the homotopy types of -complexes with fundamental group
such that and has one end. Let
and . Our main result is that (modulo two technical conditions on
) there are at most orbits of -invariants determining
"strongly minimal" complexes (i.e., those with homotopy intersection pairing
trivial). The homotopy type of a -complex with a
-group is determined by , , and the -type of
. Our result also implies that Fox's 2-knot with metabelian group is
determined up to TOP isotopy and reflection by its group.Comment: 17 page
Isometry groups with radical, and aspherical Riemannian manifolds with large symmetry I
Every compact aspherical Riemannian manifold admits a canonical series of
orbibundle structures with infrasolv fibers which is called its infrasolv
tower. The tower arises from the solvable radicals of isometry group actions on
the universal covers. Its length and the geometry of its base measure the
degree of continuous symmetry of an aspherical Riemannian manifold. We say that
the manifold has large symmetry if it admits an infrasolv tower whose base is a
locally homogeneous space. We construct examples of aspherical manifolds with
large symmetry, which do not support any locally homogeneous Riemannian
metrics
The third homotopy group as a Ļā-module
It is well-known how to compute the structure of the second homotopy group of a space, X, as a module over the fundamental group ĻāX, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method to compute the third homotopy group, ĻāX as a module over ĻāX. Moreover, we determine ĻāX as an extension of ĻāX-modules derived from Whitehead's Certain Exact Sequence. Our method is based on the theory of quadratic modules. Explicit computations are carried out for pseudo-projective 3-spaces X=SĀ¹UeĀ²UeĀ³ consisting of exactly one cell in each dimension ā¤ 3
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