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

MA4J7 Cohomology and Poincaré duality
Term II 2018-2019


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