MATH 7520 Algebraic Topology

# MATH 7520 Algebraic Topology

## Course Information

 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.

## Course Description

This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group in MATH 7510, the homology groups in MATH 7512) 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 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.

## Reference Materials

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