MATH 7590 Cohomology Theory

# MATH 7590 Cohomology Theory

## Course Information

 Course: MATH 7590 Cohomology Theory Time and Place: Tuesday & Thursday, 12:10 - 1:30 PM, in 113 Lockett Instructor: Dan Cohen Office Hours: Tuesday & Thursday, 1:30 - 2:30 PM, in 372 Lockett, and by appointment Prerequisites: MATH 7520 Algebraic Topology Grade: Based on homework and possibly in-class presentations. Homework problems are posted here. Text: Algebraic Topology, A First Course, by M. Greenberg and J. Harper 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

## Course Description

This course continues the study of algebraic topology begun in MATH 7512 and MATH 7520. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group in MATH 7512, the homology groups in MATH 7520) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., isomorphic groups). Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces: two spaces with inequivalent invariants cannot be topologically equivalent.

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.

## Reference Materials

 * 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
These are just a few. There are, of course, many others.
I have checked the books with a * out of the library.
You may borrow them from me if you wish.

Department of Mathematics
Louisiana State University
Baton Rouge, LA, 70803
Office: 372 Lockett
Phone: (225) 578-1576
Email: cohen@math.lsu.edu