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

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 - 204
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 \(H^1\). 206 - 207, 186 - 189 Two
5 Feb. 4 Quaternionic projective space. Classification of surfaces, connect sums. Cohomology of oriented surfaces, the fundamental class, explicit generators, examples. Cohomology of non-orientable surfaces over \(\ZZ/2\ZZ\), fundamental class, explicit generators. 207 - 208
6 Feb. 11 Examples of two complexes. Relative cup product. Skew (or graded) commutivity of the cup product. Review of \(R\)-modules, \(R\)-algebras, graded versions, polynomial rings, exterior algebras, tensor products of such, multiplication on such. Cross product. Künneth formula, applications, overview of proof. 208 - 216
7 Feb. 18 Checking that \(h, k\) are cohomology theories via the axioms: homotopy invariance, excision, long exact sequences, disjoint unions. Degree shifts for long exact sequences. Naturality of the cross product. Proof by induction on dimension. The completely relative Künneth formula, cross product of generators for spheres. Begin cohomology of (real) projective spaces. 216 - 220
8 Feb. 25 Finish cohomology of (real and complex) projective spaces. Manifolds, local homology, the local homology bundle \(M_R\). \(R\)-orientations and orientability. The fundamental class. The "patching together" construction (Lemma 2.27) and applications. 220 - 222, 230 - 236
9 Mar. 4 Finishing the proof of Lemma 2.27. Cap product, absolute and relative versions. Poincare duality and its generalisation to non-compact manifolds. Cohomology with compact supports, is not a cohomology theory. Directed sets and directed systems, examples. Cofinal subsystems. Directed limits, examples. \(H^*_c(\RR^n)\). 236 - 245
10 Mar. 11 Poincare duality for non-compact manifolds. Mayer-Vietoris for Poincare duality. Direct limits of diagrams and the proof of PD via "patching". Applications: orientability of codimension-one submanifolds of spheres, Euler characteristic in odd dimensions, intersection pairings. Lefschetz duality, Alexander duality (statements only). Relationship of cohomology with compact supports and relative cohomology. 245 - 250, 253 - 256 Three, Four, Five, Six, Seven