Milnor's \mu-invariants and higher dimensional link homotopy in arbitrary manifolds
by
Ulrich Koschorke
University of Siegen, Germany

Abstract for the Rokhlin Memorial 1999

In order to get a first rough understanding of the sheer overwhelming multitude of classical links Milnor introduced the concept of link homotopy in a seminal Annals of Mathematics paper in 1954. Moreover, he extracted strong link homotopy invariants from an algebraic analysis of the fundamental groups of link complements.

Using an entirely different approach (via the homotopy theory of configuration spaces, punctured manifolds and wedges of spheres) we generalize Milnor's \mu-invariants to higher dimensional (possibly singular) link maps into arbitrary manifolds. In several important cases these generalized \mu-invariants or our (stronger and more basic) \kappa-invariants are seen to classify link maps fully up to link homotopy. Also a detailed comparison is made between \mu-invariants of link maps into  S^q ×R^{m -q},  1 <= q <= m,  or other knot complements on one hand, and into  R^m  on the other hand.

Date received: May 19, 1999