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

## MA3F1 Introduction to topology Term I 2015-2016

### Schedule

Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Oct. 5 Introduction. Topological spaces, products, subspaces, quotients, disjoint unions. Homeomorphism, foundational problem. Invariance of domain. Topological invariants. Homotopies, homotopy equivalence. 1 - 3 One There are no support classes this week.
2 Oct. 12 Gluing lemma. Straight-line homotopy. Contractible spaces. Paths, concatenation, basepoints, loops. Based homotopies. The fundamental group. $$\pi_1(\RR^n) \isom 1$$. Change of basepoint. 4, 25 - 28 Two Lectures by Prof. David Mond.
3 Oct. 19 Introduction to $$\pi_1(S^1)$$ and winding number. Covering maps, covering spaces. Degree of a cover, isomorphism and automorphism of covers, deck groups. Retractions, deformation retractions. Lifting in general, lifting for the circle. 29, 56 - 60, 67, 70 Three
4 Oct. 26 Homotopies descend, concatenation descends. $$\Phi$$ is a homomorphism. $$\Phi$$ is an isomorphism. Gluing lemma, homotopy lifting property, generalizes path lifting property. 29 - 31, 60 Four Questions asked by students.
5 Nov. 2 Finish homotopy lifting property. Induced homomorphisms, functorality. Application to retractions, deformation retractions. $$\pi_1$$ is a homotopy invariant. No retraction theorem, the Brouwer fixed point theorem, variants. 31 - 34, 36 Five Questions asked by students. On the assignment it may be useful to know that $$\pi_1(X \cross Y) \isom \pi_1(X) \times \pi_1(Y)$$.
6 Nov. 9 Even versus odd, Borsuk-Ulam theorem, variants. Covers induce injections on $$\pi_1$$, degree equals index. The fundamental groups of spheres, of cartesian products. Motavation for free groups. 31 - 32, 35, 41 - 42, 61 Six Questions asked by students.
7 Nov. 16 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: factorizations. 43 - 45 Seven Questions asked by students.
8 Nov. 23 Second half of the proof of SvK: reductions, expansions, exchanges. Cells, boundaries, attaching maps, CW complexes, weak topology, subcomplexes, finite, finite dimensional, graphs, trees. Propositions A.1, A.3, A.4, A.5. 5 - 7, 45 - 46, 519 - 523 Eight
9 Nov. 30 $$\pi_1$$ of CW complexes using SvK. Presentations of groups. Presentations of $$\pi_1$$ of two-complexes, algorithm to compute presentations. 50 - 52, 97 (only first paragraph) Nine Questions asked by students.
10 Dec. 7 Classification of surfaces. Isomorphic covers, Galois correspondence. Construction of universal covers. 63 - 70 Ten

Eleven