### MATH 5311: Introduction to Geometry and Topology II

**Description:** This course is offered either as an introduction to algebraic topology or differential topology. Differential topology: review of topology; smooth manifolds; vector fields, tangent and cotangent bundles; submersions, immersions and embeddings, submanifolds; Lie groups; differential forms, orientations, integration on manifolds; De Rham cohomolgy, singular cohomology and de Rham theorem; other topics in geometry at the choice of the instructor (e.g. intrinsic Riemannian geometry of surfaces). Algebraic topology: simplical homology, singular homology; Eilenberg-Steenrod axioms; Mayer-Vietoris sequences; CW-homology; cohomology; cup products; Hom and Tensor products; Ext and Tor; the Universal Coefficient Theorems; Cech cohomology and Steenrod homology; Poincare, Alexander, Lefschetz, Alexander-Pontryagin dualities. With a change of content, this course is repeatable to a maximum of six credits.

**Prerequisites:** MATH 5310.

**Credits:** 3

### MATH 5311 - Section 1: Introduction to Geometry and Topology II

**Description:** Differential topology may be defined as the study of properties of differentiable manifolds which are invariant under diffeomorphism. We will cover the following topics: Elements of differential and Riemannian geometry; Hodge theory and De Rham cohomology; Index theory: The Chern-Gauss-Bonnet theorem, the Hirzebruch signature theorem and the Atiyah-Singer theorem; Morse theory: Morse inequalities, Calculus of variation on path spaces, Applications to Lie groups and symmetric spaces. If time allows, we will discuss other topics like the proof of Whitney's embedding theorem and cobordism theory.

**Text:** Morse Theory, Princeton University Press (1963) by J. Milnor and The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts 31, 1997 by S. Rosenberg.

**Instructor:**Fabrice Baudoin

**Offered:** Spring

**Credits:** 3

**Sections: **Spring 2017 on Storrs Campus

PSCourseID | Course | Sec | Comp | Time | Room | Instructor |
---|---|---|---|---|---|---|

24847 | 5311 | 001 | Lecture | MWF 11:15:00 AM-12:05:00 PM | MONT 314 | Baudoin, Fabrice |