Graphing injectivity for rational homotopy and homology of string-link spaces

Fix $m\geq 2$, $k\geq 1$, and $n\geq 3$. Let

$$G:\Omega^k\operatorname{Conf}(m,\mathbb R^n)\longrightarrow \operatorname{Emb}_c\left(\coprod_m\mathbb R^k,\mathbb R^{n+k}\right)$$

be the graphing map. Graphing injectivity conjecture. The induced maps on rational homotopy and rational homology are injective:

$$\pi_*(G)\otimes\mathbb Q\quad\text{and}\quad H_*(G;\mathbb Q)$$

are injective.

The conjecture strengthens the paper's integral injectivity result on the subspace of homotopy classes represented by brackets with distinct generators. It concerns the full rational homotopy and homology of the looped configuration space and the corresponding string-link space.