Course: | MATH 7520 Algebraic Topology |
Time and Place: | Monday, Wednesday, Friday, 9:30 - 10:20 a.m., in 130 Lockett |
Instructor: | Dan Cohen |
Office Hours: | Monday, Wednesday, 10:30 a.m. - 12:00 noon, in 372 Lockett, and by appointment |
Prerequisites: | MATH 7512 Topology II |
Grade: |
Based on homework...
Homework problems will be posted |
Text: |
A. Hatcher, Algebraic Topology,
Cambridge University Press. I intend to cover portions of Chapter 3 and possibly Chapter 4 in this text, and some additional topics from other sources. Some other potential sources for material covered in this course are listed below. |
The focus of this course will be on cohomology theory, dual to the homology theory developed in MATH 7512. One reason to pursue cohomology theory is that the cohomology of a space may be given a natural ring structure. This additional algebraic structure provides another topological invariant. In developing this structure, we will study several products relating homology and cohomology. These considerations will be used to study the topology of manifolds, yielding a number of duality theorems also relating homology and cohomology, with a variety of applications.
In addition to its importance within topology, cohomology theory also provides connections between topology and other subjects, including algebra and geometry. Depending on the interests of the audience, we may pursue some of these connections, such as the de Rham theorem or cohomology of groups.
Topology and Geometry, by G. Bredon | Algebraic Topology, A First Course by M. Greenberg and J. Harper | ||
Differential Forms in Algebraic Topology, by R. Bott and L. Tu | The Topology of CW Complexes, by A. Lundell and S. Weingram | ||
A Basic Course in Algebraic Topology, by W. Massey | Singular Homology Theory, by W. Massey | ||
Algebraic Topology, by E. Spanier | Homology Theory, by J. Vick |
These are just a few. There are, of course, many others.
Department of Mathematics