New lower bounds for the topological complexity of aspherical spaces

Date of Acceptance: 5/04/2015 15 pages, 4 figuresPeer reviewedPostprin

A mapping theorem for topological complexity

Peer reviewedPublisher PD

Bredon cohomology and robot motion planning

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$ can be characterised as the smallest integer $k$ such that the canonical $\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol $E(\pi\times\pi)$ denotes the classifying space for free actions and $E_{\mathcal D}(\pi\times\pi)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3, {\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.Comment: This revision contains a few additional comments, among them is Corollary 3.5.

Helium-3 in the Guaymas Basin: Evidence for injection of mantle volatiles in the Gulf of California

Helium isotope measurements in six major basins in the Gulf of California show that the deep Guaymas Basin has 3He/4He 65–70% higher than atmospheric helium, clear evidence of mantle helium injection. Smaller 3He excesses observed in the Carmen and Farallon basins may be derived from this Guaymas Basin anomaly. The 3He concentrations in the Mazatlan Basin in the mouth of the Gulf of California are similar to average eastern Pacific values, indicating that the Gulf does not provide a significant flux of 3He into the general Pacific circulation. On the basis of temperature and salinity measurements an upper limit of 0.28°C can be placed on the amount of geothermal heating observed in any of the basins. The isotopic ratio of the injected Guaymas Basin helium is found to be 3He/4He = (1.10±0.06) × 10−5, almost identical to the helium signature observed at the Galapagos Rift but somewhat lower than the average ratio in oceanic basalt glasses

Frequency up-conversion as a temperature probe of organic opto-electronic devices

Journal ArticleFrequency up-conversion is demonstrated in a polyfluorene-based conjugated polymer. Up-converted emission is observed upon excitation to the red of the 0-0 luminescence band. The emission intensity depends strongly on temperature and provides an accurate probe of the operating temperature of organic light-emitting diodes. Temperature rises of up to 30 K are observed at standard operating current densities of 225 mA/cm2. Due to the low thermal conductivity and heat capacity of the polymer film, the temperature of the film is found to rise slowly on a time-scale of seconds. Upon termination of the current, the temperature decreases again on a time-scale of minutes