Local multiplicity of continuous maps between manifolds

Let $M$ and $N$ be smooth (real or complex) manifolds, and let $M$ be equipped with some Riemannian metric. A continuous map $f\colon M\longrightarrow N$ admits a local $k$-multiplicity if, for every real number $\omega >0$, there exist $k$ pairwise distinct points $x_1,\ldots,x_k$ in $M$ such that $f(x_1)=\cdots=f(x_k)$ and $\diam\{x_1,\ldots,x_k\}<\omega$. 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 $f^*\tau N\oplus(-\tau M)$. For example, as a corollary of one criterion we deduce that for $k\geq 2$ a power of $2$, $M$ a compact smooth manifold with the integer $s:=\max\{\ell : \bar{w}_{\ell}(M)\neq 0\}$, and $N$ a parallelizable smooth manifold, if $s\geq \dim N-\dim M+1$ and $\bar{w}_{s}(M)^{k-1}\neq 0$, any continuous map $M\longrightarrow N$ 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 $N$.