654 research outputs found
Category and Topological Complexity of the configuration space
The Lusternik-Schnirelmann category cat and topological complexity TC are
related homotopy invariants. The topological complexity TC has applications to
the robot motion planning problem. We calculate the Lusternik-Schnirelmann
category and topological complexity of the ordered configuration space of two
distinct points in the product and apply the results to
the planar and spatial motion of two rigid bodies in and
respectively.Comment: 10 pages, 1 figure. Final version. To appear in Bulletin of the
Australian Mathematical Societ
On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists
The Giroux correspondence and the notion of a near force-free magnetic field
are used to topologically characterize near force-free magnetic fields which
describe a variety of physical processes, including plasma equilibrium. As a
byproduct, the topological characterization of force-free magnetic fields
associated with current-carrying links, as conjectured by Crager and Kotiuga,
is shown to be necessary and conditions for sufficiency are given. Along the
way a paradox is exposed: The seemingly unintuitive mathematical tools, often
associated to higher dimensional topology, have their origins in three
dimensional contexts but in the hands of late-onset visually impaired
topologists. This paradox was previously exposed in the context of algorithms
for the visualization of three-dimensional magnetic fields. For this reason,
the paper concludes by developing connections between mathematics and cognitive
science in this specific context.Comment: 20 pages, no figures, a paper which was presented at a conference in
honor of the 60th birthdays of Alberto Valli and Paolo Secci. The current
preprint is from December 2014; it has been submitted to an AIMS journa
Topological complexity of motion planning in projective product spaces
We study Farber's topological complexity (TC) of Davis' projective product
spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's
coming from at least two sphere factors is (much) lower than the dimension of
the manifold. This is in high contrast with the known situation for (usual)
real projective spaces for which, in fact, the Euclidean immersion dimension
and TC are two facets of the same problem. Low TC-values have been observed for
infinite families of non-simply connected spaces only for H-spaces, for finite
complexes whose fundamental group has cohomological dimension not exceeding 2,
and now in this work for infinite families of PPS's. We discuss general bounds
for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute
these invariants for specific families of such manifolds. Some of our methods
involve the use of an equivariant version of TC. We also give a
characterization of the Euclidean immersion dimension of PPS's through
generalized concepts of axial maps and, alternatively, non-singular maps. This
gives an explicit explanation of the known relationship between the generalized
vector field problem and the Euclidean immersion problem for PPS's.Comment: 16 page
Equivariant topological complexity
We define and study an equivariant version of Farber's topological complexity
for spaces with a given compact group action. This is a special case of the
equivariant sectional category of an equivariant map, also defined in this
paper. The relationship of these invariants with the equivariant
Lusternik-Schnirelmann category is given. Several examples and computations
serve to highlight the similarities and differences with the non-equivariant
case. We also indicate how the equivariant topological complexity can be used
to give estimates of the non-equivariant topological complexity.Comment: v1: 19 pages; v2: 14 pages. Final version, to appear in Algebraic &
Geometric Topolog
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