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
Primary Course Textbook
Introduction to Topological Manifolds (2nd Edition)
John M. Lee — Springer (Graduate Texts in Mathematics 202)
Main text for cell complexes (Chapter 5), fundamental groups (Chapters 7–10), covering spaces (Chapters 11–12), and homology (Chapter 13).
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 9–16) — Lee, Chapters 7–10 | ||||
| Tue 08 Sep | Course Recap | 1 | Course handover discussion and review of Lectures 1–8 foundations to align on material | — |
| Wed 09 Sep | Lecture 9 | 1 | Cell Complexes and CW Complexes (Lee, Chapter 5): basic building blocks, open and closed $n$-cells, compact convex subsets, cell decompositions | Lecture 9 Notes → |
| Thu 10 Sep | Lecture 10 | 2 | Homotopy of Paths and the Fundamental Group (Lee, Chapter 7): path homotopy, concatenation, reparametrization, path homotopy classes, group axioms for $\pi_1(X, x_0)$, base point dependence, change-of-basepoint isomorphisms | — |
| Tue 15 Sep | Lecture 11 | 1 | Induced Homomorphisms and Homotopy Invariance (Lee, Chapter 7): maps induced by continuous functions $f_*: \pi_1(X, x_0) \to \pi_1(Y, y_0)$, functoriality, homotopy invariance of $\pi_1$, simply connected spaces | — |
| Wed 16 Sep | Lecture 12 | 1 | The Circle and Covering Homotopy (Lee, Chapter 8): exponential map $\varepsilon: \mathbf{R} \to S^1$, path lifting and homotopy lifting properties for the circle, lifting loops, winding numbers | — |
| Thu 17 Sep | Lecture 13 | 2 | The Fundamental Group of the Circle and Topological Applications (Lee, Chapter 8): computation of $\pi_1(S^1) \cong \mathbf{Z}$, non-retraction theorem for the disk, Brouwer Fixed Point Theorem in dimension 2, Borsuk–Ulam Theorem for $S^2$, Fundamental Theorem of Algebra | — |
| Tue 22 Sep | Lecture 14 | 1 | Group-Theoretic Foundations for van Kampen (Lee, Chapter 9): free products of groups, words and reduced words, free groups on generating sets, group presentations by generators and relations, free abelian groups | — |
| Wed 23 Sep | Lecture 15 | 1 | The Seifert–van Kampen Theorem (Lee, Chapter 10): statement for unions of path-connected open sets, pushouts in group theory, amalgamated free products, proof strategy | — |
| Thu 24 Sep | Lecture 16 | 2 | Applications of van Kampen: Cell Attachments & Surfaces (Lee, Chapter 10): wedge sums $\bigvee S^1$, bouquets of spaces, fundamental groups of spaces formed by attaching 2-cells, fundamental groups of compact surfaces (torus, projective plane, Klein bottle, surfaces of genus $g$), Midterm review | — |
| Sat 26 Sep | Quiz 2 | 1 | Quiz 2 (Covers Fundamental Groups, $\pi_1(S^1)$, and the Seifert–van Kampen Theorem from Lectures 9–16) | Scheduled |
| 28 Sep – 02 Oct | Mid-Semester Exam | 2 | Mid-Semester Examination Week (30%) — covers material from Lectures 1–16 (includes Gandhi Jayanthi on 02 Oct) | KSoM Exam Schedule |
| Part III: Covering Spaces & Group Actions (Lectures 17–26) — Lee, Chapters 11–12 | ||||
| Tue 06 Oct | Lecture 17 | 1 | Covering Maps and Basic Properties (Lee, Chapter 11): definition, evenly covered neighborhoods, local homeomorphisms, standard examples ($\mathbf{R} \to S^1$, $S^n \to \mathbf{RP}^n$, torus coverings) | — |
| Wed 07 Oct | Lecture 18 | 1 | The Lifting Problem for Covering Maps (Lee, Chapter 11): unique path lifting, homotopy lifting property, uniqueness of lifts | — |
| Thu 08 Oct | Lecture 19 | 2 | The Lifting Criterion and Monodromy Action (Lee, Chapter 11): general lifting criterion, injectivity of $p_*$, characteristic subgroup $p_*(\pi_1(\widetilde{X}))$, monodromy action on fibers | — |
| Tue 13 Oct | Lecture 20 | 1 | Universal Covering Spaces (Lee, Chapter 11): simply connected coverings, universal property, semi-local simple connectivity, local path-connectedness | — |
| Wed 14 Oct | Lecture 21 | 1 | Construction and Existence of Universal Covers (Lee, Chapter 11): explicit construction using path homotopy classes, basis for the covering topology, verification of covering properties | — |
| Thu 15 Oct | Lecture 22 | 2 | The Classification of Covering Spaces (Lee, Chapter 11): covering homomorphisms, equivalence of coverings, the Galois correspondence between connected coverings and subgroups of $\pi_1(X, x_0)$ | — |
| 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 23 | 2 | The Automorphism Group of a Covering (Lee, Chapter 12): deck transformations, normal (regular) coverings, canonical isomorphism $\operatorname{Deck}(p) \cong \pi_1(X, x_0)/p_*(\pi_1(\widetilde{X}, \widetilde{x}_0))$ | — |
| Tue 27 Oct | Lecture 24 | 1 | Quotients by Group Actions (Lee, Chapter 12): group actions on topological spaces, properly discontinuous actions, free actions, orbit spaces $X/G$, quotient projections as covering maps | — |
| Wed 28 Oct | Lecture 25 | 1 | Coverings of Manifolds and Orbit Spaces (Lee, Chapter 12): manifolds as quotients, lens spaces $L(p, q)$, flat torus, Klein bottle, proper group actions | — |
| Thu 29 Oct | Lecture 26 | 2 | Covering Space Applications & Review (Lee, Chapters 11–12): subgroups of free groups via graph coverings, problem session, and Quiz 3 | — |
|
Part IV: Introduction to Homology Theory (Lectures 27–35) — Lee, Chapter 13
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 27 | 1 | Singular Chain Complexes (Lee, Chapter 13): motivation for homology, geometric chains, singular $n$-simplices, free abelian chain groups $C_n(X)$, boundary operator $\partial$, proof that $\partial^2 = 0$ | — |
| Wed 04 Nov | Lecture 28 | 1 | Singular Homology Groups (Lee, Chapter 13): cycles $Z_n(X)$, boundaries $B_n(X)$, homology groups $H_n(X)$, homology of a point, path components and $H_0(X) \cong \mathbf{Z}^{\pi_0(X)}$, reduced homology $\widetilde{H}_n(X)$ | — |
| Thu 05 Nov | Lecture 29 | 2 | Homotopy Invariance of Homology (Lee, Chapter 13): chain maps, chain homotopies, the prism operator, proof that homotopic continuous maps induce identical homomorphisms on homology | — |
| Tue 10 Nov | Lecture 30 | 1 | Homology and the Fundamental Group (Lee, Chapter 13): the Hurewicz homomorphism $h: \pi_1(X, x_0) \to H_1(X)$, proof of the isomorphism $H_1(X) \cong \pi_1(X, x_0)_{\text{ab}}$ for path-connected spaces | — |
| Wed 11 Nov | Lecture 31 | 1 | Exact Sequences and Relative Homology (Lee, Chapter 13): chain complexes of pairs, relative chain groups $C_n(X, A)$, the Snake Lemma, connecting homomorphism $\partial_*$, long exact sequence of a pair $(X, A)$ | — |
| Thu 12 Nov | Lecture 32 | 2 | The Mayer–Vietoris Theorem (Lee, Chapter 13): Mayer–Vietoris sequence for open covers, connecting homomorphisms, homology of spheres $S^n$, suspensions and wedges | — |
| Tue 17 Nov | Lecture 33 | 1 | Homology of Cell Complexes (Lee, Chapter 13): cellular boundary formula, degree of sphere maps, homology groups of CW complexes, compact surfaces, and projective spaces | — |
| Wed 18 Nov | Lecture 34 | 1 | Classical Applications of Homology (Lee, Chapter 13): Brouwer Fixed Point Theorem in all dimensions, Invariance of Domain, degree theory, Hairy Ball Theorem, Borsuk–Ulam Theorem | — |
| Thu 19 Nov | Lecture 35 | 2 | Course Summary, Perspectives, and Final Review: synthesis of homotopy, fundamental groups, covering spaces, and homology, outlook toward cohomology and higher homotopy in Algebraic Topology II | — |
| 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.