Minimal intersection of curves on surfaces

Details Minimal intersection of curves on surfaces Minimal intersection of curves on surfaces

2007-06-16

arxiv.org/abs/0706.2439v2

Mathematics

Geometric Topology

49 pages, 9 figures

Scientific paper

In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the classes has a simple representative we give a combinatorial group theory description of the terms of the Lie bracket and prove that this bracket has as many terms, counted with multiplicity, as the minimal number of intersection points of a and b. In other words the bracket with a simple element has no cancellation and determines minimal intersection numbers. We show that analogous results hold for the Lie bracket (also discovered by Goldman) of unoriented curves. We give three applications: a factorization of Thurston's map defining the boundary of Teichmuller space, various decompositions of the underlying vector space of conjugacy classes into ad invariant subspaces and a connection between bijections of the set of conjugacy classes of curves on a surface preserving the Goldman bracket and the mapping class group.

Affiliated with

Also associated with

No associations

Rating

Minimal intersection of curves on surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Minimal intersection of curves on surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Minimal intersection of curves on surfaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-468422