Local multiplicity of continuous maps between manifolds

Abstract

Let $M$ and $N$ be smooth (real or complex) manifolds, and let $M$ be equipped with some Riemannian metric. A continuous map $f:M⟶N$ admits a local $k$-multiplicity if, for every real number $\omega >0$, there exist $k$ pairwise distinct points $x_1,\dots ,x_k$ in $M$ such that $f(x_1)=\cdots =f(x_k)$ and $\diam\{x_1,\ldots,x_k\}<ω$. In this paper we systematically study the existence of local $k$-mutiplicities and derive criteria for the existence of local $k$-multiplicity in terms of Stiefel--Whitney classes and Chern classes of the vector bundle $f^{\ast }\tau N\oplus (-\tau M)$. For example, as a corollary of one criterion we deduce that for $k\ge 2$ a power of $2$, $M$ a compact smooth manifold with the integer $s:=\max \{\ell :{\bar{w}}_{\ell }(M)\ne 0\}$, and $N$ a parallelizable smooth manifold, if $s\ge \dim N-\dim M+1$ and ${\bar{w}}_s(M{)}^{k-1}\ne 0$, any continuous map $M⟶N$ admits a local $k$-multiplicity. Furthermore, as a special case of this corollary we recover, when $k=2$, the classical criterion for the non-existence of an immersion $M\looparrowright N$ between manifolds $M$ and $N$.