On spaces of long knots

Fix integers $p\le d$. The space of long knots (of type $(p,d)$) is the space of compactly supported embeddings of $R^p$ into $R^d$. In codimension at least three, i.e. when $p \le d-3$, work of Boavida de Brito–Weiss in embedding calculus shows that its homotopy type is closely related to that of the space of operad maps from $E_p$ to $E_d$. The rational homotopy groups of the latter were determined by Fresse–Turchin–Willwacher in terms of graph homology.

In this talk, I will discuss how one can access the homotopy type, localised away from 2 and in a range of degrees, of spaces of long knots when $p \le d-3$ using methods from pseudoisotopy theory and orthogonal calculus. For $p = d-2$, when embedding calculus fails to fully adress this homotopy type, I will also describe more recent joint work with João Lobo Fernandes computing its rational homotopy groups in a range of degrees.