Lifting Generic Maps to Embeddings. The Double Point Obstruction

Given a generic PL map or a generic smooth fold map f:Nⁿ→ Mᵐ, where m≥ n and 2(m+k)≥ 3(n+1), we prove that f lifts to a PL or smooth embedding N→ M× Rᵏ if and only if its double point locus (f× f)⁻¹(ΔM)ΔN admits an equivariant map to Sᵏ⁻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)× R³. We also discuss several criteria for lifting of maps N→ M to embeddings in M× R, elaborating on Poenaru's observations. 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× MΔM.