Curves on surfaces

This is a course about very low dimensional topology, specifically about 1-manifolds immersed or embedded in 2-manifolds. Some possible topics with some suggested references are below. (links should work on campus, or off-campus, using PassKey)

Geometry of the curve complex

S. Hensel, P. Przytycki, R. Webb. "1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs." J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 755–762. MR3336835.
U. Hamenstӓdt. "Geometry of the complex of curves and of Teichmüller space." in Handbook of Teichmüller theory. Vol. I, 447–467, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Zürich, 2007. MR2349677
S. Schleimer. "Notes on the complex of curves." unpublished notes.

Gabai's Simple Loop Theorem

D. Gabai. "The simple loop conjecture." J. Differential Geom. 21 (1985), no. 1, 143–149. MR806708

Intersection numbers and self-intersection numbers

J. Hass and P. Scott. "Intersections of curves on surfaces." Israel J. Math. 51 (1985), no. 1-2, 90–120. MR804478
M. Chas. "The Goldman bracket and the intersection of curves on surfaces." in Geometry, groups and dynamics, 73–83, Contemp. Math., 639, Amer. Math. Soc., Providence, RI, 2015. MR3379820

Cube complexes from curves on surfaces

T. Aougab and J. Gaster. "Curves intersecting exactly once and their dual cube complexes." Groups Geom. Dyn. 11 (2017), no. 3, 1061–1101. MR3692906
J. Gaster. "Infima of length functions and dual cube complexes." Algebr. Geom. Topol. 17 (2017), no. 2, 1041–1057. MR3623681

Automorphisms of surfaces

One of the reasons to study curves on surfaces is to understand automorphisms of surfaces and the mapping class group of a surface. Two books on this topic:

A. Casson and S. Bleiler. "Automorphisms of Surfaces after Nielsen and Thurston." Online here.
B. Farb and D. Margalit. "A Primer on Mapping Class Groups." Online here.

Schedule

Date	Speaker	Topic	Reference
23 Aug	Jason	Intro to the course, and some hyperbolic geometry	Casson and Bleiler
28 Aug	Jason	Some hyperbolic geometry	Bruno Martelli, An introduction to Geometric Topology, especially chapter 6.
30 Aug	Jason	More hyperbolic geometry.
4 Sep	Jason	The bigon criterion.
6 Sep	John	Separability in surface groups	Peter Scott, Subgroups of surfaces groups are almost geometric.
13 Sep	Philip	Minimally self-intersecting curves	J. Hass and P. Scott. "Intersections of curves on surfaces." Israel J. Math. 51 (1985), no. 1-2, 90–120. MR804478
20 Sep	Chaitanya	Introduction to Goldman bracket	M. Chas. "The Goldman bracket and the intersection of curves on surfaces." in Geometry, groups and dynamics, 73–83, Contemp. Math., 639, Amer. Math. Soc., Providence, RI, 2015. MR3379820 and "Minimal intersection of curves on surfaces," Geom. Dedicata 144 (2010), 25–60. MR250416.
27 September	Jim	Gabai's simple loop theorem	D. Gabai. The simple loop conjecture.
16 October	John	The simple loop theorem for Seifert spaces	J. Hass. Minimal surfaces in manifolds with S¹ actions and the simple loop conjecture for Seifert fibered spaces.

Jason Manning's home page.

Last Updated 2018-10-17