21 research outputs found
Fractional Dehn twists in knot theory and contact topology
Fractional Dehn twists give a measure of the difference between the relative
isotopy class of a homeomorphism of a bordered surface and the Thurston
representative of its free isotopy class. We show how to estimate and compute
these invariants. We discuss the the relationship of our work to stabilization
problems in classical knot theory, general open book decompositions, and
contact topology. We include an elementary characterization of overtwistedness
for contact structures described by open book decompositions.Comment: We have removed an incorrect assumption about properties of
meridional disks of Heegaard decompositions of S^3 and have added a
conjecture about stabilizations of knots in S^
CAT(0) groups with specified boundary
We specify exactly which groups can act geometrically on CAT(0) spaces whose
visual boundary is homeomorphic to either a circle or a suspension of a Cantor
set.Comment: This is the version published by Algebraic & Geometric Topology on 4
May 200
Hyperbolic (1,2)-knots in S^3 with crosscap number two and tunnel number one
A knot in S^3 is said to have crosscap number two if it bounds a
once-punctured Klein bottle but not a Moebius band. In this paper we give a
method of constructing crosscap number two hyperbolic (1,2)-knots with tunnel
number one which are neither 2-bridge nor (1,1)-knots. An explicit infinite
family of such knots is discussed in detail.Comment: 31 pages, 11 figures. To appear in Topology and its Applications
(2008
Obstructions to nonpositive curvature for open manifolds
We study algebraic conditions on a group G under which every properly
discontinuous, isometric G-action on a Hadamard manifold has a G-invariant
Busemann function. For such G we prove the following structure theorem: every
open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy
equivalent to a finite complex of codimension >2 is an open regular
neighborhood of a subcomplex of the same codimension. In this setting we show
that each tangential homotopy type contains infinitely many open K(G,1)
manifolds that admit no complete nonpositively curved metric even though their
universal cover is the Euclidean space. A sample application is that an open
contractible manifold W is homeomorphic to a Euclidean space if and only if the
product of W and a circle admits a complete Riemannian metric of nonpositive
curvature.Comment: 29 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