KSM3E02
Ongoing Semester
Algebraic Topology I — Fall 2026
Course at the Kerala School of Mathematics. Algebraic Topology develops algebraic invariants to distinguish and classify topological spaces up to homotopy equivalence. The course studies cell complexes, paths and homotopy, the fundamental group and its categorical properties, the Seifert–van Kampen theorem, the comprehensive theory of covering spaces and deck transformations, and concludes with an introduction to simplicial and singular homology theory.
Course Instruction Handover
This course was initiated and taught through August 2026 (Lectures 1–8 and Quiz 1) by Aritra Bhowmick. I am taking over the course from Lecture 9 onwards starting in September 2026. All previous lecture notes and quiz materials remain preserved below.
Instructor & Contact
Instructor: Pranav Haridas
Office: F-5
Office Hours: By appointment
Teaching Assistant: Sandra P M
Initial Instructor (Aug 2026): Aritra Bhowmick
Class Schedule:
- Tuesday: 12:00 – 13:00 (1 hour)
- Wednesday: 12:00 – 13:00 (1 hour)
- Thursday: 10:00 – 12:00 (2 hours)
Evaluation Scheme
- Quizzes (Best 2 taken) 30%
- Mid-Semester Examination 30%
- End-Semester Examination 40%
Quiz Policy:
Multiple quizzes will be conducted periodically throughout the semester. Only the best two scores will be considered toward the 30% quiz evaluation.
Textbooks & References
Secondary Reference
Introduction to Topological Manifolds
John M. Lee — Springer (Graduate Texts in Mathematics)
Topology (2nd Edition)
James R. Munkres — Prentice Hall
A Basic Course in Algebraic Topology
William S. Massey — Springer (Graduate Texts in Mathematics)
Lecture & Assessment Schedule
Note: The following schedule is tentative and will be updated as and when lectures and assessments take place.
| Date | Lecture | Hours | Topic | Notes & Resources |
|---|---|---|---|---|
| Part I: CW Complexes & Foundations of Homotopy (Lectures 1–8) — Instructor: Aritra Bhowmick | ||||
| Tue 04 Aug | Lecture 1 | 1 | Quotient topology, homotopy, and homotopy equivalence of spaces and maps | Class Note 1 (PDF) |
| Wed 05 Aug | Lecture 2 | 1 | Homotopy extension property, homotopy lifting, composition and homotopies | Class Note 2 (PDF) |
| Thu 06 Aug | Lecture 3 | 2 | Contractible spaces, disjoint union, attaching spaces, mapping cylinders, and mapping cones | Class Note 3 (PDF) |
| Tue 11 Aug | Lecture 4 | 1 | Examples of homotopy equivalence: wedge sums, topologist's sine curve, Hawaiian earring, and local contractibility | Class Note 4 (PDF) |
| Thu 13 Aug | Lecture 5 | 2 | Relative homotopy, pointed spaces, basepoint-preserving homotopy, and attaching cells | Class Note 5 (PDF) |
| Tue 18 Aug | Lecture 6 | 1 | Relative CW complexes, skeletons, and the weak topology | Class Note 6 (PDF) |
| Wed 19 Aug | Lecture 7 | 1 | Topological properties of CW complexes: Hausdorffness ($T_2$) and normal spaces | Class Note 7 (PDF) |
| Thu 20 Aug | Lecture 8 | 2 | Closure finiteness of CW complexes, cellular maps, and subcomplexes | Class Note 8 (PDF) |
| Thu 20 Aug | Quiz 1 | 1 | Covers material from Lectures 1–8 | Quiz 1 (PDF) | Solutions (PDF) |
| Part II: The Fundamental Group & The Seifert–van Kampen Theorem (Lectures 9–17) | ||||
| Tue 08 Sep | Lecture 9 | 1 | Paths and Homotopy of Paths; The Fundamental Group $\pi_1(X, x_0)$ and Group Axioms | Lecture 9 Notes → |
| Wed 09 Sep | Lecture 10 | 1 | Induced homomorphisms, functoriality of $\pi_1$, and change-of-basepoint isomorphisms | — |
| Thu 10 Sep | Lecture 11 | 2 | The Fundamental Group of the Circle $S^1$; Path lifting, homotopy lifting, and winding numbers | — |
| Tue 15 Sep | Lecture 12 | 1 | Retractions, deformation retractions, and the Brouwer Fixed Point Theorem in dimension 2 | — |
| Wed 16 Sep | Lecture 13 | 1 | The Fundamental Theorem of Algebra and the Borsuk–Ulam Theorem for $S^2$ | — |
| Thu 17 Sep | Lecture 14 / Quiz 2 | 2 | Free products of groups, presentations, and amalgamated free products; Quiz 2 (covering Lectures 9–13) | Quiz 2 |
| Tue 22 Sep | Lecture 15 | 1 | The Seifert–van Kampen Theorem: statement, pushout property, and proof strategy | — |
| Wed 23 Sep | Lecture 16 | 1 | Applications of van Kampen's Theorem: wedges of circles, bouquets of spheres, and spaces with abelian fundamental groups | — |
| Thu 24 Sep | Lecture 17 / Quiz 3 | 2 | Attaching 2-cells; Fundamental groups of compact surfaces; Midterm Review; Quiz 3 (covering Lectures 14–16) | Quiz 3 |
| 28 Sep – 02 Oct | Mid-Semester Exam | 2 | Mid-Semester Examination Week (30%) — covers material from Lectures 1–17 (includes Gandhi Jayanthi on 02 Oct) | KSoM Exam Schedule |
| Part III: Covering Spaces & Group Actions (Lectures 18–27) | ||||
| Tue 06 Oct | Lecture 18 | 1 | Covering spaces: definition, evenly covered neighborhoods, and standard examples ($\mathbb{R} \to S^1$, $S^n \to \mathbb{R}P^n$) | — |
| Wed 07 Oct | Lecture 19 | 1 | The Homotopy Lifting Property, unique path lifting, and lifting of homotopies | — |
| Thu 08 Oct | Lecture 20 | 2 | General lifting criterion, injectivity of $p_*$, and the subgroup $p_*(\pi_1(\tilde{X}, \tilde{x}_0)) \le \pi_1(X, x_0)$ | — |
| Tue 13 Oct | Lecture 21 | 1 | Universal covering spaces: semi-locally simple connectivity and the existence theorem | — |
| Wed 14 Oct | Lecture 22 | 1 | Explicit construction of the universal cover via path homotopy classes | — |
| Thu 15 Oct | Lecture 23 / Quiz 4 | 2 | The Galois correspondence between connected covering spaces and subgroups of $\pi_1(X, x_0)$; Quiz 4 (covering Lectures 18–22) | Quiz 4 |
| Tue 20 Oct | Holiday | — | Mahanavami (Institute Holiday — No Class) | Holiday |
| Wed 21 Oct | Holiday | — | Vijaya Dashami (Institute Holiday — No Class) | Holiday |
| Thu 22 Oct | Lecture 24 | 2 | Deck transformations (covering automorphisms), normal coverings, and the isomorphism $\text{Deck}(p) \cong \pi_1(X)/p_*(\pi_1(\tilde{X}))$ | — |
| Tue 27 Oct | Lecture 25 | 1 | Group actions on topological spaces: properly discontinuous actions and orbit spaces $X/G$ | — |
| Wed 28 Oct | Lecture 26 | 1 | Covering projections arising from group actions; examples: lens spaces $L(p, q)$ and the Klein bottle | — |
| Thu 29 Oct | Lecture 27 / Quiz 5 | 2 | Classification of covering spaces: applications and comprehensive examples; Quiz 5 (covering Lectures 24–26) | Quiz 5 |
|
Part IV: Introduction to Homology Theory (Lectures 28–36)
Note: If the topics from Part III require additional time or spill over, Part IV (or parts of it) will be moved to the Algebraic Topology II course.
|
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| Tue 03 Nov | Lecture 28 | 1 | Motivation for Homology: geometric cycles vs boundaries; affine simplices and $\Delta$-complexes | — |
| Wed 04 Nov | Lecture 29 | 1 | Simplicial Homology: chain groups $\Delta_n(X)$, the boundary operator $\partial_n$, and proof that $\partial^2 = 0$ | — |
| Thu 05 Nov | Lecture 30 | 2 | Simplicial homology groups $H_n^\Delta(X)$; explicit computations for $S^1, S^2$, the torus $T^2$, and the projective plane $\mathbb{R}P^2$ | — |
| Tue 10 Nov | Lecture 31 | 1 | Singular Homology: singular $n$-simplices, singular chain complex $C_\bullet(X)$, and singular homology groups $H_n(X)$ | — |
| Wed 11 Nov | Lecture 32 | 1 | Elementary properties of singular homology: path components ($H_0(X) \cong \mathbb{Z}^{\pi_0(X)}$), reduced homology $\tilde{H}_n(X)$, and one-point spaces | — |
| Thu 12 Nov | Lecture 33 / Quiz 6 | 2 | Homotopy invariance of singular homology: chain homotopy and the prism operator; Quiz 6 (covering Lectures 28–32) | Quiz 6 |
| Tue 17 Nov | Lecture 34 | 1 | Relative homology groups $H_n(X, A)$, short exact sequence of chains, the Snake Lemma, and the long exact sequence of a pair | — |
| Wed 18 Nov | Lecture 35 | 1 | The Excision Theorem, the Mayer–Vietoris sequence, and homology of spheres $S^n$ | — |
| Thu 19 Nov | Lecture 36 | 2 | Applications of Homology: Brouwer Fixed Point Theorem in all dimensions, Invariance of Domain, degree of sphere maps, Hairy Ball Theorem, and Course Wrap-up | — |
| Fri 20 Nov | Last Instructional Day | Last instructional day of Fall 2026 semester at KSoM | ||
| 30 Nov – 05 Dec | End-Semester Exam | 3 | Comprehensive Final Examination (40%) | KSoM Exam Schedule |
Course Logistics & Policies
- Prerequisites: General Topology (topological spaces, quotient topology, connectedness, path-connectedness, compactness) and Abstract Algebra (groups, homomorphisms, quotient groups, group actions).
- Lecture Notes: Detailed lecture notes for the lectures from September onwards will be published progressively on this webpage. Aritra Bhowmick's notes for Lectures 1–8 are accessible via the links in the schedule above.
- Collaboration: Collaborative discussions on homework and understanding concepts are strongly encouraged; written submissions must be the student's own work.