We prove the Acyclic Models Theorem, which constructs natural chain homotopies and chain equivalences between free and acyclic functors.

Read MoreWe prove that on the category of finite cell complexes, there is, up to isomorphism, at most one homology theory. We then study free chain complexes.

Read MoreWe define natural transformations and functor categories, and then go on to introduce the Eilenberg-Steenrod Axioms for a homology theory.

Read MoreWe define cellular homology and show it agrees with singular homology. We prove the Cellular Boundary Formula and compute the homology of projective spaces.

Read MoreWe show that for "nice" pairs of topological spaces, relative homology coincides with reduced homology. We then prove the Relative Homeomorphism Theorem.

Read MoreWe define adjunction spaces and show that these are pushouts in the category of topological spaces. We then introduce cell complexes.

Read MoreWe consider sequential colimits of closed embeddings, and prove the Jordan-Brouwer Separation Theorem and the Invariance of Domain Theorem.

Read MoreWe define diagrams in a category. We then introduce colimits and filtered colimits, and prove that homology commutes with taking filtered colimits.

Read MoreWe define the homological degree of a map from a sphere to itself. We prove the "Hairy Ball Theorem", and prove that an odd map has odd degree.

Read MoreWe prove the excision axiom for singular homology, and deduce the Mayer-Vietoris long exact sequence. We use this to compute the homology of spheres.

Read MoreWe introduce barycentric subdivision. We show that barycentric subdivision is a chain map, and that the induced map on homology is the identity.

Read MoreWe define reduced homology and relative homology. This allows us to see singular homology as a functor on the category of pairs of topological spaces.

Read MoreWe prove the Snake Lemma, and use this to deduce the long exact sequence in homology associated to a short exact sequence of chain complexes.

Read MoreWe introduce the category of chain complexes and chain maps. We then define exactness, a notion that is at the heart of homological algebra.

Read MoreWe prove the Hurewicz Theorem: the first homology group of a path connected space is isomorphic to the abelianisation of the fundamental group.

Read MoreWe investigate then zeroth homology groups of a path-connected topological space. We then prove that singular homology satisfies the homotopy axiom.

Read MoreSingular simplices are continuous maps from a simplex into a topological space. We introduce the singular chain complex and define singular homology.

Read MoreWe show that the free product with amalgamation is a pushout in the category of groups. We use this to prove the Seifert-van Kampen Theorem.

Read MoreWe compute the fundamental group of the circle. We then introduce pushouts and universal properties, and prove that a pushout is unique if it exists.

Read MoreWe define the fundamental group and show that it is a functor from the homotopy category of pointed topological spaces to the category of groups.

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