Geometry/Topology First-Year Course
Texts: Hatcher, Spivak, Madsen and Tornehave, Milnor
FALL: Introduction to topology
Text suggestions: Hatcher, Bredon
- Weeks 1-2:
Topological spaces.
Examples: spheres, projective spaces,
surfaces (as identification
spaces of polyhedra, and as group
quotients). Homotopy equivalence.
More generally, spaces constructed as
cell complexes, yielding further
examples:
projective spaces, Grassmannians, flag
varieties.
Topological manifolds.
- Week 3:
Fundamental group; Van Kampen, and
computations with it.
Examples: Riemann surfaces, real
projective space, ...
- Weeks 4-5:
Covering spaces; examples drawn from
Riemann surfaces. Galois theory of
covering
spaces.
- Week 6: Topological groups, with
examples from classical groups.
- Weeks 7-8:
Principal bundles as generalizations of
covering spaces. Definition of
vector
bundles. Frame bundle, tangent bundle,
and cotangent bundle of embedded
submanifolds of $R^n.$ Grassmannians
and their tautological bundles.
- Week 9:
Classifying spaces (if time permits).
WINTER: Introduction to differentiable manifolds
Texts: Spivak v.1 (plus e.g. Milnor's Morse Theory
for Riemannian geometry).
Also, Madsen and Tornehave, "From
calculus to cohomology".
- Week 1: Differentiable manifolds,
definition and examples
Examples: Manifolds defined from
regular values, projective spaces
- Week 2: Smooth vector bundles; tangent
vectors as derivations and the
tangent,
cotangent bundles
Examples: P^1, Riemann surfaces,
projective spaces and Grassmannians
- Week 3: Higher rank tensors (associated
bundles via transition functions),
vector fields and flows
Examples: top forms, calculus ODE's,
vector fields from symmetries
- Weeks 4-5: Lie derivatives, Lie bracket,
Lie groups and Lie algebras
Examples: Cartan's formula, Heisenberg
group and algebra, SL_2(R), SU(2)
- Weeks 5-6: Integral manifolds and the
Frobenius theorem
Examples: integrability for PDE's;
when's a vector field a grad
- Week 6: Differential forms, and the de
Rham complex (no cohomology yet)
Examples: div-grad-curl
- Week 7: Orientability, integration,
Stokes's theorem
Examples: RP^2, calculus, Gauss's law
- Week 8: Riemannian metrics; geodesics as
minimizers.
Examples: submanifolds of Euclidean space
- Week 9: Exponential map; tubular
neighborhood theorem; second
variation and Jacobi fields if time permits.
Examples: Lie groups, S^2
SPRING: Cohomology
Text: Hatcher again; can use end of
Spivak v.1 and Madsen + Tornehave for
de Rham theory
- Week 1: de Rham cohomology; integration
as map from $H^n$ to scalars
- Week 2: Mayer-Vietoris; computation of
many examples.
- Week 3: Poincar\'e duality
- Week 4: Singular homology (abrupt change
of gears) and pairing with de Rham
cohomology on a smooth manifold.
- Week 5: Singular cohomology.
Mayer-Vietoris for singular homology and cohomology.
- Week 6: Cohomology of cell complexes,
simplicial cohomology, some
notions of equivalence of all these flavors
- Week 7: Cech cohomology, Weil's proof of
Cech-de-Rham theorem; sketch of
proof that
singular homology is dual to de Rham.
- Week 8: Cup product in singular and de
Rham theories; relationship to
intersection
theory.
- Week 9: Sheaves.