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

In this paper, we study the space of compactly supported embeddings between Euclidean spaces, ${\mathrm{E}\mathrm{m}\mathrm{b}}_c({\mathbb{R}}^j,{\mathbb{R}}^n)$. By utilizing hairy graphs, we construct elements in the homotopy groups ${\pi }_{\bullet }({\overline{\mathrm{E}\mathrm{m}\mathrm{b}}}_c({\mathbb{R}}^j,{\mathbb{R}}^n))\otimes \mathbb{Q}$ 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 ${\mathrm{E}\mathrm{m}\mathrm{b}}_c({\mathbb{R}}^{n-2},{\mathbb{R}}^n)$ are infinite-dimensional in infinitely many degrees when $n\ge 5$ is odd.