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

MA3F1 Introduction to topology
Term I 2017-2018


Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Oct. 2 Introduction and overview. Topological spaces, bases, products, subspaces, examples. Homeomorphism, foundational problem, invariance of domain, topological invariants. Disjoint union, quotients, our friend the square, the Mobius band. NA One There is no lecture on Monday, and no support classes this week. Questions asked by students on 2017-10-04.
2 Oct. 9 Pairs of spaces, pointed spaces. Retracts, deformation retracts. No retract theorem. Straight-line homotopy, homotopy, gluing lemma, homotopy equivalence. Contractible spaces, spheres are not contractible. Paths, concatenation, homotopy rel endpoints. Loops, the fundamental group. \(\pi_1(\RR^n) \isom 1\). Introduction to \(\pi_1(S^1)\) and winding number. 1 - 4, 25 - 27 Two Questions asked by a student on 2017-10-11.
3 Oct. 16 Basepoints. Covering maps, covering spaces. Isomorphism of covers, degree of a cover, deck groups. The rose with \(n\) petals, its covers. Lifting. Homotopies descend, concatenations descend. \(\Phi\) is a homomorphism. 28 - 29, 56 - 60, 67, 70 Three Questions asked by students on 2017-10-18.
4 Oct. 23 \(\Phi\) is an isomorphism. Covering maps have the (unique) homotopy lifting property. 29 - 31, 60 Four
5 Oct. 30 Induced homomorphisms, functorality. Application to retractions, deformation retractions. \(\pi_1\) is a homotopy invariant. No retraction theorem, the Brouwer fixed point theorem, variants. The fixed point property for a space. Even versus odd. 31 - 34, 36 Five
6 Nov. 6 Borsuk-Ulam theorem. Covers induce injections on \(\pi_1\), degree equals index. The fundamental groups of spheres, of cartesian products. Motivation for free groups. 31 - 32, 35, 41 - 42, 61 Six Questions asked by a student on 2017-11-07.
7 Nov. 13 Free products, words, empty, reduced. Proof of associativity. Hawaiian earring. Statement of Seifert-van Kampen. \( \pi_1(S^1 \vee S^1) = \ZZ \ast \ZZ\). First half of the proof of SvK: factorisations. 43 - 45 Seven Question asked by a student on 2017-11-13.
8 Nov. 20 Second half of the proof of SvK: reductions, expansions, exchanges. SvK and \(\pi_1(\RP^2)\). Cells, boundaries, attaching maps, CW complexes, weak topology, subcomplexes, finite, finite dimensional, graphs, trees. 5 - 7, 45 - 46 Eight
9 Nov. 27 Propositions A.1, A.3, A.4, A.5. Local properties. Warsaw circle. \(\pi_1\) of CW complexes using SvK. Presentations of groups. Presentations of \(\pi_1\) of two-complexes, algorithm to compute presentations. 519 - 523, 50 - 52, 97 (only first paragraph) Nine
10 Dec. 4 Classification of surfaces. Construction of universal covers. 63 - 70 Ten