Gromov's macroscopic dimension conjecture

In this note we construct a closed 4-manifold having torsion-free fundamental group and whose universal covering is of macroscopic dimension 3. This yields a counterexample to Gromov's conjecture about the falling of macroscopic dimension.Comment: This is the version published by Algebraic & Geometric Topology on 14 October 200

Simplification Techniques for Maps in Simplicial Topology

This paper offers an algorithmic solution to the problem of obtaining "economical" formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain explicit combinatorial descriptions of Steenrod k-th powers exclusively in terms of face operators

Cohomotopy sets of 4-manifolds

Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.Comment: 20 pages, 6 figures; this version to appear in the FreedmanFest (G&T Monographs, Volume 18

A spatial memory signal shows that the parietal cortex has access to a craniotopic representation of space

Humans effortlessly establish a gist-like memory of their environment whenever they enter a new place, a memory that can guide action even in the absence of vision. Neurons in the lateral intraparietal area (LIP) of the monkey exhibit a form of this environmental memory. These neurons respond when a monkey makes a saccade that brings the spatial location of a stimulus that appeared on a number of prior trials, but not on the present trial, into their receptive fields (RFs). The stimulus need never have appeared in the neuron’s RF. This memory response is usually weaker, with a longer latency than the neuron’s visual response. We suggest that these results demonstrate that LIP has access to a supraretinal memory of space, which is activated when the spatial location of the vanished stimulus can be described by a retinotopic vector from the center of gaze to the remembered spatial location

A geometric interpretation of the homotopy groups of the cobordism category

The classifying space of the embedded cobordism category has been identified in by Galatius, Tillmann, Madsen, and Weiss as the infinite loop space of a certain Thom spectrum. This identifies the set of path components with the classical cobordism group. In this paper, we give a geometric interpretation of the higher homotopy groups as certain cobordism groups where all manifolds are now equipped with a set of orthonormal sections in the tangent bundle. We also give a description of the fundamental group as a free group with a set of geometrically intuitive relations.Comment: 23 page

Effect of mixing on polymerizations in batch reactors

The effects of mixing on polymerizations in batch reactors are examined theoretically for initiations by thermal decomposition of catalyst and by absorption of ionizing radiation. Mathematical expressions for predicting the first three moments of the dead polymer size distribution are presented. Two extreme mixing states. perfect and no mixing. are considered. It is shown that in batch reactors mixing in any direction in which nonuniform initiations exist increases the polymerization reaction rate and the number average molecular weight, and decreases the weight average and the polydispersity. Experimentally, the effects of mixing were studied in a solution polymerization system, in a batch reactor. Two states of mixing were studied, perfect and no mixing for a catalyst initiated polymerization. The experimental work verified that the polymerization rate for the perfect mixing state is greater than for the no mixing state at zero hours reaction time, for a total reactor volume/volume of catalyst solution (τ) = 35.7. It was also shown that for a total reactor volume/volume of catalyst solution (τ) = 1, that no or negligible mixing effects exist

Bihomogeneity of solenoids

Solenoids are inverse limit spaces over regular covering maps of closed manifolds. M.C. McCord has shown that solenoids are topologically homogeneous and that they are principal bundles with a profinite structure group. We show that if a solenoid is bihomogeneous, then its structure group contains an open abelian subgroup. This leads to new examples of homogeneous continua that are not bihomogeneous.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-1.abs.htm

Super coset space geometry

Super coset spaces play an important role in the formulation of supersymmetric theories. The aim of this paper is to review and discuss the geometry of super coset spaces with particular focus on the way the geometrical structures of the super coset space G/H are inherited from the super Lie group G. The isometries of the super coset space are discussed and a definition of Killing supervectors - the supervectors associated with infinitesimal isometries - is given that can be easily extended to spaces other than coset spaces.Comment: 49 pages, 1 figure, AFK previously published under the name A. F. Schunc

Tidal estimation in the Atlantic and Indian Oceans, 3 deg x 3 deg solution

An estimation technique was developed to extrapolate tidal amplitudes and phases over entire ocean basins using existing gauge data and the altimetric measurements provided by satellite oceanography. The technique was previously tested. Some results obtained by using a 3 deg by 3 deg grid are presented. The functions used in the interpolation are the eigenfunctions of the velocity (Proudman functions) which are computed numerically from a knowledge of the basin's bottom topography, the horizontal plan form and the necessary boundary conditions. These functions are characteristic of the particular basin. The gravitational normal modes of the basin are computed as part of the investigation; they are used to obtain the theoretical forced solutions for the tidal constituents. The latter can provide the simulated data for the testing of the method and serve as a guide in choosing the most energetic functions for the interpolation