On the metastable Mabillard-Wagner conjecture

Abstract

The purpose of this note is to attract attention to the following conjecture (metastable $r$-fold Whitney trick) by clarifying its status as not having a complete proof, in the sense described in the paper. Assume that $D=D_1\bigsqcup \dots \bigsqcup D_r$ is disjoint union of $r$ disks of dimension $s$, $f:D\to B^d$ a proper PL map such that $f\partial D_1\cap \dots \cap f\partial D_r=\varnothing$, $rd\ge (r+1)s+3$ and $d\ge s+3$. If the map $$f^r:\partial (D_1\times \dots \times D_r)\to (B^d{)}^r-\{(x,x,\dots ,x)\in (B^d{)}^r|x\in B^d\}$$ extends to $D_1\times \dots \times D_r$, then there is a PL map $\overline{f}:D\to B^d$ such that $$\overline{f}=f\text{on}{D}_r\cup \partial D\text{and}\overline{f}{D}_1\cap \dots \cap \overline{f}{D}_r=\varnothing .$$

5 pages