We survey: the Hurewicz Theorem, the Whitehead Theorem, approximating spaces by cell complexes, and the general uniqueness result for homology theories,

Read MoreWe prove the homotopy LES for fibrations. We define weak fibrations and show any fibre bundle is a weak fibration. We finish by proving Serre’s Theorem.

Read MoreWe define exactness in the pointed homotopy category. We prove the Puppe Sequence and derive the long exact sequence axiom for homotopy groups.

Read MoreWe define the higher homotopy groups and prove they are abelian for n ≥ 2. We then prove that the homotopy groups form a local system of groups.

Read MoreWe show that loop spaces and reduced suspensions are (co)group objects in the pointed homotopy category, and that they form an adjoint pair.

Read MoreIn this lecture we define a group object in a category, together with the dual notion of a cogroup object, and then investigate some examples.

Read MoreWe introduce limits in category theory (which are dual to colimits), and define adjoint functors. We prove that adjoint functors preserve (co)limits.

Read MoreWe construct a duality isomorphism from the Čech cohomology to the relative singular homology. In the compact case this is known as Poincaré Duality.

Read MoreWe introduce Euclidean neighbourhood retracts, and define the Čech cohomology of a pair of compact subsets in a Euclidean neighbourhood retract.

Read MoreWe explain what it means for a manifold to be orientable, and construct the fundamental class of an orientable closed manifold. We then define the cap product.

Read MoreWe define topological manifolds and presheaves, and investigate the top-dimensional homology groups of a manifold.

Read MoreWe prove the Thom Isomorphism Theorem for vector and disk bundles. We prove that a Thom class always exists for orientable vector bundles.

Read MoreWe prove the Leray-Hirsch Theorem, which is a powerful tool for computing the additive structure of the cohomology of a total space of a fibre bundle.

Read MoreWe define fibre bundles and fibrations, and look at some examples, including: covering spaces, vector bundles, Hopf fibrations and homogeneous spaces.

Read MoreWe prove the Alexander-Whitney Formula, which gives an explicit Eilenberg-Zilber morphism, and show the cross product is a homomorphism of graded rings.

Read MoreWe prove diagonal approximations are unique up to chain homotopy. We define the cross product, and show the cup product is graded commutative.

Read MoreWe define the cup product on the singular cochain complex with coefficients in a ring, and show this turns singular cohomology into a graded ring.

Read MoreWe introduce the Ext funtor, and prove the Dual Universal Coefficients Theorem. We prove a Künneth formula for cohomology for spaces of finite type.

Read MoreWe define contravariant functors and cochain complexes. We introduce singular cohomology and show it satisfies the Eilenberg-Steenrod Axioms.

Read MoreWe prove the Eilenberg-Zilber Theorem using the Acyclic Models Theorem. We use this to deduce the (topological) Künneth Formula.

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