### 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:

- Categories and Functors
- Homotopy and Fundamental groups
- Covering spaces
- Singular homology
- Axiomatic homology theory
- Miscellaneous applications
- 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