The goal of this course is to provide a basic introduction into singular homology and cohomology and related constructions and applications. There is a sequel, MAT 542, that covers homotopy theory, fibrations, and further more advanced topics in algebraic topology. Another course to consider after MAT 541 is MAT 566, which covers vector bundles, characteristic class, and topology of smooth manifolds.

Please do some of the following questions from Hatcher. The numbers in bold are recommended but some of the non-bold are very nice, too. Just pick what you like but please do a variety of topics.

section 2.2 p.155:

p.184: 4

section 3.1 p.204:

section 3.2 p.228:

Please do some of the following questions from Hatcher; pick what you like but please do a variety of topics.

As a warm-up question, compute the cohomology ring (with integer coefficients) of the complex projective space; use nondegeneracy of cup product, as we did for real projective spaces.

p. 229 4, 6, 7, 8, 10;

p.258 6, 7-9, 15, 16, 20, 22, 25, 26, 32, 33.

- Basic constructions
- Homotopies & homotopy equivalences, homotopy type; retractions, deformation retractions
- CW-complexes, definition, examples; simplicial complexes
- Operations on spaces: products, quotients, wedge sums, suspension, etc

- Homology
- Singular homology & simplicial homology, constructions
- Chain complexes, chain maps, chain homotopies; exact sequences; the Euler characteristic
- Homotopy invariance of singular homology
- Relative homology, long exact sequence of a pair
- The excision theorem
- Equivalence of simplicial and singular homology
- Homology of CW-complexes via cellular homology
- Computations: surfaces, spheres, projective spaces, lens spaces, etc
- Mayer-Vietoris sequence, more calculations
- Applications: Brouwer fixed point thm, degrees, Jordan curve thm, invariance of domain, etc
- Eilenberg-Steenrod axioms

- Cohomology
- Simplicial and singular cohomology groups
- Hom and Ext, universal coefficient theorems
- Relative cohomology, exact sequences, isomorphism between simplicial and singular cohomology
- Cup product, calculations (cohomology ring of projective spaces, etc)
- Kunneth formulas
- Poincare duality

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