Will J. Merry

Algebraic Topology II Lecture Notes

46. Epilogue

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

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44. The Puppe Sequence

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

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40. Adjoint functors

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

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39. The Duality Theorem

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

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38. Čech cohomology

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

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33. Fibre bundles

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

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