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Geometry Topology Syllabus

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".

WINTER: Introduction to AlgebraicTopology

Text suggestions: Hatcher, Bredon

SPRING: Cohomology

Text: Hatcher again; can use end of Spivak v.1 and Madsen + Tornehave for de Rham theory

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