18.937 - Topics in Geometric Topology - Spring 2006

D. Auroux - Mondays & Wednesdays, 9:30-11 in 2-136.

Lecture notes:

• Wed Feb 8: overview of the course.
• Mon Feb 13: configuration spaces, braids, pure braids; P_n vs. P_{n-1}.
• Tue Feb 21: Artin presentations of B_n and P_n.
• Wed Feb 22: braids as homeomorphisms of the plane; action of B_n on the free group F_n; the center of B_n.
• Mon Feb 27: the image of B_n in Aut(F_n); B_n(S^2); braids and links; sawteeth and Alexander's theorem.
• Wed Mar 2: Markov's theorem.
• Mon Mar 7: positive braids and the Garside embedding theorem.
• Wed Mar 9: Elrifai-Morton's solution to the word problem; left-canonical form.
• Mon Mar 13: the conjugacy problem; band generators; braid cryptography.
• Wed Mar 15: mapping class groups; Map(S_{g,r}) vs. Map_r(S_g) and Map(S_g).
• Mon Mar 20: Map_n(S) vs. Map(S); mapping class groups of the sphere.
• Wed Mar 22: mapping class groups are generated by Dehn twists.
• Mon Apr 3: presentations of mapping class groups; branched coverings; liftability.
• Wed Apr 5: the lifting homomorphism; relations among Dehn twists; presentation of Map_g.
• Mon Apr 10: open book structures on 3-manifolds; stabilization of open books; contact structures.
• Wed Apr 12: constructing contact structures from open books.
• Wed Apr 19: constructing open books from contact structures; plane curves and braid monodromy.
• Mon Apr 24: braid monodromy for projective curves; monodromy factorizations, Hurwitz equivalence; Hurwitz curves.
• Wed Apr 26: Hurwitz curves and braid monodromy factorizations; isotopy problem; complex projective surfaces.
• Mon May 1: generic projections of complex projective surfaces and their branch curves.
• Wed May 3: the Zariski-Van Kampen theorem; symmetric group monodromy, Chisini conjecture; examples of braid monodromies.
• Mon May 8: symplectic geometry; symplectic branched coverings; monodromy invariants of symplectic 4-manifolds.
• Wed May 10: Lefschetz fibrations and their monodromy; Thurston and Gompf's theorems.
• Mon May 15: Lefschetz pencils and their monodromy; the pencil of conics on CP^2; Gompf and Donaldson's theorems; pencils vs. branched covers: lifting the monodromy
• Wed May 17: classification of genus 1 and 2 Lefschetz fibrations; fiber sums; classification of Lefschetz fibrations up to fiber sums.

Course description

This course will focus on braid groups, mapping class groups, and their applications to the topology of plane curves and complex surfaces.

The first half of the course will be mostly group-theoretic and loosely based on the book "Braids, Links, and Mapping Class Groups" by Joan Birman (Princeton University Press). It will cover the following topics:

• Braid groups and mapping class groups: definitions and basic properties.
• Braids and links.
• Presentations of braid groups and mapping class groups.
• Branched covers, liftable braids and lifting homomorphisms.
• Algorithmics of the braid group: Garside property; word and conjugacy problems.

The second half of the course will focus on applications to algebraic geometry (especially the study of plane curves and complex surfaces) and symplectic topology. We will in particular review Moishezon and Teicher's work on the braid monodromy of plane curves, as well as recent results on the topology of Lefschetz fibrations. The main topics will be:

• The braid monodromy of a complex plane curve; examples.
• Fundamental groups of plane curve complements: the Zariski-Van Kampen theorem.
• Lefschetz fibrations, branched covers, and their monodromy.
• Calculation techniques for the braid monodromy of algebraic surfaces.
• Hurwitz problems, isotopy problems, and stabilization properties.