# Syllabus Math 525

## Math 525. Algebraic Topology I Instructor Syllabus

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.