224 research outputs found
A universal characterization of higher algebraic K-theory
In this paper we establish a universal characterization of higher algebraic
K-theory in the setting of small stable infinity categories. Specifically, we
prove that connective algebraic K-theory is the universal additive invariant,
i.e., the universal functor with values in spectra which inverts Morita
equivalences, preserves filtered colimits, and satisfies Waldhausen's
additivity theorem. Similarly, we prove that non-connective algebraic K-theory
is the universal localizing invariant, i.e., the universal functor that
moreover satisfies the "Thomason-Trobaugh-Neeman" localization theorem.
To prove these results, we construct and study two stable infinity categories
of "noncommutative motives"; one associated to additivity and another to
localization. In these stable infinity categories, Waldhausen's S. construction
corresponds to the suspension functor and connective and non-connective
algebraic K-theory spectra become corepresentable by the noncommutative motive
of the sphere spectrum. In particular, the algebraic K-theory of every scheme,
stack, and ring spectrum can be recovered from these categories of
noncommutative motives.
In order to work with these categories of noncommutative motives, we
establish comparison theorems between the category of spectral categories
localized at the Morita equivalences and the category of small
idempotent-complete stable infinity categories. We also explain in detail the
comparison between the infinity categorical version of Waldhausen K-theory and
the classical definition.
As an application of our theory, we obtain a complete classification of the
natural transformations from higher algebraic K-theory to topological
Hochschild homology (THH) and topological cyclic homology (TC). Notably, we
obtain an elegant conceptual description of the cyclotomic trace map.Comment: Various revisions and correction
Topological Hochschild homology of Thom spectra and the free loop space
We describe the topological Hochschild homology of ring spectra that arise as
Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for
stable spherical fibrations. To do this, we consider symmetric monoidal models
of the category of spaces over BF and corresponding strong symmetric monoidal
Thom spectrum functors. Our main result identifies the topological Hochschild
homology as the Thom spectrum of a certain stable bundle over the free loop
space L(BX). This leads to explicit calculations of the topological Hochschild
homology for a large class of ring spectra, including all of the classical
cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p
and HZ.Comment: 58 page
The thick-thin decomposition and the bilipschitz classification of normal surface singularities
We describe a natural decomposition of a normal complex surface singularity
into its "thick" and "thin" parts. The former is essentially metrically
conical, while the latter shrinks rapidly in thickness as it approaches the
origin. The thin part is empty if and only if the singularity is metrically
conical; the link of the singularity is then Seifert fibered. In general the
thin part will not be empty, in which case it always carries essential
topology. Our decomposition has some analogy with the Margulis thick-thin
decomposition for a negatively curved manifold. However, the geometric behavior
is very different; for example, often most of the topology of a normal surface
singularity is concentrated in the thin parts.
By refining the thick-thin decomposition, we then give a complete description
of the intrinsic bilipschitz geometry of in terms of its topology and a
finite list of numerical bilipschitz invariants.Comment: Minor corrections. To appear in Acta Mathematic
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Finite covers of random 3-manifolds
A 3-manifold is Haken if it contains a topologically essential surface. The
Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite
fundamental group has a finite cover which is Haken. In this paper, we study
random 3-manifolds and their finite covers in an attempt to shed light on this
difficult question. In particular, we consider random Heegaard splittings by
gluing two handlebodies by the result of a random walk in the mapping class
group of a surface. For this model of random 3-manifold, we are able to compute
the probabilities that the resulting manifolds have finite covers of particular
kinds. Our results contrast with the analogous probabilities for groups coming
from random balanced presentations, giving quantitative theorems to the effect
that 3-manifold groups have many more finite quotients than random groups. The
next natural question is whether these covers have positive betti number. For
abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show
that the probability of positive betti number is 0.
In fact, many of these questions boil down to questions about the mapping
class group. We are lead to consider the action of mapping class group of a
surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show
that if the genus of S is large, then this action is very mixing. In
particular, the action factors through the alternating group of each orbit.
This is analogous to Goldman's theorem that the action of the mapping class
group on the SU(2) character variety is ergodic.Comment: 60 pages; v2: minor changes. v3: minor changes; final versio
On the Whitehead spectrum of the circle
The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss and
Williams shows that the homotopy groups in low degrees of the space of
homeomorphisms of a closed Riemannian manifold of negative sectional curvature
can be expressed as a functor of the fundamental group of the manifold. To
determine this functor, however, it remains to determine the homotopy groups of
the topological Whitehead spectrum of the circle. The cyclotomic trace of B
okstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an
expression for these homotopy groups in terms of the equivariant homotopy
groups of the homotopy fiber of the map from the topological Hochschild
T-spectrum of the sphere spectrum to that of the ring of integers induced by
the Hurewicz map. We evaluate the latter homotopy groups, and hence, the
homotopy groups of the topological Whitehead spectrum of the circle in low
degrees. The result extends earlier work by Anderson and Hsiang and by Igusa
and complements recent work by Grunewald, Klein, and Macko.Comment: 52 page
3-manifolds which are spacelike slices of flat spacetimes
We continue work initiated in a 1990 preprint of Mess giving a geometric
parameterization of the moduli space of classical solutions to Einstein's
equations in 2+1 dimensions with cosmological constant 0 or -1 (the case +1 has
been worked out in the interim by the present author). In this paper we make a
first step toward the 3+1-dimensional case by determining exactly which closed
3-manifolds M^3 arise as spacelike slices of flat spacetimes, and by finding
all possible holonomy homomorphisms pi_1(M^3) to ISO(3,1).Comment: 10 page
Dutch translation and cross-cultural validation of the Adult Social Care Outcomes Toolkit (ASCOT)
Background: The Adult Social Care Outcomes Toolkit was developed to measure outcomes of social care in England. In this study, we translated the four level self-completion version (SCT-4) of the ASCOT for use in the Netherlands and performed a cross-cultural validation.
Methods: The ASCOT SCT-4 was translated into Dutch following international guidelines, including two forward and back translations. The resulting version was pilot tested among frail older adults using think-aloud interviews. Furthermore, using a subsample of the Dutch ACT-study, we investigated test-retest reliability and construct validity and compared response distributions with data from a comparable English study.
Results: The pilot tests showed that translated items were in general understood as intended, that most items were reliable, and that the response distributions of the Dutch translation and associations with other measures were comparable to the original English version. Based on the results of the pilot tests, some small modifications and a revision of the Dignity items were proposed for the final translation, which were approved by the ASCOT development team. The complete original English version and the final Dutch translation can be obtained after registration on the ASCOT website (http://www.pssru.ac.uk/ascot).
Conclusions: This study provides preliminary evidence that the Dutch translation of the ASCOT is valid, reliable and comparable to the original English version. We recommend further research to confirm the validity of the modified Dutch ASCOT translation
The K-theoretic Farrell-Jones Conjecture for hyperbolic groups
We prove the K-theoretic Farrell-Jones Conjecture for hyperbolic groups with
(twisted) coefficients in any associative ring with unit.Comment: 33 pages; final version; to appear in Invent. Mat
Higher algebraic -groups and -split sequences
In this paper, we use -split sequences and derived equivalences
to provide formulas for calculation of higher algebraic -groups (or mod-
-groups) of certain matrix subrings which cover tiled orders, rings related
to chains of Glaz-Vasconcelos ideals, and some other classes of rings. In our
results, we do not assume any homological requirements on rings and ideals
under investigation, and therefore extend sharply many existing results of this
type in the algebraic -theory literature to a more general context.Comment: 20 page
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