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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 (topology): Topological spaces, fundamental group, covering spaces, principal bundles, vector bundles, frame bundles, tangent and cotangent bundles, classifying spaces.

Winter (differentiable manifolds): differentiable manifolds; implicit function theorem and statement of 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 (without cohomology). Orientation, integration, Riemannian metrics, geodesics, exponential map.

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: Introduction to Topology

Text suggestions: Hatcher, Bredon

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".

SPRING: Cohomology

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

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