Math 525. Topology

1. DeRham cohomology (for open subsets of the plane R²) and winding numbers
2. covering spaces and the fundamental group
3. singular homology and the Eilenberg-Steenrod axioms it satisfies.

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.

1. Path integrals, winding numbers, DeRham cohomology in dimensions 0 and 1 and Mayer-Vietoris, fixed point theorems, Jordan curve theorem.

2. 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.

3. Singular homology, Eilenberg-Steenrod axioms for singular homology, homology and fundamental group of spheres and tori, fixed point and separation theorems in higher dimensions.

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Department of Mathematics University of Illinois at Urbana-Champaign 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350    Fax (217) 333-9576 office@math.uiuc.edu

Syllabus Revised 6/21/99