Course: | MATH 7520 Algebraic Topology |
Time and Place: | Tuesday & Thursday, 1:40 - 3:00 PM, in 135 Lockett |
Instructor: | Dan Cohen |
Office Hours: | Monday & Wednesday, 1:30 - 2:30 PM, in 372 Lockett,
and by appointment |
Prerequisites: | MATH 7200 and MATH 7510, or the equivalents The exposure to algebraic topology provided by MATH 7512 would be useful, but not absolutely essential. |
Grade: |
Based on homework and possibly in-class presentations.
Homework problems will be posted here. |
Text: |
Elements of Algebraic Topology, by J. R. Munkres,
Perseus Books, 1984 We will probably cover the first four chapters in the text, and some additional topics from other sources. Some other sources for material covered in this course are listed below. |
The focus of this course will be on homology theory (which complements the study of algebraic topology begun in MATH 7512). To a topological space, we will associate a sequence of abelian groups, called the homology groups. These homology groups are often more accessible than the fundamental group, so sometimes provide an easier means for distinguishing between topological spaces. We will concretely study simplicial and singular homology, the homology of CW-complexes, and related topic such as homology with coefficients, Mayer-Vietoris sequences, degrees of maps, and Euler characteristics. Geometric examples, including surfaces, projective spaces, lens spaces, etc., will be used to illustrate the techniques. We will also discuss a number of applications, including Brouwer and Lefschetz Fixed Point Theorems, and the Jordan Curve Theorem.
A continuation of this course will be offered in Spring 2003. There, we will study cohomology (dual to homology), and duality on (compact) manifolds.
Topology and Geometry, by G. Bredon | 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