Projected and near-projected embeddings

A stable smooth map $f:N\to M$ is called “$k$-realizable” if its composition with the inclusion $M\subset M\times {\mathbb{R}}^k$ is $C^0$-approximable by smooth embeddings; and a “$k$-prem” if the same composition is $C^{\mathrm{\infty }}$-approximable by smooth embeddings, or equivalently if $f$ lifts vertically to a smooth embedding $N\to M\times {\mathbb{R}}^k$. It is obvious that if $f$ is a $k$-prem, then it is $k$-realizable. We refute the long-standing conjecture that the converse is always true. Namely, for each $n=4k+3\ge 15$ there exists a stable smooth immersion $S^n\to {\mathbb{R}}^{2n-7}$ that is $3$-realizable but is not a $3$-prem. We also prove the converse in a wide range of cases. A $k$-realizable stable smooth fold map $N^n\to {\mathbb{R}}^{2n-q}$ is a $k$-prem if $q\le n$ and $q\le 2k-3$;orif$q<n/2$and$k=1$;orif$q\in{2k-1,,2k-2}$and$k\in{2,4,8}$and$n$ is sufficiently large.