Sem 1 2023: Algebraic Topology

Lectures: Tuesday 09-11 and Friday 11:10-13:10 Office Hours: Thursday 14-15

Topics covered in this course:

1. Categories and Functors
2. Homotopy and Fundamental groups
3. Covering spaces
4. Singular homology
5. Axiomatic homology theory
6. Miscellaneous applications
7. Cellular homology

Lectures

• 01.08: Lecture 1: Introduction, Categories and functors
• 04.08: Lecture 2: Point-set topology recollection: subspace, products, quotients
• 08.08: Lecture 3: Homotopy
• 11.08: Lecture 4: Fundamental group
• 18.08: Lecture 5: Seifert-van Kampen theorem
• 22.08: Lecture 6: Fundamental groups: computations
• 25.08: Lecture 7: Covering spaces
• 01.09: Lecture 8: Monodromy action; Universal covers (10-13:10)
• 05.09: Lecture 9: Deck transformation action; Classification of coverings
• 08.09: Lecture 10: Fundamental groups and covering spaces: computations and applications (10-13)
• 12.09: Lecture 11: Simplicial homology; Singular homology
• Mid-Semester break
• 26.09: Lecture 12: Basics of homological algebra
• 29.09: Lecture 13: Properties of singular homology
• 03.10: Lecture 14: Eilenberg-Steenrod axioms; Formal properties of homology theories
• 06.10: Lecture 15: Computations of some homology groups
• 10.10: Lecture 16: Some applications of homology theory
• 13.10: Lecture 17: Applications of homology theory (contd.)
• 17.10: Lecture 18: Simplicial approximation theorem; CW complexes
• 20.10: Lecture 19: Cellular homology
• 27.10: Lecture 20: Some examples and computations
• 31.10: Lecture 21: Local degrees; Euler characteristic
• 03.11: Problem session
• 07.11: Problem session

Exercises

• Assignment 1(Due 04.09)
• Assignment 2(Due 29.09)
• Assignment 3(Due 21.10)
• Assignment 4(Due 06.11)

Exams

• Midsem: Sep 22, 2023
• Endsem: Nov 22, 2023

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. Haugseng, Algebraic Topology I, 2022
• J. J. Rotman, An Introduction to Homological Algebra, 2009