Lifting generic maps to embeddings. The double point obstruction

Abstract

Given a generic PL map or a generic smooth fold map $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\to M\times {\mathbb{R}}^k$ if and only if its double point locus $\{(x,y)\in N\times N\mid f(x)=f(y),x\ne y\}$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1990 question of P. Petersen and obtain some other applications. We also discuss several criteria for lifting of a non-degenerate PL map or a $C^0$-stable smooth map $f:N^n\to M^m$, where $m\ge n$, to an embedding in $M\times \mathbb{R}$, elaborating on V. Poénaru's observations. In particular, the existence of such a lift is determined by the equivariant homotopy type of the diagram consisting of the three projections from the triple point locus $\{(x,y,z)\in N\times N\times N\mid f(x)=f(y)=f(z),x\ne y\ne z\ne x\}$ to the double point locus. The three Appendices, which can be read independently of the rest of the paper, are devoted to stable and generic maps. Appendix B introduces an elementary theory of stable PL maps. Appendix C extends the 2-multi-0-jet transversality theorem over the usual compactification of $M\times M\setminus \hat{I}{”}_M$.

46 pages. v9: Minor changes. In older versions, some content has moved between this paper and the companion paper arXiv:2011.01402