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- PL Morse theory
Introduction to PL Morse theory. Affine polyhedral complexes and the Morse lemma. Application: the Solomon-Tits theorem. Arc complex of a punctured surface.
- Finiteness properties of groups
Proof, using PL Morse theory, that the arc complex of a disk with punctures on the boundary is a sphere, and otherwise it is contractible. Definition of various finiteness properties of groups. The Eilenberg-Ganea conjecture. The Kervaire-Laudenbach conjecture.
Mapping class groups have type VF. Kernels of right angled Artin groups. Potential counterexamples to Eilenberg-Ganea. A controlled topology conjecture that would imply that Eilenberg-Ganea is false.