\( \newcommand{\EE}{\mathbb{E}} \newcommand{\RR}{\mathbb{R}} \newcommand{\Simple}{\mathcal{S}} \newcommand{\Curves}{\mathcal{C}} \newcommand{\Farey}{\mathcal{F}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\MCG}{\operatorname{MCG}} \newcommand{\PML}{\operatorname{PML}} \newcommand{\Homeo}{\operatorname{Homeo}} \newcommand{\Teich}{\operatorname{Teich}} \newcommand{\PML}{\operatorname{PML}} \)

The mapping class group
and the curve complex, Term I, 2008


Week Date of Tuesday Tuesday Friday Lecture notes Comments
3 Oct. 14 Outline of course. Classification of surfaces. Loops and arcs. Cutting, gluing. Reflections, twists, half-twists. Tuesday Friday How many Mobius bands can you embed in $\RR^3$?
4 Oct. 21 $\Homeo(S)$, $\MCG(S)$, $\Simple(S)$. Sphere and disk. Annulus and pants. Geometric intersection number. Bigon criterion. Tuesday Friday Why is the $\alpha$-cover of S an annulus?
5 Oct. 28 $\Simple(T)$, $\PML(S)$ and $\PML(T)$. $\Teich(T)$. The hyperelliptic element. Tuesday Friday Friday lecture cut short due to technical difficulties.
6 Nov. 4 $\MCG(T)$. Alexander method. Braid relation. Elliptic, parabolic, hyperbolic. Examples. $\Teich(S)$. Nielsen-Thurston classification. Braid relation. Tuesday Friday The hyperelliptic generates the center of $\MCG(T)$.
7 Nov. 11 Relations, hyperelliptic in MCG(S_2). Statement of Dehn-Lickorish theorem. Genus zero case. Surgery of arcs. Surgery of loops. Finish proof of theorem. Tuesday Friday $2g + 1$ twists suffice.
8 Nov. 18 $\Curves(S)$. Pants decompositions. Filling. Distance. Statement of Ivanov's theorem. Special cases. Hyperelliptics. Farey graph, $\Farey$. Tuesday Friday Harmonic vs hyperbolic Farey graph.
9 Nov. 25 $\Aut(\Farey)$. Link of a simplex. Crawling. Duality. Action of twists on $\Farey$. Large and small links. Arc complex. Pentagon lemma. Tuesday Friday Combinatorial data determines topological.
10 Dec. 2 Ivanov in genus zero. Pentagon in $S_{1,2}$. Second pentagon lemma. Tuesday Friday What does $C(S_{0,5})$ look like?
11 Dec. 9 Finished second pentagon. Combinatorial duals. Bracelets. Bicolored Farey graph. Finish Ivanov's theorem. A hint of Masur-Minsky I. $\delta$-hyperbolicity. Systoles. Tuesday Friday How many different colored pentagons are there?