Math 544: Topology and Geometry of Manifolds

Ethan Devinatz

Autumn 1999, MWF 1:30-2:20

Manifolds are arbitrary-dimensional generalizations of curves and surfaces; that is, they are spaces that locally look like Euclidean space (Rn) but globally may not, just as the sphere locally looks like the plane. They are the basic subject matter of differential geometry, but also play a role in many other fields of mathematics, including algebraic topology, algebraic geometry, partial differential equations, and number theory. They are also increasingly important in applications such as mathematical physics and computer graphics. The fall quarter will concentrate on the topology of manifolds, i.e., properties that are invariant under continuous deformations.  The main tool here is the fundamental group, which is a group associated to every manifold in such a way that topologically equivalent manifolds have isomorphic groups. The topics to be covered include topological spaces; subspace, product, and quotient topologies; compactness and connectedness; the classification of closed surfaces; homotopy and the fundamental group; fundamental groups of surfaces; the theory and classification of covering spaces.

Text: Introduction to Topological Manifolds, notes by John M. Lee, to be handed out in class.  Recommended supplementary text: An Introduction to Topology and Homotopy, by Allan J. Sieradski.

Prerequisites:

Set Theory: Basic facts of "naive set theory'': properties of the real numbers and Rn, functions, equivalence relations, countability, order relations, the well-ordering theorem. References: Rudin, Principles of Mathematical Analysis, Chapter 1; Sieradski, Introduction to Topology and Homotopy, Chapter 1.

Algebra: Elementary group theory: homomorphisms, isomorphisms, subgroups, normal subgroups, cosets, quotient groups.  References: Sieradski, Introduction to Topology and Homotopy, Chapter 11; Hungerford, Abstract Algebra: An Introduction, Chapter 7.

Analysis: Open and closed sets, continuous maps, convergence, metric spaces, compact and connected sets.  References: Rudin, Principles of Mathematical Analysis, Chapters 2, 3, 4; Sieradski, Introduction to Topology and Homotopy, Chapters 2, 3.1-3.2.

If you aren't familiar with one or two of these topics, you should spend some time reading the reference books listed above.  If, however, you find a lot of items on the list  unfamiliar, you should consider taking a 400-level course instead.