### 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:** In this course we shall study some basic and useful tools in algebraic topology, and emphasize its connections to differential geometry, algebraic geometry, and partial differential equations. More specifically we shall introduce Mayer-Vietoris sequence and apply it to compute the cohomology groups. We shall study the spectral sequences, including the applications of Serre, Gysin, H. C. Wang to fiber bundles and A. Borel's result on the Dolbeault cohomology.
Instructor: Damin Wu

**Prerequisites:** MATH 5310, or, contact the instructor for a permission number.

**Offered:** Spring

**Credits:** 3

**Sections: **Spring 2018 on Storrs Campus

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

10371 | 5311 | 001 | Lecture | MWF 11:15:00 AM-12:05:00 PM | MONT314 | Wu, Damin |