Projected and Near-Projected Embeddings

Abstract

A stable smooth map f : N → M is called k-realizable if its composition with the inclusion M ⊂ M × ℝk is C0-approximable by smooth embeddings; and a k-prem if the same composition is C∞-approximable by smooth embeddings, or, equivalently, if f lifts vertically to a smooth embedding N ↪ M × ℝk. It is obvious that if f is a k-prem, then it is k-realizable. We refute the so-called “prem conjecture” that the converse holds. Namely, for each n = 4k+3 ≥ 15 there exists a stable smooth immersion Sn ↬ ℝ2n−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 Nn → M2n−q is a k-prem if q ≤ n and q ≤ 2k−3; or if q < n/2 and k = 1; or if q ∈ {2k−1, 2k−2} and k ∈ {2, 4, 8} and n is sufficiently large.