255,283 research outputs found
Central extensions of groups of sections
If q : P -> M is a principal K-bundle over the compact manifold M, then any
invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a
Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms
modulo exact forms. In the present paper we analyze the integrability of this
extension to a Lie group extension for non-connected, possibly
infinite-dimensional Lie groups K. If K has finitely many connected components
we give a complete characterization of the integrable extensions. Our results
on gauge groups are obtained by specialization of more general results on
extensions of Lie groups of smooth sections of Lie group bundles. In this more
general context we provide sufficient conditions for integrability in terms of
data related only to the group K.Comment: 54 pages, revised version, to appear in Ann. Glob. Anal. Geo
Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.
We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time
Topology of spaces of knots in dimension 3
This paper is a computation of the homotopy type of K, the space of long
knots in R^3, the same space of knots studied by Vassiliev via singularity
theory. Each component of K corresponds to an isotopy class of long knot, and
we `enumerate' the components via the companionship trees associated to the
knot. The knots with the simplest companionship trees are: the unknot, torus
knots, and hyperbolic knots. The homotopy-type of these components of K were
computed by Hatcher. In the case the companionship tree has height, we give a
fibre-bundle description of those components of K, recursively, in terms of the
homotopy types of `simpler' components of K, in the sense that they correspond
to knots with shorter companionship trees. The primary case studied in this
paper is the case of a knot which has a hyperbolic manifold contained in the
JSJ-decomposition of its complement.Comment: 22 pages, 16 figure
Components of Gr\"obner strata in the Hilbert scheme of points
We fix the lexicographic order on the polynomial ring
over a ring . We define \Hi^{\prec\Delta}_{S/k},
the moduli space of reduced Gr\"obner bases with a given finite standard set
, and its open subscheme \Hi^{\prec\Delta,\et}_{S/k}, the moduli
space of families of #\Delta points whose attached ideal has the standard set
. We determine the number of irreducible and connected components of
the latter scheme; we show that it is equidimensional over ; and
we determine its relative dimension over . We show that analogous
statements do not hold for the scheme \Hi^{\prec\Delta}_{S/k}. Our results
prove a version of a conjecture by Bernd Sturmfels.Comment: 49 page
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