Sem 1 2025: Algebraic Topology

Lectures: Mondays and Wednesdays 09-11 in AG 77; Office Hours: By appointment

TA: Aniket Chakraborty; Office Hours with TA: Wednesdays 3-4 PM

Mid-semester examination: 06 October, 09:00-12:00 in Mathematics Seminar Room (A-369)

Topics covered in this course:

1. Homotopy theory
2. Fundamental groupoids (and groups)
3. Covering Spaces
4. Singular homology
5. Singular cohomology

Lectures

• 18.08: Lecture 1: Homotopy
• 20.08: Lecture 2: Fundamental groupoid
• 25.08: Lecture 3: Homotopy groups: Definition
Assignment 1: Due on 04.09
• 01.09: Lecture 4: Seifert-van Kampen theorem
• 03.09: Lecture 5: Covering spaces
• 08.09: Lecture 6: Monodromy action
• 10.09: Lecture 7: Universal covering
Assignment 2: Due on 22.09
• 15.09: Lecture 8: Classification of coverings, Fundamental groups of graphs
• 17.09: Lecture 9: Simplicial homology
• 22.09: Lecture 10: Singular homology
• 24.09: Lecture 11: Eilenberg-Steenrod axioms
• 29.09: Lecture 12: Some computations and applications of homology groups
• 01.10: Lecture 13: CW complexes and cellular homology
Midsem Exam: 06.10
• 13.10: Lecture 14: Euler characteristic and Lefschetz number
• 15.10: Lecture 15: Singular cohomology
Assignment 3: Due on 30.10
• 22.10: Lecture 16: Cup product

References

• A. Hatcher, Algebraic Topology, 2001
• C. Löh, Algebraic Topology, An introductory course, Wintersemester 2018/19
• J. P. May, A Concise Course in Algebraic Topology, 1999
• T. tom Dieck, Algebraic Topology, 2008
• B. Gray, Homotopy Theory: An Introduction to Algebraic Topolopy, 1975
• J. W. Vick, Homology Theory: An Introduction to Algebraic Topology, 1994
• R. Brown, Topology and Groupoids, 2006
• R. Haugseng, Algebraic Topology I, 2022
• J. J. Rotman, An Introduction to Homological Algebra, 2009