The procedure to remove double intersections of a map called the Whitney trick is one of the main tools in the topology of manifolds. The analogues of Whitney trick for $r$-tuple intersections of a map 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 intersections have positive dimension. 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 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 $\bar fD_1\cap\ldots\cap \bar fD_r=\emptyset$.
翻译：移除名为 Whitney 的地图的双倍交叉点的程序是图层学中的主要工具之一。 地图 $r$ 的 Whitney 的 惠特尼 的 trick 类比于 $r$- tuple 交叉点是 1960 年代 的 “ 在空气中 ” 。 然而, 只是在最近才声明、 并应用它们来获得有趣的结果。 在这里, 当通用位置 $r$- tuple 交叉点具有正维度时, 我们证明并应用 $r 的 惠特尼 的 trick 。 理论。 让 $D== D_ 1\ times\ ldotts\ times D_ r= dbar$x, d_ d_ d_ d_ d$\ d_ r\ x\ f_ d\\\\ cal_ f_ d_ d_ d_ dex\ f_ d_ d_ d_ d_ dr\ x\\\ dex a f_ d_ d_ dr_ fr_ d_ d_ dr_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ axxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ a_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_ d_xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx_
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