Mathematics

# MATH 7221: Topology 2

Lecture - 4 credits

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- Continues MATH 5121.
- Introduces homology and cohomology theory.
- Studies singular homology, homological algebra (exact sequences, axioms), Mayer-Vietoris sequence, CW-complexes and cellular homology, calculation of homology of cellular spaces, and homology with coefficients.
- Moves on to cohomology theory, universal coefficients theorems, Bockstein homomorphism, Knnneth formula, cup and cap products, Hopf invariant, Borsuk-Ulam theorem, and Brouwer and Lefschetz-Hopf fixed-point theorems.
- Ends with a study of duality in manifolds including orientation bundle, Poincar&#233; duality, Lefschetz duality, Alexander duality, Euler class, Lefschetz numbers, Gysin sequence, intersection form, and signature.

Continues MATH 5121.

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