|Course:||MATH 7590 Cohomology Theory|
|Time and Place:||Tuesday & Thursday, 12:10 - 1:30 PM, in 113 Lockett|
Tuesday & Thursday, 1:30 - 2:30 PM, in 372 Lockett,
and by appointment
|Prerequisites:||MATH 7520 Algebraic Topology|
Based on homework and possibly in-class presentations.
Homework problems are posted here.
||Algebraic Topology, A First Course, by M. Greenberg and
Mathematics Lecture Notes Series, Perseus Books, 1981
We will cover Parts III and IV in this text, and some additional topics from other sources.
Some other sources for material covered in this course are listed here
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.
Specific topics we will cover include:
|Cohomology Theory||Singular cohomology, the universal coefficient theorem, cross products, the Künneth formula, cup and cap products.|
|Duality on Manifolds||Orientation of manifolds, Poincaré duality, Alexander duality, Lefschetz duality, the Lefschetz fixed point theorem, intersection numbers|
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 cohomology of groups or the De Rham theorem.
|*||Topology and Geometry, by G. Bredon|
|Chapter 3 of Algebraic Topology, by A. Hatcher, available his website|
|*||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