Multiplicity of continuous maps between manifolds

TL;DR

This paper establishes lower bounds on the multiplicity of continuous maps between manifolds using characteristic classes, with specific results for maps from real projective spaces to Euclidean spaces.

Contribution

It introduces a method to estimate the multiplicity of maps between manifolds via characteristic classes of associated vector bundles, providing new bounds.

Findings

Bound on multiplicity for maps from real projective spaces to Euclidean spaces

Characteristic classes guarantee multiplicity bounds

Specific estimates for certain dimensions

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 $f^*TN-TM$ that guarantee a bound on the multiplicity of $f$. In particular, we prove some non-trivial bound on the multiplicity for a continuous map of a real projective space of certain dimension into a Euclidean space.