Topologically Nontrivial Counterexamples to Sard’s Theorem

Abstract

We prove the following dichotomy: if n = 2, 3 and f ∈ C 1 (S n+1 , S n ) is not homotopic to a constant map, then there is an open set Ω ⊂ S n+1 such that rank df = n on Ω and f (Ω) is dense in S n , while for any n ≥ 4, there is a map f ∈ C 1 (S n+1 , S n ) that is not homotopic to a constant map and such that rank df < n everywhere. The result in the case n ≥ 4 answers a question of Larry Guth.

Lower bounds on mapping content and quantitative factorization through trees

“…Remark 1.7. In the case n ≤ 1 or m = 0, mappings from [0, 1] n+m with vanishing or small H n,m ∞ are easy to characterize, while in the case n > 2 the problem appears difficult due to the constructions of topologically non-trivial low-rank Lipschitz mappings in [16,9]. Thus, the case n = 2 that we address here is simply the first remaining open problem in a long list.…”

C1 mappings in R5 with derivative of rank at most 3 cannot be uniformly approximated by C2 mappings with derivative of rank at most 3

“…. • σ j i of similarity transformations τ j and σ j that are used at the very end of the proof of Lemma 5.1 in [2]. The Cantor set E F is the same as the Cantor set C in the proof of Lemma 5.1 in [2].…”

“…The mapping F is obtained through an iterative construction, described in detail in [2]. We shall present here a sketch of that construction.…”

“…) onto the m-dimensional skeleton of the grid: first, we project the outside of the unit cube onto the boundary of the cube using the nearest point projection π, then in each of the N closed cubes of the grid we use the mapping R defined in the proof of Lemma 8. Even though π is not smooth, this composition turns out to be smooth (see [2,Lemma 5.3]).…”

“…Similarly, some additional work is necessary to glue F with scaled copies of F in each of the balls B i in a differentiable way. These are purely technical difficulties, the details are provided in [2].…”