$$\newcommand{\cross}{\times} \newcommand{\isom}{\cong} \newcommand{\RP}{\mathbb{RP}} \newcommand{\EE}{\mathbb{E}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ext}{\operatorname{Ext}}$$

## MA4J7 Cohomology and Poincaré duality Term II 2018-2019

### Schedule

Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Jan. 7 Introduction and examples. Chain complexes. $$\Hom$$. Functors, categories, co- and contravariance. Examples. Sequences - homological degree, exactness, shortness, sums thereof. Cochain complexes, cochains, cocycles, coboundaries. The universal coefficient theorem. 190 - 193
2 Jan. 14 The universal coefficient theorem. $$\Hom(\cdot, G)$$ is right exact. $$\Ext$$ measures the failure of $$\Hom(\cdot, G)$$ to be left exact. Free resolutions, definition of $$\Ext$$, examples. UCT is natural. Cohomology with $$\ZZ$$ coefficients, with PID coefficients, with field coefficients. Examples of cochains. Reduced and relative cohomology, the connecting homomorphisms. 193 - 200 One
3 Jan. 21 Relative cochains are "simpler" than relative chains. Induced homomorphisms and homotopy invariance. Excision for cohomology using UCT and the five-lemma. Bundles, trivial bundles, unit tangent bundles. Simplical and cellular cohomology. Mayer-Vietoris, absolute and relative versions. 200 - ???
4 Jan. 28 Cup product on cochains. Example computations. Graded Leibniz rule. Cup product on cohomology and induced homomorphisms. Interpretation of $$H^0$$. Graphs, spanning trees, and interpretation of $$H^1$$. Two
5 Feb. 4
6 Feb. 11
7 Feb. 18
8 Feb. 25
9 Mar. 4
10 Mar. 11