Math 109c: Introduction to Geometry and Topology

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Instructor: Alexander Gorodnik
Office: 172 Sloan
Phone: 395-4325
E-mail: gorodnik "at" caltech "dot" edu

Office Hours: MW 1:30-2:30pm and by appointment

TA: Dongping Zhuang
E-mail: dongping at caltech dot edu
Office hour: Tu. 4-5pm (Sloan 385)

Course Objective: Math 109c is the final part of the 109 sequence (Introduction to geometry and topology). In this course, we cover the fundamentals of differential topology:

• Topological and smooth manifolds, tangent and vector bundles, vector fields, immersions and submersions, transversality, homotopy and stability, Sard's theorem.
• Differential forms, de Rham cohomology, and integration on manifolds.
• Intersection theory (Lefschetz fixed point theorem, Poincare-Hopf theorem).
• Elements of Riemannian geometry.

Prerequisites: Ma 109ab, and Ma 108 must be taken previously or concurrently.

Books: The main references for this course are

• K. Burns and M. Gidea, Differential Geometry and Topology: With a View to Dynamical Systems, Chapman & Hall/CRC, 2005.
• W. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press, 2002.

• V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974.

• J. Milnor, Topology from the differentiable viewpoint, Princeton University Press, Princeton, NJ, 1997.

Grading Policy: Weekly homework (50%), a midterm (20%) and a final (30%). Late work will only be accepted with a valid and a well-documented explanation. Please, contact the TA to submit late homework.

Homework

• Homework 1 (due Wed., April 5, by noon)
• Homework 2 (due Wed., April 12, by noon)
• Homework 3 (due Wed., April 19, by noon)
• Homework 4 (due Wed., April 26, by noon)
• Midterm - pick up from the HW box (due Wed., May 3, by noon)
• Homework 5 (due Wed., May 10, by noon)
• Homework 6 (due Wed., May 17, by noon)
• Homework 7 (due Wed., May 24, by noon)
• Final Exam - pick up from the HW box (due Friday, June 2, 2006, 10am)