  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 $N$.

Algebraic Topology;Metric Geometry