Mathematics – Geometric Topology
Scientific paper
2008-01-25
Mathematics
Geometric Topology
30 pages, 7 figures, 1 table, typos were corrected
Scientific paper
We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X contains an embedded representative if and only if the Goldman Lie bracket of X with the third power of X is zero. The proof uses combinatorial group theory and Chas' combinatorial description of the bracket recast here in terms of an exposition of the Cohen-Lustig algorithm. Using results of Ivanov, Korkmaz and Luo there are corollaries characterizing which permutations of conjugacy classes are related to diffeomorphisms of the surfaces. The problem is motivated by a group theoretical statement from the sixties equivalent to the Poincare conjecture due to Jaco and Stallings and by a question of Turaev from the eighties. Our main theorem actually counts the minimal possible number of self-intersection points of representatives of a conjugacy class X in terms of the bracket of X with the third power of X.
Chas Moira
Krongold Fabiana
No associations
LandOfFree
An algebraic characterization of simple closed curves on surfaces with boundary 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 An algebraic characterization of simple closed curves on surfaces with boundary, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An algebraic characterization of simple closed curves on surfaces with boundary will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-412728