# A k-component link defines a map T^k --> Conf_k S^3. Does the homotopy type capture Milnor's invariants?

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.
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