Given a generic PL map or a generic smooth immersion $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\hookrightarrow M\times\mathbb R^k$ if and only if its double point locus $(f\times f)^{-1}(\Delta_M)\setminus\Delta_N$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1990 question of P. Petersen on whether the universal coverings of the lens spaces $L(p,q)$, $p$ odd, lift to embeddings in $L(p,q)\times\mathbb R^3$. We also show that if a non-degenerate PL map $N\to M$ lifts to a topological embedding in $M\times\mathbb R^k$ then it lifts to a PL embedding in there. 查看全文>>