Spaces of polynomials without 3-fold real roots. (Q1411708)

Let \(P_n^d(\mathbb{R})\) denote the space of all monic real coefficient polynomials of degree \(d\) that have no real roots of multiplicity \(\geq n\). The authors study the homotopy type of this space for \(n=3\). The loop space \(\Omega S^{n-1}\) has, up to homotopy, a natural CW complex structure whose \(k(n-2)\)-skeleton is called the James \(k\)-th stage filtration and is denoted \(J_k(\Omega S^{n-1})\). It is known that there is a homotopy equivalence \(P_n^d(\mathbb{R})\simeq J_{[d/n]}(\Omega S^{n-1})\) for \(n\geq 4\) (induced by a map called the jet embedding); the authors ask if this holds for \(n=3\). They answer their question affirmatively when \(1\leq d \leq 17\). In addition, they show that the jet embedding is \([d/3]\)-connected and induces an integral homology equivalence (for \(n=3\) and all \(d\geq 1\)).