This paper begins the study of relations between Riemannian geometry and
global properties of contact structures on 3-manifolds. In particular we prove
an analog of the sphere theorem from Riemannian geometry in the setting of
contact geometry. Specifically, if a given three dimensional contact manifold
(M,\xi) admits a complete compatible Riemannian metric of positive 4/9-pinched
curvature then the underlying contact structure \xi is tight; in particular,
the contact structure pulled back to the universal cover is the standard
contact structure on S^3. We also describe geometric conditions in dimension
three for \xi to be universally tight in the nonpositive curvature setting.Comment: 29 pages. Added the sphere theorem, removed high dimensional material
and an alternate approach to the three dimensional tightness radius estimate
To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientationpreserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space. C 2013 American Institute of Physics.
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen’s iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.
Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer µ, the "triple linking number" of the title, which is well-defined modulo the greatest common divisor of p, q and r.To each such link L we associate a geometrically natural characteristic map g L from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of g L . Maps of the 3-torus to the 2-sphere were classified up to homotopy by Lev Pontryagin in 1941. A complete set of invariants is given by the degrees p, q and r of their restrictions to the 2-dimensional coordinate subtori, and by the residue class of one further integer ν, an "ambiguous Hopf invariant" which is well-defined modulo twice the greatest common divisor of p, q and r.
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