Lifting Generic Maps to Embeddings. The Double Point Obstruction

Given a generic PL map or a generic smooth fold map f : N n → M m $$f\colon N^n\to M^m$$ , where m ≥ n $$m\ge n$$ and 2 ( m + k ) ≥ 3 ( n + 1 ) $$2(m+k)\ge 3(n+1)$$ , we prove that f lifts to a PL or smooth embedding N → M × ℝ k $$N\to M\times \mathbb {R}^k$$ if and only if its double point locus { ( x , y ) ∈ N × N ∣ f ( x ) = f ( y ) , x ≠ y } $$\{(x,y)\in N\times N\mid f(x)=f(y),\,x\ne y\}$$ admits an equivariant map to S k −1 $$S^{k-1}$$ . As a corollary we answer a 1990 question of P. Petersen and obtain some other applications. We also discuss several criteria for lifting a non-degenerate PL map or a C 0 $$C^0$$ -stable smooth map f : N n → M m $$f\colon N^n\to M^m$$ , where m ≥ n $$m\ge n$$ , to an embedding in M × ℝ $$M\times \mathbb {R}$$ , 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 ) ∈ N × N × N ∣ f ( x ) = f ( y ) = f ( z ) , x ≠ y ≠ z ≠ x } $$\{(x,y,z)\in N\times N\times N\mid f(x)=f(y)=f(z),\,x\ne y\ne z\ne x\}$$ to the double point locus. The three Appendices, which can be read independently of the rest of this chapter, 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 × M ∖ Δ M $$M\times M\setminus \Delta _M$$ .