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 $f^{\ast }TN-TM$ 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.

Cited by in corpus (2)

• Regular embeddings of manifolds and topology of configuration spaces
• Geometric coincidence results from multiplicity of continuous maps