Multiplicity of continuous maps between manifolds
Abstract
We consider a continuous map $f :M\to N$ between two manifolds and try to estimate its multiplicity from below, i.e. find a $q$tuple of pairwise distinct points $x_1,..., x_q\in M$ such that $f(x_1) = f(x_2) = ... = f(x_q)$. We show that there are certain characteristic classes of vector bundle $f^*TNTM$ that guarantee a bound on the multiplicity of $f$. In particular, we prove some nontrivial bound on the multiplicity for a continuous map of a real projective space of certain dimension into a Euclidean space.
 Publication:

arXiv eprints
 Pub Date:
 February 2010
 arXiv:
 arXiv:1002.0660
 Bibcode:
 2010arXiv1002.0660K
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry;
 Mathematics  Metric Geometry;
 55M20;
 55M30;
 55M35;
 55R25;
 55R80;
 57R45