Mathematical Sciences Geometry/Topology Seminar

Abstract: The purpose of this lecture is to describe a connection between certain choices of free groups, and pure braid groups with the homotopy groups of the 2-sphere. The results are non-computational in nature. This work is joint with Jie Wu.

There is a "non-standard" homomorphism of the free group on n-letters F(n) to the (n+1)-st pure braid group P(n+1), f:F(n) ---> P(n+1). Some properties of this homomorphism are as follows:

1. The homomorphism f is an embeddding by appealing to the structure of the Vassiliev invariants of pure braids.
2. A natural subquotient of F(n) is isomorphic to the (n+1)-st homotopy group of the 2-sphere.
3. Illustrations of links which arise by "closing" pure braids, and which represent elements in homotopy groups will be given. For example, a choice of link obtained from a pure 3-stranded braid which represents the classical Hopf map from the 3-sphere to the 2-sphere is given by the Borromean rings.
4. Further properties will be addressed.