151 research outputs found
Algebraic K-theory over the infinite dihedral group: an algebraic approach
We prove that the Waldhausen nilpotent class group of an injective index 2
amalgamated free product is isomorphic to the Farrell-Bass nilpotent class
group of a twisted polynomial extension. As an application, we show that the
Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family
of virtually cyclic subgroups to the family of finite-by-cyclic subgroups
Topological rigidity and H_1-negative involutions on tori
We prove there is only one involution (up to conjugacy) on the n-torus which
acts as on the first homology group when is of the form
, is of the form , or is less than . In all other cases we prove
there are infinitely many such involutions up to conjugacy, but each of them
has exactly fixed points and is conjugate to a smooth involution. The key
technical point is that we completely compute the equivariant structure set for
the corresponding crystallographic group action on in terms of
the Cappell -groups arising from its infinite dihedral
subgroups. We give a complete analysis of equivariant topological rigidity for
this family of groups.Comment: 50 pages, to appear in Geometry & Topolog
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 space of Anosov diffeomorphisms
We consider the space \X of Anosov diffeomorphisms homotopic to a fixed
automorphism of an infranilmanifold . We show that if is the 2-torus
then \X is homotopy equivalent to . In contrast,
if dimension of is large enough, we show that \X is rich in homotopy and
has infinitely many connected components.Comment: Version 2: referee suggestions result in a better expositio
On three-manifolds dominated by circle bundles
We determine which three-manifolds are dominated by products. The result is
that a closed, oriented, connected three-manifold is dominated by a product if
and only if it is finitely covered either by a product or by a connected sum of
copies of the product of the two-sphere and the circle. This characterization
can also be formulated in terms of Thurston geometries, or in terms of purely
algebraic properties of the fundamental group. We also determine which
three-manifolds are dominated by non-trivial circle bundles, and which
three-manifold groups are presentable by products.Comment: 12 pages; to appear in Math. Zeitschrift; ISSN 1103-467
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
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
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
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
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
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