## Syllabus for

Math 525. Algebraic Topology I

**Fundamental group and covering spaces** [first half of semester]

- Definition of the fundamental group.
- Covering spaces and lifts of maps.
- Computing the fundamental group via covering spaces.
- Applications, such as the Fundamental Theorem of Algebra and the Brouwer fixed point theorem in 2d.
- Deforming spaces: retraction and homotopy equivalence.
- Quotient topology and cell complexes.
- Homotopy extension property and applications to homotopy equivalence.
- Fundamental groups of CW complexes.
- Van Kampen's Theorem.
- Covering spaces and subgroups of the fundamental group.
- Universal covers.
- The definitive lifting criterion, classification of covering spaces.
- Covering transformations and regular covers.

**Homology** [second half of semester]

- Delta complexes and their cellular homology.
- Singular homology.
- Homotopic maps and homology.
- The long exact sequence of the pair.
- Relative homology and excision.
- Equality of cellular and singular homology.
- Applications, such as degree of maps of spheres, invariance of dimension, and the Brouwer fixed point theorem.
- Homology of CW complexes.
- Homology and the fundamental group: the Hurewicz theorem.
- Euler characteristic.
- Homology with coefficients.
- Intro to categories and axiomatic characterization of homology theories.
- Further applications, such as the Jordan curve theorem, wild spheres, invariance of domain.

Examples of more detailed pacings for this course can be found at

- http://www.math.uiuc.edu/~nmd/classes/2009/525/index.html
- http://www.math.uiuc.edu/~rezk/math-525-fal10.html

All of this material may be covered on the 525 comp exam.

**Textbook: **

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002.
Freely downloadable at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

The content of the course is essentially Chapters 0-2.