Scheduling
Fridays 12–2 PM EST.
Week 1 (May 21): Preliminary review (point-set)
Topologies, continuity, homeomorphisms, subspaces and products, closures, open/closed/clopen, bases, retracts
Compactness, metric spaces, completeness, boundedness
Connectedness (path-connected & “locally” versions), totally disconnected
Separation axioms, Hausdorff spaces, normal, regular
The tube lemma
Common counterexamples, such as the topologist’s sine curve
Week 2 (May 28): The fundamental group
Van Kampen
Homotopic maps and 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
Mayer-Vietoris, Künneth, UCT
Cellular homology (direct definition)
Relative homology
Facts for manifolds (dim 3 and 4), Poincaré 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 and degree theory
Degree theory
Lefschetz, Brouwer, hairy ball theorem, Borsuk-Ulam
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
Qualifying exams: Monday and Tuesday, August 16–17
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
Mayer-Vietoris
Universal coefficient theorem
Künneth formula
Poincaré duality
Lefschetz fixed point formula
Hopf index theorem
Čech cohomology and de Rham cohomology
Equivalence between singular, Čech and de Rham cohomology