Scheduling
Fridays 122 PM EST

Week 1 (May 21): Preliminary Review (PointSet)
 Topologies, continuity, homeomorphisms, subspaces and products, closures, open/closed/clopen, bases, retracts
 Compactness, metric spaces, completeness, boundedness
 Connectedness (pathconnected & “locally” versions), totally disconnected
 mSeparation axioms, Hausdorff spaces, normal, regular
 The tube lemma
 Common counterexamples (sine curve)

Week 2 (May 38): The Fundamental Group
 Van Kampen
 Homotopic maps vs conjugacy, change of base point
 Nullhomotopic maps

Week 3 (June 4): Covering Spaces
 Definitions, HLP, deck transformations, regular covers, universal cover
 Classifications of covers, applications to subgroups of free groups
 Lifting criterion, Galois correspondence

Week 4 (June 11): CW Complexes, Pushouts
 Cellular chain complex
 Euler characteristic
 Hurewicz, LES in homotopy
 Simplicial complexes

Week 5 (June 18): Homology
 MayerVietoris, Kunneth, UCT
 Cellular homology (direct definition)
 Relative homology
 Facts for manifolds (dim 3 and 4), Poincare duality

Week 6 (June 25): Surfaces
 Definition of manifolds (sphere, torus, projective spaces)
 Connect sum, fundamental polygons and gluing
 Classification of surfaces, boundary components

Week 7 (July 2): Fixed Points & Degree Theory
 Degree theory
 Lefschetz, Brouwer, Hairy Ball Theorem, BorsukUlam
 Week 8 (July 9): Buffer
 Week 9 (July 16): Buffer
 Week 10 (July 23): Buffer
 Week 11 (July 30): Buffer
 Week 12 (August 6): Buffer
 Week 13 (August 13): Timed practice exam
 Quals: Monday and Tuesday, August 1617
Topics
 Fundamental groups
 Covering spaces
 Higher homotopy groups.
 Fibrations and the long exact sequence of a fibration
 Singular homology and cohomology
 Relative homology
 CW complexes and the homology of CW complexes.
 MayerVietoris
 Universal coefficient theorem
 Kunneth formula
 Poincare duality
 Lefschetz fixed point formula
 Hopf index theorem
 Cech cohomology and de Rham cohomology.
 Equivalence between singular, Cech and de Rham cohomology