39 research outputs found
On fiber diameters of continuous maps
We present a surprisingly short proof that for any continuous map , if , then there exists no bound on
the diameter of fibers of . Moreover, we show that when , the union of
small fibers of is bounded; when , the union of small fibers need not
be bounded. Applications to data analysis are considered.Comment: 6 pages, 2 figure
Estimating the higher symmetric topological complexity of spheres
We study questions of the following type: Can one assign continuously and
-equivariantly to any -tuple of distinct points on the sphere
a multipath in spanning these points? A \emph{multipath} is a
continuous map of the wedge of segments to the sphere. This question is
connected with the \emph{higher symmetric topological complexity} of spheres,
introduced and studied by I. Basabe, J. Gonz\'alez, Yu. B. Rudyak, and D.
Tamaki. In all cases we can handle, the answer is negative. Our arguments are
in the spirit of the definition of the Hopf invariant of a map by means of the mapping cone and the cup product.Comment: This version has minor corrections compared to what published in AG