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

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

Some Previous Concepts Lecture # 1 Some Previous Concepts Set Groups Vector spaces and linear transformations MTH 477 - Algebraic Topology by Dr Sohail Iqbal

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Relative Singular Homology Lecture # 23 Relative Singular Homology Deformation retract Reduced homology (cont.) Excision Mayer-Vietoris theorem Applications Source: http://www.faculty.idc.ac.il/arik/site/skeleton-based.asp MTH 477-Algebraic Topology by Dr Sohail Iqbal

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: http://www.faculty.idc.ac.il/arik/site/skeleton-based.asp MTH 477-Algebraic Topology by Dr Sohail Iqbal

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

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

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

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

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

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

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

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: www.solitaryroad.com Wikipedia Algebraic topology by Allen Hatcher Differential geometry Barret O’Neil Topology by Mukares MTH 477-Algebraic Topology by Dr Sohail Iqbal

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