Eliminating higher-multiplicity intersections in the metastable dimension range

Abstract: The procedure to remove double intersections called the Whitney trick is one of the main tools in the topology of manifolds. The analogues of Whitney trick for $r$-tuple intersections were `in the air' since 1960s. However, only recently they were stated, proved and applied to obtain interesting results. Here we prove and apply the $r$-fold Whitney trick when general position $r$-tuple intersection has positive dimension. A continuous map $f\colon M \to B^d$ from a manifold with boundary to the $d$-dimensional ball is called proper, if $f^{-1}(\partial B^d)=\partial M$. Theorem. Let $D=D_1\sqcup\ldots\sqcup D_r$ be disjoint union of $k$-dimensional disks, and $f:D\to B^d$ a proper map such that $f\partial D_1\cap\ldots\cap f\partial D_r=\emptyset$, and 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 continuously to $D_1\times\ldots\times D_r$. If $rd\ge (r+1)k+3$, then there is a proper map $\bar f:D\to B^d$ such that $\bar f=f$ on $\partial D$ and $\bar fD_1\cap\ldots\cap \bar fD_r=\emptyset$.