Title: Eliminating higher-multiplicity intersections in the metastable dimension range,
Abstract: The $r$-fold analogues of Whitney trick were `in the air' since 1960s.
However, only in this century they were stated, proved and applied to obtain
interesting results, most notably by Mabillard and Wagner. Here we prove and
apply a version of the $r$-fold Whitney trick when general position $r$-tuple
intersections have positive dimension.
Theorem. Assume that $D=D_1\sqcup\ldots\sqcup D_r$ is disjoint union of
$k$-dimensional disks, $rd\ge (r+1)k+3$, and $f:D\to B^d$ a proper PL (smooth)
map such that $f\partial D_1\cap\ldots\cap f\partial D_r=\emptyset$. If the map
$$f^r:\partial(D_1\times\ldots\times D_r)\to
(B^d)^r-\{(x,x,\ldots,x)\in(B^d)^r\ |\ x\in B^d\}$$ extends to
$D_1\times\ldots\times D_r$, then there is a proper PL (smooth) map $\overline
f:D\to B^d$ such that $\overline f=f$ on $\partial D$ and $\overline
fD_1\cap\ldots\cap \overline fD_r=\emptyset$.