Connections Between Braid Groups, Homotopy Theory, and Low Dimensional Topology.

Topology of Arrangements and Applications
October 06, 2004 09:30 AM to 10:30 AM

Speakers: Frederick Cohen

Summary:
An elementary homomorphism from a free group to the pure braid group
yields interesting connections between braid groups, homotopy theory, and
low dimensional topology. This map induces a map on the Lie algebra
obtained from the descending central series. Further, this map induces a
morphism of simplicial groups. All of these maps are shown to be
injective.
Brunnian braids are discussed. The analogous maps of Lie algebras induced
on the filtration quotients of the mod-p descending central series is
again an injection. Using these facts it turns out that the homotopy
groups of this simplicial group, those of the 2-sphere, are isomorphic to
natural subquotients of the pure braid group. In addition, the mod-p
analogues give a connection between the classical unstable Adams spectral
sequence, and the mod-p analogues of Vassiliev invariants of pure braids.

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