MATH 7520 Algebraic Topology

MATH 7520 Algebraic Topology

Fall 2009

Course Information

Course: MATH 7520 Algebraic Topology
Time and Place: Monday, Wednesday, Friday, 10:40 - 11:30 a.m., in 111 Lockett     <--   note the room change
Instructor: Dan Cohen
Office Hours: tentatively Monday, Wednesday, Friday, 11:40 a.m. - 12:30 p.m., in 372 Lockett, and by appointment
Prerequisites: MATH 7512 Topology II
Grade: Based on homework...
Homework problems will be posted at
A. Hatcher, Algebraic Topology, Cambridge University Press.
I intend to cover portions of Chapters 3 and 4 in this text, and some additional topics from other sources.
Some other potential sources for material covered in this course are listed at

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 and cohomology 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.

Here are some more:
Topological Methods in Algebraic Geometry, by F. Hirzebruch
Characteristic Classes, by J. Milnor
Notes on Cobordism Theory, by R. Stong

Department of Mathematics
Louisiana State University
Baton Rouge, LA, 70803
Office: 372 Lockett
Phone: (225) 578-1576

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