Question

A k-component link defines a map T^k --> Conf_k S^3. Does the homotopy type of this map capture the Milnor invariants?

Some special cases:

• k=2, no, it's null homologous, but you can look instead at the map T^2 --> Conf_2 R^3, which captures linking number.
• k=3, Melvin et al. proved it does.

Answer 1

It's been a while since I've thought about this but I think Koschorke answered much of your question back in 1997 "A generalization of Milnor's mu-invariants to higher-dimensional link maps" Topology 36 (1997), no 2. 301--324. Scanning through the paper I see he recovers many of the mu invariants but not all. He lists it as an open question (6.3) if the homotopy class of the map T^k --> C_k R^3 is a complete link homotopy invariant of the link.

Related:

Brian Munson put these Koschorke "linking maps" into the context of the Goodwillie calculus in a recent arXiv paper. I've wondered for a while if you could use these types of maps to create a direct construction of the Cohen-Wu correspondence between the homotopy groups of S^2 and their corresponding simplicial quotient object made from the brunnian braid groups.