7,973 research outputs found
Some recent approaches in 4-dimensional surgery theory
It is well-known that an n-dimensional Poincar\'{e} complex , ,
has the homotopy type of a compact topological -manifold if the total
surgery obstruction vanishes. The present paper discusses recent
attempts to prove analogous result in dimension 4. We begin by reviewing the
necessary algebraic and controlled surgery theory. Next, we discuss the key
idea of Quinn's approach. Finally, we present some cases of special fundamental
groups, due to the authors and to Yamasaki
Controlled surgery and -homology
This paper presents an alternative approach to controlled surgery
obstructions. The obstruction for a degree one normal map with control map to complete controlled
surgery is an element , where are topological manifolds of dimension . Our proof uses
essentially the geometrically defined -spectrum as described by
Nicas (going back to Quinn) and some well known homotopy theory. We also
outline the construction of the algebraically defined obstruction, and we
explicitly describe the assembly map in terms of forms in the case . Finally, we
explicitly determine the canonical map
The quadratic form E_8 and exotic homology manifolds
An explicit (-1)^n-quadratic form over Z[Z^{2n}] representing the surgery
problem E_8 x T^{2n} is obtained, for use in the Bryant-Ferry-Mio-Weinberger
construction of 2n-dimensional exotic homology manifolds.Comment: This is the version published by Geometry & Topology Monographs on 22
April 200
Sequential motion planning of non-colliding particles in Euclidean spaces
In terms of Rudyak's generalization of Farber's topological complexity of the
path motion planning problem in robotics, we give a complete description of the
topological instabilities in any sequential motion planning algorithm for a
system consisting of non-colliding autonomous entities performing tasks in
space whilst avoiding collisions with several moving obstacles. The Isotopy
Extension Theorem from manifold topology implies, somewhat surprisingly, that
the complexity of this problem coincides with the complexity of the
corresponding problem in which the obstacles are stationary.Comment: 10 pages; Final version, to appear in Proc. Amer. Math. So
Controlled surgery with trivial local fundamental groups
We provide a proof of the controlled surgery sequence, including stability,
in the special case that the local fundamental groups are trivial. Stability is
a key ingredient in the construction of exotic homology manifolds by Bryant,
Ferry, Mio and Weinberger, but no proof has been available. The development
given here is based on work of M. Yamasaki.Comment: 5 page
Applications of controlled surgery in dimension 4: Examples
The validity of Freedman's disk theorem is known to depend only on the
fundamental group. It was conjectured that it fails for nonabelian free
fundamental groups. If this were true then surgery theory would work in
dimension four. Recently, Krushkal and Lee proved a surprising result that
surgery theory works for a large special class of 4-manifolds with free
nonabelian fundamental groups. The goal of this paper is to show that this also
holds for other fundamental groups which are not known to be good, and that it
is best understood using controlled surgery theory of Pedersen--Quinn--Ranicki.
We consider some examples of 4-manifolds which have the fundamental group
either of a closed aspherical surface or of a 3-dimensional knot space. A more
general theorem is stated in the appendix
- …