Topological complexity of motion planning and Massey products

We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces $X$ for which the topological complexity \TC(X) (defined to be the genus of the free path fibration on $X$) is greater than the zero-divisors cup-length plus one.Comment: 11 pages; minor revisions and 1 added reference; to appear in the Proceedings of the M. M. Postnikov Memorial Conferenc

Homologies are infinitely complex

We show that for any k>1, stratified sets of finite complexity are insufficient to realize all homology classes of codimension k in all smooth manifolds. We also prove a similar result concerning smooth generic maps whose double-point sets are co-oriented.Comment: 5 pages; v2: Very minor corrections, to appear in Topol. Methods Nonlinear Ana

Sphalerons with Two Higgs Doublets

We report on work studying the properties of the sphaleron in models of the electroweak interactions with two Higgs doublets in as model-independent a way as possible: by exploring the physical parameter space described by the masses and mixing angles of the Higgs particles. If one of the Higgs particles is heavy, there can be several sphaleron solutions, distinguished by their properties under parity and the behaviour of the Higgs field at the origin. In general, these solutions are not spherically symmetric, although the departure from spherical symmetry is small.Comment: Talk given at Strong and Electroweak Matter, Marseille, 14-17 June 2000. Uses World Scientific proceedings class ws-p8-50x6-00.cl

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

On realizing homology classes by maps of restricted complexity

We show that in every codimension greater than one there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) which cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word `immersion' is replaced by `map with some restricted set of multi-singularities'.Comment: 13 pages; Final version, to appear in Bull. Lond. Math. So

Do Two Temperature Debris Disks Have Multiple Belts?

We present a study of debris disks whose spectra are well modelled by dust emission at two different temperatures. These disks are typically assumed to be a sign of multiple belts, which in only a few cases have been confirmed via high resolution observations. We first compile a sample of two-temperature disks to derive their properties, summarised by the ratios of the warm and cool component temperatures and fractional luminosities. The ratio of warm to cool temperatures is constant in the range 2-4, and the temperatures of both warm and cool components increases with stellar mass. We then explore whether this emission can arise from dust in a single narrow belt, with the range of temperatures arising from the size variation of grain temperatures. This model can produce two-temperature spectra for Sun-like stars, but is not supported where it can be tested by observed disk sizes and far-IR/mm spectral slopes. Therefore, while some two-temperature disks arise from single belts, it is probable that most have multiple spatial components. These disks are plausibly similar to the outer Solar System's configuration of Asteroid and Edgeworth-Kuiper belts separated by giant planets. Alternatively, the inner component could arise from inward scattering of material from the outer belt, again due to intervening planets. In either case, we suggest that the ratio of warm/cool component temperatures is indicative of the scale of outer planetary systems, which typically span a factor of about ten in radius.Comment: accepted to MNRA

A mapping theorem for topological complexity

Peer reviewedPublisher PD