Free online courseAlgebraic Topology Course

Course content

• Video class: 01 Introduction

  0h20m

• 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

  0h17m

• Exercise: What New Concept Was Introduced in Relative Singular Homology?

• Video class: 04 Eilenberg-Steenrod Axioms

  0h30m

• Exercise: What is the primary use of the Eilenberg-Steenrod axioms in homology theory?

• Video class: 05 Homology of spheres from the axioms

  0h44m

• Exercise: What is the homology of a circle (S1)?

• Video class: 06 Exemplary computation of an induced map

  0h25m

• Exercise: What does the reflection map on spheres induce?

• Video class: 07 The snake lemma

  0h39m

• Exercise: What is the Snake Lemma in singular homology?

• Video class: 08 The long exact sequence axiom in homology

  0h37m

• 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

  0h08m

• Exercise: What conclusion does the Five Lemma in homological algebra lead to?

• Video class: 11 The excision axiom

  0h47m

• 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

  0h07m

• 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

  0h20m

• Exercise: Which axiom does singular homology satisfy in relation to topological space decomposition?

• Video class: 15 Singular homology in degree 0

  0h18m

• Exercise: What does the zeroth singular homology reveal about a topological space?

• Video class: 16 The Hurewicz isomorphism

  0h48m

• Exercise: What does the Hurriwich Theorem in Degree One relate in topology?

• Video class: 17 The triple sequence

  0h14m

• Exercise: What is the concept behind the triple sequence in topology?

• Video class: 18 Relative homology as absolute homology

  0h47m

• Exercise: What is the relationship between reduced and unreduced homology for a point?

• Video class: 19 Mayer-Vietoris sequence

  0h16m

• Exercise: What is the purpose of the Mayer-Vietoris sequence in algebraic topology?

• Video class: 20 Mayer-Vietoris sequence for pushouts

  0h08m

• Exercise: Understanding the Excision Axiom in Homology