Duration of the online course: 24 hours and 16 minutes
5
(1)
Algebraic topology is where geometry meets algebra: you translate the features of spaces into algebraic objects that can be computed and compared. This free online course guides you through that translation with singular homology as the central tool, helping you move from pictures and intuition to rigorous, reusable methods. If you want a clearer way to distinguish spaces, detect holes, and understand when two shapes are essentially the same, this course gives you the framework to do it with confidence.
You start by building the language of singular homology from the ground up, including the role of simplices and the chain-level viewpoint that turns continuous problems into algebra. From there, the course develops relative homology so you can analyze a space together with a subspace, a perspective that becomes indispensable in real computations and in proofs that compare constructions. Along the way, you learn why axioms matter: the Eilenberg-Steenrod approach shows which properties characterize a homology theory and how these properties unlock results such as the homology of spheres and functorial behavior under maps.
A major theme is learning to navigate the machinery that makes homology powerful rather than mysterious. You work with induced maps, exactness, and core homological tools such as the snake lemma and the 5-lemma, gaining the ability to pass information through diagrams and conclude isomorphisms when direct computation would be difficult. Key structural principles like homotopy invariance clarify which features of a space are truly topological, while excision and the small simplices perspective explain why local modifications often do not change homology, paving the way for practical decompositions.
As your understanding matures, the course emphasizes computational strategy and conceptual connections. Degree zero homology provides a crisp algebraic lens on connectedness, while the Hurewicz viewpoint links homology with more geometric invariants and offers intuition for what homology is measuring. You also meet the triple sequence and see how relative homology can be reframed as absolute homology in suitable settings, improving flexibility when you tackle examples.
To tie everything together, Mayer-Vietoris becomes a central method: it shows how to compute the homology of a complicated space from simpler overlapping pieces, including versions suited to pushouts. By the end, you will be better equipped to read and write proofs in algebraic topology, understand standard theorems in a principled way, and approach new spaces with a reliable toolbox for breaking them apart and extracting invariant information.
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
Digital certificate of course completion (Free)
Exercises to train your knowledge
100% free, from content to certificate
Ready to get started?Download the app and get started today.
Install the app now
to access the course
Over 5,000 free courses
Programming, English, Digital Marketing and much more! Learn whatever you want, for free.
Study plan with AI
Our app's Artificial Intelligence can create a study schedule for the course you choose.
From zero to professional success
Improve your resume with our free Certificate and then use our Artificial Intelligence to find your dream job.
You can also use the QR Code or the links below.
Free CourseBasics of Algebra
5
(6)
37m
5 exercises
Free CourseAlgebra Introduction
5
(5)
1h18m
6 exercises
Free CourseLinear Algebra Course
5
(2)
10h49m
46 exercises
Free CourseLinear algebra
4.91
(120)
27h55m
35 exercises
Free CourseAlgebra
4.84
(57)
3h18m
25 exercises
Free CourseIntermediate Algebra
4.82
(22)
54h38m
43 exercises
Free CoursePrealgebra
3.63
(54)
38h54m
44 exercises
Free CourseLinear Algebra Essentials
New
3h00m
16 exercises
Free CourseUndergraduate Linear Algebra Full Course
New
29h54m
34 exercises
Free CourseAlgebraic geometry
New
38h05m
50 exercises
Thousands of online courses in video, ebooks and audiobooks.
To test your knowledge during online courses
Generated directly from your cell phone's photo gallery and sent to your email
Download our app via QR Code or the links below::.
Carry knowledge in your pocket.
Download the Cursa app.
There are hundreds of free courses available, with a free certificate of completion that is saved in your mobile image gallery.
Download the app to access the Course Completion Certificate for Free.
+ 10 million
students
Free and Valid
Certificate
60 thousand free
exercises
4.8/5 rating in
app stores
Free courses in
video and ebooks
Course comments: Algebraic Topology Course
Venkata sree Rama murty maddula
if u add the applications of it in classical and quantum computing very helpful