Lifting generic maps to embeddings. The double point obstruction
Sergey A. Melikhov
TL;DR
<3-5 sentence high-level summary> The paper develops a rigorous obstruction theory for lifting a generic map f: N^n → M^m to an embedding N ↪ M×R^k, tying realizability to the existence of an equivariant map from the double point locus Δ_f to S^{k-1} under a Z/2-action. Under the dimension constraint 2(m+k) ≥ 3(n+1) and m ≥ n, the authors prove that such a lift exists if and only if the equivariant obstruction vanishes, and provide both PL and smooth versions along with a Main Lemma and detailed PL and smooth constructions. The results yield corollaries including resolving Petersen’s problem in many cases, and they connect to stability theory, 2-jet/1-jet transversality, and stable maps via Appendices. The work also develops a comprehensive framework for stable PL maps and provides intricate lifting lemmas (Main Lemma) that enable a constructive Whitney-trick-like approach to eliminate self-intersections while preserving equivariant data. These contributions advance the understanding of when maps can be lifted to embeddings in product spaces and illuminate the geometry of double/triple point loci in high dimensions.
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 $M\times M\setminusΔ_M$.