Syllabus for
Math 525. Topology
There are three general areas of algebraic topology covered in this course:
- DeRham cohomology (for open subsets of the plane R2) and winding numbers
Path integrals, winding numbers, DeRham cohomology in dimensions 0 and 1 and Mayer-Vietoris, fixed point theorems, Jordan curve theorem. - covering spaces and the fundamental group
Fundamental group, Hurewicz theorem (in dimension 1), covering spaces, group actions, relations of the fundamental group and deck transformations, classification for existence of covering spaces, van Kampen Theorem. - singular homology and the Eilenberg-Steenrod axioms it satisfies.
Singular homology, Eilenberg-Steenrod axioms for singular homology, homology and fundamental group of spheres and tori, fixed point and separation theorems in higher dimensions.
Each of these topics covers about a third of the course. The exam for this course will consist of one question from each of these topics and a fourth question which combines ideas from two or more of these areas.
Suggested reading:
One possible source of the material for the first part of the syllabus is Fulton's book Algebraic Topology - A First Course. In this case, chapters 1-6 and 10 are suitable for the DeRham material, chapters 11-17 for the covering space materials. Another source for the covering space materials is chapter 8 of Munkre's Topology - A First Course.
For DeRham theory one can also see the first five sections of Bott and Tu's book Differential Forms and Algebraic Topology but this does not treat the properties of winding numbers which is done in Fulton.
For the singular homology many good sources are available including G. E. Bredon's Topology and Geometry (chapter IV, sections 1-7, 15-18), the first two chapters of Vick's Homology Theory, an Introduction to Algebraic Topology or chapters 4-6 of Rotman's An introduction to Algebraic Topology.