1. Algebraic Topology Dr. SOHAIL IQBAL MTH 477 For Master of MathematicsBy
Dr. SOHAIL IQBAL
Assistant Professor
Department of Mathematics, CIIT Islamabad
MTH 477-Algebraic Topology by Dr Sohail Iqbal

2. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 32
Review
MTH 477-Algebraic Topology by Dr Sohail Iqbal

3. Some Previous Concepts
Lecture # 1
Some Previous Concepts
Set
Groups
Vector spaces and linear transformations
MTH Algebraic Topology by Dr Sohail Iqbal

4. A review of category theory
Lecture # 2
A review of category theory
What is roughly a category and functor?
Category
Functor
MTH 477-Algebraic Topology by DrSohail Iqbal

5. MTH 477-Algebraic Topology by DrSohail Iqbal
Lecture # 3
Homotopy
Topology, some basics
MTH 477-Algebraic Topology by DrSohail Iqbal

6. MTH 477-Algebraic Topology by DrSohail Iqbal
Lecture # 4
Homotopy Continued
Homotopy category
Quotient topology
MTH 477-Algebraic Topology by DrSohail Iqbal

7. MTH 477-Algebraic Topology by DrSohail Iqbal
Lecture # 5
Path connectedness
The functor 𝜋 0
MTH 477-Algebraic Topology by DrSohail Iqbal

8. MTH 477-Algebraic Topology by DrSohail Iqbal
Lecture # 6
Simplexes
Affine spaces
MTH 477-Algebraic Topology by DrSohail Iqbal

9. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 7
Fundamental group
Simply connected spaces
MTH 477-Algebraic Topology by Dr Sohail Iqbal

10. MTH 477-Algebraic Topology by DrSohail Iqbal
Lecture # 8
Simplexes
The functor 𝜋 1
Calculation of the fundamental group
MTH 477-Algebraic Topology by DrSohail Iqbal

11. Fundamental group of circle
Lecture # 9
Fundamental group of circle
Topological group
MTH 477-Algebraic Topology by Dr Sohail Iqbal

12. Simplicial Homology-I
Lecture # 10
Simplicial Homology-I
Simplexes
Simplicial complexes
Triangulation
MTH 477-Algebraic Topology by Dr Sohail Iqbal

13. Simplicial Homology-II
Lecture # 11
Simplicial Homology-II
Simplicial homology
MTH 477-Algebraic Topology by Dr Sohail Iqbal

14. Simplicial Homology-III
Lecture # 12
Simplicial Homology-III
Calculation of simplicial homology
Homology: An Idea Whose Time Has Come
B. Cipra, SIAM News, Vol. 42(10), (2009)
MTH 477-Algebraic Topology by Dr Sohail Iqbal

15. Simplicial Homology-IV
Lecture # 13
Simplicial Homology-IV
Calculation of simplicial homology
and connectedness
Source: Rashid bin muhammad
MTH 477-Algebraic Topology by Dr Sohail Iqbal

16. Homological algebra-I (Basics of rings, fields and modules)
Lecture # 14
Homological algebra-I
(Basics of rings, fields and modules)
Rings and fields
Modules
Source: Rashid bin muhammad
MTH 477-Algebraic Topology by Dr Sohail Iqbal

17. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 15
Short exact sequences
Five-Lemma
MTH 477-Algebraic Topology by Dr Sohail Iqbal

18. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 16
Chain complexes
MTH 477-Algebraic Topology by Dr Sohail Iqbal

19. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 17
Homology groups of simplicial pairs
Simplicial pairs
Collapsing
MTH 477-Algebraic Topology by Dr Sohail Iqbal

20. Calculation and definition of homology
Lecture # 18
Calculation and definition of homology
Euler characteristic
Homology of torus
Axiomatic definition of homology
MTH 477-Algebraic Topology by Dr Sohail Iqbal

21. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 19
Singular homology
Dimension axiom for singular homology
MTH 477-Algebraic Topology by Dr Sohail Iqbal

22. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 20
Singular homology
Dimension axiom for singular homology
The functor 𝐻 𝑛
Path connectedness and homology
MTH 477-Algebraic Topology by Dr Sohail Iqbal

23. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 21
Homotopy And Homology
Homotopy and homology
MTH 477-Algebraic Topology by Dr Sohail Iqbal

24. Relative Singular Homology
Lecture # 22
Relative Singular Homology
Relative singular homology
Relative homotopy axiom
Reduced homology
MTH 477-Algebraic Topology by Dr Sohail Iqbal

25. Relative Singular Homology
Lecture # 23
Relative Singular Homology
Deformation retract
Reduced homology (cont.)
Excision
Mayer-Vietoris theorem
Applications
Source:
MTH 477-Algebraic Topology by Dr Sohail Iqbal

26. Fundamental group and homology
Lecture # 24
Fundamental group and homology
Free product of groups
Van Kampen theorem (Fundamental group continued)
Comparison of 𝐻 1 and 𝜋 1
Equivalence of simplicial and singular homology (an overview)
Source:
MTH 477-Algebraic Topology by Dr Sohail Iqbal

27. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture# 25
CW Complexes
(Cellular homology)
Some topological results
Projective spaces
Attaching a cell
MTH 477-Algebraic Topology by Dr Sohail Iqbal

28. Homology of attaching cells (Cellular homology)
Lecture # 26
Homology of attaching cells
(Cellular homology)
Homology and attaching cells
Calculating homology using “attaching cells”
Homology of real projective plane
Homology of complex projective spaces
MTH 477-Algebraic Topology by Dr Sohail Iqbal

29. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture#27
CW complexes
(Cellular homology)
Manifolds
MTH 477-Algebraic Topology by Dr Sohail Iqbal

30. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 28
Cellular homology
CW complexes
Degree of map 𝑆 𝑛 → 𝑆 𝑛
MTH 477-Algebraic Topology by Dr Sohail Iqbal

31. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 29
Cellular homology
Cellular homology continued
Homology with coefficients
Covering spaces
MTH 477-Algebraic Topology by Dr Sohail Iqbal

32. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 30
Covering spaces
MTH 477-Algebraic Topology by Dr Sohail Iqbal

33. MTH 477-Algebraic Topology by Dr Sohail Iqbal
Lecture # 31
Covering spaces
Group actions
Covering spaces and fundamental group
MTH 477-Algebraic Topology by Dr Sohail Iqbal

34. MTH 477-Algebraic Topology by Dr Sohail Iqbal
References
An introduction to algebraic topology by Joseph J. Rotman
Algebraic topology by David R Wilkins
Algebraic topology by Allen hatcher
Lecture notes on algebraic geometry by John Jones
Notes for a first year graduate course in Algebraic topology by Rob Thompson
Homology theory by Kay Werndli
Some of the images are taken from different sources including the following sources:
Wikipedia
Algebraic topology by Allen Hatcher
Differential geometry Barret O’Neil
Topology by Mukares
MTH 477-Algebraic Topology by Dr Sohail Iqbal

35. MTH 477-Algebraic Topology by Dr Sohail Iqbal
End of the course
MTH 477-Algebraic Topology by Dr Sohail Iqbal