Duration of the online course: 24 hours and 16 minutes
5
(1)
Explore the fascinating world of algebraic topology with this comprehensive online course. Dive into the foundational concepts starting with an introduction to algebraic topology, laying the groundwork for understanding the intricate structures and principles that guide this mathematical domain.
Delve into the core subjects, including the definition of singular and relative singular homology, and grasp the Eilenberg-Steenrod axioms that serve as critical touchstones in homology theory. Learn to apply these axioms to compute homologies of spheres and understand important concepts like the snake lemma and the long exact sequence axiom.
Unravel the mysteries of homotopy invariance and the 5-lemma, explore the excision and dimension axioms, and gain insights into further nuances of the Eilenberg-Steenrod axioms. The course encourages students to explore singular homology in degree zero, understand relative homology as absolute homology, and appreciate the Hurewicz isomorphism.
The Mayer-Vietoris sequences and their application for pushouts are examined in depth, along with the suspension isomorphism and computations related to real-world objects like the Klein bottle. Delve into sophisticated mathematical theorems such as the Borsuk-Ulam and Ham-Sandwich Theorems. Discover the usefulness of cellular chain complexes, their computations, and how they relate back to singular homology.
Develop a deep understanding of the Euler characteristic, lens spaces, and projective modules. Grasp the fundamental theorem of homological algebra, alongside tensor products and their computations. Obtain thorough knowledge of the Tor functor and the universal coefficient theorem, and apply the Künneth theorem and acyclic models theorem to solve complex algebraic problems.
By the end of the course, students will have gained a robust understanding of algebraic topology, equipped with tools and techniques to tackle a wide range of problems with confidence.
Video class: 01 Introduction
20m
Exercise: What is the main task of algebraic topology?
Video class: 02 Definition of singular homology
1h03m
Exercise: What is an n-simplex?
Video class: 03 Definition of relative singular homology
17m
Exercise: What New Concept Was Introduced in Relative Singular Homology?
Video class: 04 Eilenberg-Steenrod Axioms
30m
Exercise: What is the primary use of the Eilenberg-Steenrod axioms in homology theory?
Video class: 05 Homology of spheres from the axioms
44m
Exercise: What is the homology of a circle (S1)?
Video class: 06 Exemplary computation of an induced map
25m
Exercise: What does the reflection map on spheres induce?
Video class: 07 The snake lemma
39m
Exercise: What is the Snake Lemma in singular homology?
Video class: 08 The long exact sequence axiom in homology
37m
Exercise: What is the main consequence of the Snake Lemma in singular homology?
Video class: 09 The homotopy invariance of singular homology
1h00m
Exercise: What is the concept of homotopic invariance in singular homology?
Video class: 10 The 5-lemma
08m
Exercise: What conclusion does the Five Lemma in homological algebra lead to?
Video class: 11 The excision axiom
47m
Exercise: What is an essential aspect of the excision axiom in singular homology?
Video class: 12 The lemma of small simplices
1h10m
Exercise: What is the primary purpose of barycentric subdivision in the proof of the small simplices lemma?
Video class: 13 The dimension axiom
07m
Exercise: What is the result of the nth singular homology of a single point according to the dimension axiom?
Video class: 14 Further remarks on the Eilenberg-Steenrod axioms
20m
Exercise: Which axiom does singular homology satisfy in relation to topological space decomposition?
Video class: 15 Singular homology in degree 0
18m
Exercise: What does the zeroth singular homology reveal about a topological space?
Video class: 16 The Hurewicz isomorphism
48m
Exercise: What does the Hurriwich Theorem in Degree One relate in topology?
Video class: 17 The triple sequence
14m
Exercise: What is the concept behind the triple sequence in topology?
Video class: 18 Relative homology as absolute homology
47m
Exercise: What is the relationship between reduced and unreduced homology for a point?
Video class: 19 Mayer-Vietoris sequence
16m
Exercise: What is the purpose of the Mayer-Vietoris sequence in algebraic topology?
Video class: 20 Mayer-Vietoris sequence for pushouts
08m
Exercise: Understanding the Excision Axiom in Homology
24 hours and 16 minutes of online video course
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Course comments: Algebraic Topology Course
Venkata sree Rama murty maddula
if u add the applications of it in classical and quantum computing very helpful