Algebraic structures related to closed curves on surfaces.

1. Algebraic structures related to closed curves on surfaces.

2. Definition • A surface S is a topological space, such that • S is Hausdorff • For every point s in S there exist a neighborhood of s homeomorphic to a neighborhood of a point in a closed half-plane of R2

3. A surface can be characterized by • Genus (number of “handles”) • Number of boundary components (“holes”)

20. Orientation of a surface • A surface is oriented if it is “two sided” or equivalently, if it does not contain a Möbius strip.

21. The Klein bottle, another non-orientable surface

22. Based oriented curves on surface

23. Fundamental group of a surface Choose p in S. (oriented curves based at p) quotiented by (homotopic relative to p) This is a group. Denote it by π

24. Fundamental group of a surface If the surface S has non-empty boundary then it is a free group with 2.genus + b-1 generators. If S has empty boundary and genus g then the group can be written as a group with 2g generators and one relation.

25. Free homotopy class of closed curves on a surface: (closed curves on S)/homotopy.This is a set. Denote it by π*

26. Definition • If a and b are free homotopy classes of curves on a surface then the minimal intersection number of a and b is the minimum number of intersection points of representatives of a and b. • The minimal self-intersection number of a is the minimum number of self-intersection points of representatives of a

27. S is a fixed oriented surface • π denotes the fundamental group of S • π* denotes the set of free homotopy classes of elements of π (this is the set of conjugacy classes of elements of π ) • Denote by V(π*) the vector space generated by π*.

28. The Goldman bracket (Goldman, 86) [ , ]:V(π*) x V(π*) →V(π*)

29. The Goldman bracket (Goldman, 86) [ , ]:V(π*) x V(π*) →V(π*) [b1 b2 , x]= - b1 b2 x + b2 b1 x

30. The Goldman bracket cont. [b1 b2 , x]= b2 b1 x - b1 b2 x

31. The Goldman bracket is well defined

32. For every triple of elements,a, b and c in V(π*), [[a,b],c] + [[b,c],a] + [[c,a],b] = 0 (Jacobi identity)

33. Theorem (Goldman, 86) • The bracket is well defined. • It is skew-symmetric [a,b]=-[b,a] • Satisfies the Jacobi identity, [[a,b],c]+[[b,c],a]+[[c,a],b]=0 In other words: V(π*), the vector space generated by free homot. classes of curves on an orientable surface has a Lie algebra structure.

34. What can we say about the Lie algebra of curves on surfaces? • (C.) There exists a combinatorial presentation of the Lie algebra when the surface has non-empty boundary. • (C.) Counts minimal intersection of two curves when one of the curves is simple. • (C. - Krongold) Counts minimal number of self intersection points of a curve. In particular, characterizes simple closed curves (when S has non-empty boundary.) • (C. - Sullivan) Generalization of Lie bracket for manifolds of dimension larger than two.

35. Recall: The center of a Lie algebra L is the set of elements a of b such that [a,b] =0 for all b in L. Theorem: (Etingof) The center of the Lie algebra of curves on surfaces consist in • 0 if the surface has empty boundary • Linear combination of classes of curves parallel to the boundary if surface has non-empty bondary. • Proof: One way is clear

36. Given the Goldman Lie algebra, can one recover the surface? • Compute the genus of the surface? (Use characterization of simple closed curves and that 3.genus -3+b? is the maximal number of simple closed curves that do not intersect) • Number of boundary components? (Compute the center and use Etingof’s Theorem.)

37. Remark: In the case of surfaces with boundary, there is one-to-one correspondence between Cyclically reduced cyclic words on the generators and inverses of generators of the fundamental group of S Free homotopy classes of curves on S a c b a b a

38. Glue edges of polygonsGet a surface

39. The generators of the fundamental group

40. The words aab and ab “talk” about their intersection points

41. Theorem:Reinhart (60’s)(boundary)Cohen-Lustig (80’s)(boundary)Lustig (80’s)(no boundary) • The minimal intersection points of two free homotopy classes of curves on a surface can be counted using the cyclic words labeling the classes.

42. To compute the bracket [aab,ab]aab ba = aabba baa ab = baaab

43. Theorem (C.) : The Lie algebra has a purely combinatorial presentation. Moreover, you can compute the Lie bracket of a pair of words at http://www.math.sunysb.edu/moira/

44. Theorem (C - Krongold) • If W is a cyclically reduced word which is not the power of another word then the number of terms of the bracket [W2 ,W3] is equal to 2.3 times the minimal self-intersection number of W.

45. The Lie bracket is a refinement of the intersection product of curves. Questions: How good is it this refinement? Does it really identify all non-removable intersection points? Or is there cancellation? - b1 b2 x + b2 b1 x Intersection number = 1-1=0

46. No, the bracket of intersecting pairs can be zero [aab,ab]= - aab ba + baa ab=0

47. Is the number of terms of the bracket counted with multiplicity, equal to the minimum intersection number of a pair of conjugacy classes of curves?

48. However, Theorem (Goldman, 86) If the bracket of two conjugacy classes of curves is zero and one of the classes contains a simple representative, then the classes have disjoint representatives.

49. Theorem (C.) The Goldman bracket of two curves, one of them simple, has as many terms as the intersection number.