Infinite-dimensionality of the rational homotopy groups of the space of long embeddings of codimension 2

Abstract

In this paper, we study the space of compactly supported embeddings between Euclidean spaces, $\mathrm{Emb}_c(\mathbb{R}^j, \mathbb{R}^n)$. By utilizing hairy graphs, we construct elements in the homotopy groups $π_{\bullet}(\overline{\mathrm{Emb}}_c(\mathbb{R}^j, \mathbb{R}^{n})) \otimes \mathbb{Q}$ corresponding to certain uni-trivalent graphs in the model. We then prove that these elements are nontrivial. Consequently, we show that the rational homotopy groups of $\mathrm{Emb}_c(\mathbb{R}^{n-2}, \mathbb{R}^n)$ are infinite-dimensional in infinitely many degrees when $n \ge 5$ is odd.