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

## MA3F1 Introduction to topology Term I 2014-2015

### Schedule

Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Sep. 29 Introduction. Topological spaces, products, subspaces, quotients. Homeomorphisms. Invariance of domain. Peano curves. Homotopic functions, homotopy equivalent spaces. 1 - 3 One
2 Oct. 6 Homeomorphism and homotopy invariants. The questions of topology. Straight line homotopy. Contractible spaces. Paths, basepoints, loops, and concatenation. Based homotopies. The fundamental group. $$\pi_1(\RR^n) = 1$$. 4, 25 - 27 Two
3 Oct. 13 Change of basepoint. Introduction to $$\pi_1(S^1)$$. Covering maps and spaces, degree of a cover, isomorphism of covers, deck groups. Examples. 28, 56 - 58, 67, 70 Three
4 Oct. 20 $$\Phi$$ is a homomorphism. $$\Phi$$ is an isomorphism. Homotopy lifting property. 29 - 31, 60 Four
5 Oct. 27 Gluing lemma. Finish homotopy lifting property. Applications of $$\pi_1(S^1)$$. Retractions, deformation retractions, induced homomorphisms, "functoriality". Permutation representation of fundamental group of the base space. 31 - 34, 36, 68 - 70 Five
6 Nov. 3 No retraction from ball to sphere, the Brouwer fixed point theorem. The fundamental groups of spheres, of products. $$\pi_1$$ is a homotopy invariant. Covers induce injections on $$\pi_1$$, degree equals index. Free products, reduced words. 31 - 32, 35, 37, 41 - 42, 61 Six
7 Nov. 10 Classification of orientable surfaces, connect sum. $$T \# P = 3P$$. 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
8 Nov. 17 Second half of the proof of SvK: reductions, expansions, exchanges. Cells, boundaries, attaching maps, CW complexes, subcomplexes, graphs, trees. 5, 7, 45 - 46, 519 - 523 Eight The eighth example sheet is closely related to pages 83 - 87 in Hatcher.
9 Nov. 24 $$\pi_1$$ of CW complexes using SvK. Presentations of groups, examples. Presentations of $$\pi_1$$ of two-complexes, examples. 50 - 52, 97 Nine Posted a new version of ninth example sheet on 2014-11-26.
10 Dec. 1 Attaching three-cells, fundamental group determined by two-skeleton. Isomorphic covers, Galois correspondence. Local properties of topological spaces, construction of universal covers, examples. 62 - 68 Ten