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

MA4J7 Cohomology and Poincaré duality
Term II 2019-2020

Schedule

Week Date of Monday Topics Pages in Hatcher Example sheet Comments
1 Jan. 6 Introduction and examples. Categories, (covarient) functors, natural transformations. Examples. Rings, modules, graded modules, sequences, and chain complexes. The homology functor factors. \(\Hom_R(P, \ast)\) is covariant. \(\Hom_R(\ast, Q)\) is contravariant. 162 - 165, 190 - 191 Example sheet 1
2 Jan. 13 Fundamental theorem of \(R\)-modules (for \(R\) a PID). Chains, cycles, boundaries, homology. Exact, short, split sequences. Sections. Cochains, cocycles, coboundararies, cohomology. The universal coefficient theorem (UCT). Definition of, properties of, naturality of \(h\). \(\Hom(\cdot, G)\) is right exact. \(\Ext\) measures the failure of \(\Hom(\cdot, G)\) to be left exact. Free resolutions, definition of \(\Ext\), examples. Functoriality of \(\Ext\) (Lemma 3.1). 191 - 195
3 Jan. 20 Finish the proof of UCT. UCT is a functor. Hands on cohomology of \(\RR^n\), of \(S^1\), of \(\TT^2\). Reduced cohomology. Triples of spaces, long exact sequence for relative cohomology. Degree shift. Relative cochains are "simpler" than relative chains. The map \(h\) is a natural transformation between connecting homomorphisms. Review of the five lemma. Long exact sequences, induced homomorphisms, homotopy invariance, excision for cohomology. 129, 195 - 202 Example sheet 2
4 Jan. 27 Axioms for cohomology - homotopy invariance, long exact sequences, excision, and disjoint unions. (We follow Hatcher in omitting the dimension axiom.) Singular cohomology is a cohomology theory. Simplical cohomology is isomorphic to singular. Statement of Proposition 3.17. Review of CW complexes. Two definitions of cellular cohomology; natural isomorphism between them. Cellular cohomology is isomorphic to singular. Dualising split short exact sequences. Review of relative Mayer-Vietoris for homology: excisive covers, subordinate chain complexes, chain homotopy equivalences. Dualise to obtain relative Mayer-Vietoris for cohomology. 202 - 204 Answers to questions asked by students on 2020-01-23.
5 Feb. 3 Cup product on cochains. Example computations. Graded Leibniz rule. Induced cup product on cohomology, induced homomorphisms. \(R\)-algebras, graded \(R\)-algebras, cohomology rings. Example of the torus. Graded (skew) commutivity of cup product at level of cohomology classes. Cup products in two-complexes, for real projective plane. Hatcher's introduction to cohomology. Cohomology of real, complex, and quaternionic projective spaces (statements). 186 - 189, 206 - 208, 210 - 211 Example sheet 3 Answers to questions asked by students on 2020-02-04.
6 Feb. 10 Relative cup product. Tensor products of modules, of \(R\)-algebras. Exterior algebras. Cross product. Statement of Künneth formula, applications, overview of proof. Natural transformation of theories giving isomorphism of theories. Mayer–Vietoris for cohomology theories. Mapping telescopes, transferring the proof from finite to infinite dimensional complexes. 209, 212 - 217
7 Feb. 17 Checking that \(h, k\) are cohomology theories via the axioms: homotopy invariance, excision, long exact sequences, disjoint unions. Degree shifts for long exact sequences. Tensor products and direct products. Naturality of the cross product. The completely relative Künneth formula, cross product of generators for spheres. Cohomology rings for real, complex, and quaternionic projective spaces (statement). 217 - 220 Example sheet 4 Thursday lecture missed due to strike.
8 Feb. 24 Proof of cohomology of (real) projective spaces, using naturality of cross product. 220 - 222 Monday and Tuesday lectures missed due to strike.
9 Mar. 2 All lectures missed due to strike.
10 Mar. 9 All lectures missed due to strike.