Diagrams for primitive cycles in spaces of pure braids and string links (Q6567991)

In this paper, the authors consider a graphing operation from an iterated loop space of configuration spaces in Euclidean space, to a space of string links (also in Euclidean space). Their primary result is that this map is injective on homotopy groups, at least on a large submodule spanned by certain iterated Whitehead products.\N\NThe authors conjecture the map is injective rationally, i.e. from the rational homotopy groups of configuration spaces in Euclidean space. They have informed me that Victor Turchin has proven this result recently.\N\NIt should be noted that these graphing maps are related to the theory of pseudo-isotopy embeddings. Assuming the manifold \(M\) is a submanifold of \(N\), an embedding of \(M \times I\) into another manifold \(N \times I\) which is the identity on both the ``initial face'' \(M \times \{0\}\) and \(\partial M \times I\) is called a pseudo-isotopy embedding. The space of pseudo-isotopy embeddings is the total space of a bundle over the space of embeddings of \(M\) in \(N\), and the fiber is the space of embeddings of \(M \times I\) in \(N \times I\) with fixed boundary conditions.\N\NThe graphing map considered by the authors is the connecting map for the homotopy LES of this bundle. Thus, their results on injectivity could be interpreted as a connectivity result on the space of pseudo-isotopy embeddings. Presumably, one could deduce these results from Tom Goodwillie's dissertation, with perhaps a bit of additional work. Goodwillie's dissertation was the first instance of functor calculus in the literature, the embedding calculus associated to pseudo-isotopy embedding spaces.