Easter term 2021
Wednesdays 13.30-15.30 on Teams, 28 April-16 June
I consider myself a geometer, but as I am interested in manifolds I also often care about their topology. In modern algebraic topology that seems to be regarded as very mundane, and I have no pretension to be modern or even to have much mastery over the traditional parts of the field. My aim is to share some pieces of algebraic topology that I have encountered and enjoyed and/or found useful such as
Brisk review of (co)chain complexes and (co)homology, singular (co)homology, long exact sequences, universal coefficients, cup products.
Poincaré duality, intersection and torsion linking forms and their geometric interpretations. Künneth formula for cohomology of a product and Leray-Hirsch theorem for cohomology of (some) fibre bundles.
Thom isomorphism theorem, and its use in geometric interpretation of Poincar\é duality. Homotopy groups, Hurewicz theorem, weak homotopy equivalences.
CW complexes and Whitehead's theorem. Fibrations and their long exact sequence of homotopy groups. Cellular (co)homology. Eilenberg-MacLane spaces.
Proof that Hn(X; G) <-> [X, K(G,n)] by naive obstruction theory. Principal G-bundles and characteristic classes, with first Chern class of complex line bundles as illustration of different approaches. Euler class and obstruction to existence of non-vanishing sections of oriented bundles.
Classifying spaces. Primary obstruction classes. Characterisation of Chern classes by Whitney sum formula.
Existence of Charn classes and cohomology of BU(n). Stiefel-Whitney classes, Pontrjagin classes and cohomology of BO(n). Axioms of Steenrod squares and construction via cup-i products.
Cohomology of SQ(X) = (X x X x S^oo)/Z_2 and Steenrod squares. Wu's theorems on relationship between Steenrod squares and Stiefel-Whitney classes.