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# DEPARTMENT OF MATHEMATICS

## Geometry Topology Syllabus

##### Math 440-1,2,3 Geometry/Topology
• Texts: Hatcher, Spivak, Madsen and Tornehave, Milnor
• The course will emphasize examples throughout the year.
##### FALL (Differentiable Topology):

Differentiable manifolds; implicit function theorem and Sard's theorem; smooth vector bundles, tangent vectors, tensors, vector fields and flows. Lie derivatives, Lie groups and Lie algebras. Integral manifolds, Frobenius's theorem. Differential forms and the de Rham complex.

##### WINTER (Introduction to Algebraic Topology):

The fundamental group of a space, covering spaces, and the Van-Kampen theorem. Singular homology, Mayer-Vietoris, degree and Euler characteristic. Spring (cohomology): de Rham cohomology, Poincare' duality, singular cohomology. Cohomolgy of cell complexes, simplicial cohomology, Cech cohomology. Cup product; sheaves.

##### SPRING (Cohomology):

de Rham cohomology, Mayer-Vietoris, Poincare' duality, singular homology and cohomology. Cohomolgy of cell complexes, simplicial cohomology, Cech cohomology, equivalences between cohomology theories. Cup product; sheaves.

##### FALL: Differentiable Topology

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: Implicit function theorem and statement of Sard's theorem
• Week 3: Smooth vector bundles; tangent vectors as derivations and the tangent, cotangent bundles Examples: $P^1$, Riemann surfaces, projective spaces and Grassmannians
• Week 4: Higher rank tensors (associated bundles via transition functions), vector fields and flows Examples: top forms, calculus ODE's, vector fields from symmetries
• Weeks 5-6: Lie derivatives, Lie bracket, Lie groups and Lie algebras
• Examples: Cartan's formula, Heisenberg group and algebra, $SL_2(R)$, $SU(2)$
• Weeks 6-7: Integral manifolds and the Frobenius theorem Examples: integrability for PDE's; when's a vector field a grad
• Week 7: Differential forms, and the de Rham complex (no cohomology yet) Examples: div-grad-curl
• Week 8: Orientability, integration, Stokes's theorem Examples: $RP^2$, calculus, Gauss's law
• Week 9: Riemannian metrics; geodesics as minimizers. Examples: submanifolds of Euclidean space
• Week 10: Exponential map; tubular neighborhood theorem; second variation and Jacobi fields if time permits. Examples: Lie groups, $S^2$
##### WINTER: Introduction to Algebraic 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).
##### 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: Poincaré 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.