\( \newcommand{\cross}{\times} \newcommand{\isom}{\cong} \newcommand{\RP}{\mathbb{RP}} \newcommand{\EE}{\mathbb{E}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \)

MA3F1 Introduction to topology
Term I 2024-2025

Schedule

Week Date of Monday Topics Pages in Hatcher Lecture notes Example sheet Comments
1 Sep. 30 Introduction and overview. Topological spaces, bases, products, subspaces, examples. Homeomorphisms, foundational problem, invariance of domain, topological invariants. Disjoint union, quotients, group actions, cut-and-paste. Our friend the square, the annulus, the Mobius band. Homotopies, gluing lemma. 1 Monday Tuesday Thursday One
2 Oct. 7 Our friend the square. Homotopy equivalence. Paths, "reeling in", homotopy rel endpoints. Concatenation of paths and homotopy. Loops, space of loops, concatenation of loops is a loop, concatenation does not have indentity, inverses, and is not associative. The fundamental group. Multiplication is well-defined, closed, has identity, inverses, and associativity. Basepoints, basepoint isomorphism. Covering spaces, covering maps, stack of pancakes, covering degree. Isomorphism of covering spaces. Graphs, the rose with two petals. 1 - 4, 21 - 28, 56 - 60 Monday Tuesday Thursday Two Note that I have updated the third example sheet - please do Problem 3.3 and turn it in Friday, noon, in week three.
3 Oct. 14 Deck transformations, deck groups. Carpark example. Homotopies descend, concatenations descend. \(\pi_1(S^1)\) and winding paths. Definition of \(\Phi\). \(\Phi\) is a homomorphism. Deck transforms for cover of the circle, straight-line homotopy. Homotopy lifting and covering spaces. Homotopy lifting property for covering spaces (1.30). \(\Phi\) is an isomorphism (1.7). 29 - 31, 60, 71 - 72 Monday Tuesday Thursday Three Fixed typo in Thursday lecture notes - image of lift is in \(Z\).
4 Oct. 21 Covering maps have the (unique) homotopy lifting property. Special case of path lifting. Constant paths lift to constant paths. Fixed point property. \(B^n\) has the fixed point property. The tripod space. Induced homomorphisms, functorality. The fundamental group is a homeomorphism invariant. 30 - 31, 60, 34 Monday Tuesday Thursday Four
5 Oct. 28 The fundamental group is a homotopy invariant. Retracts, deformation retracts. No retract theorem. Brouwer fixed point theorem (for \(n = 2\)). "Tablecloth theorem", Perron-Frobenius theorem (for \(n = 3\)), fundamental theorem of algebra. Null-homotopies. Even versus odd functions. 31 - 34, 36 Monday Tuesday Thursday Five
6 Nov. 4 Even and odd functions, Borsuk-Ulam theorem. The fundamental groups of spheres, of cartesian products. The "Galois correspondence". Covers induce injections on \(\pi_1\). Index equals degree. Motivation for free groups. Pointed union. Beginning free products: words, empty word, length, concatenation, associativity, reductions. 31 - 32, 35, 41 - 42, 61 Monday Tuesday Thursday Six
7 Nov. 11 The free products of groups is a group. Earring space. Statement of Seifert-van Kampen. \( \pi_1(S^1 \vee S^1) = \ZZ \ast \ZZ\). Factorisations. Diagrams of spaces, diagrams of groups. Reductions, expansions, exchanges. 49, 43 - 45 Monday Tuesday Thursday Seven
8 Nov. 18 Finish proof of Seifert-van Kampen. CW complexes: cells, boundaries, attaching maps, skeleta. Examples. Non-examples: Cantor set, Earring space, topologist's circle. Properties of subspaces (precompact, connected, path-connected, Hausdorff, normal, contractible, simply-connected). Local properties. Subcomplexes. CW complexes are nice (A1, A3, A4, A5). Spheres and projective spaces. One-skeleton determines path-conectivity, two-skeleton determines fundamental group. Generators and relations, presentations of groups. Finite generation, finite presentations. Fundamental group of the rose. Fundamental group of the two-torus. 45 - 46, 5 - 7, 519 - 523, 39 - 40, 97 (only first paragraph) Monday Tuesday Thursday Eight
9 Nov. 25 HERE. Spheres are not contractible. \(\pi_1\) of CW complexes using SvK. Presentations of \(\pi_1\) of two-complexes, algorithm to compute presentations. 50 - 52 Nine
10 Dec. 2 Classification of surfaces. Construction of universal covers. 63 - 70