|Course:||MATH 7590 Cohomology Theory|
|Time and Place:||Tuesday & Thursday, 12:10 - 1:30 PM, in 381 Lockett|
|Office Hours:|| tentatively
Monday - Thursday, 8:40 - 9:30 AM, in 372 Lockett,
and by appointment
|Prerequisites:||MATH 7520 Algebraic Topology|
Based on homework and possibly in-class presentations.
Homework problems will be posted here.
||Elements of Algebraic Topology, by J. R. Munkres,
Perseus Publishing, 1984
I intend to cover portions of Chapters 5 through 8 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 7520. 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, and of course time, 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|
|Algebraic Topology, by A. Hatcher||*||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|
Department of Mathematics