Will J. Merry

Algebraic Topology I Lecture Notes

22. Free chain complexes

We 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.

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20. Cellular homology

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

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15. The degree

We 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.

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10. Chain complexes

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

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9. The Hurewicz Theorem

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

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8. The homotopy axiom

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

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7. Singular homology

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

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4. The fundamental group

We 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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