**Instructor:**Olga Plamenevskaya, office 2-112 Math Tower, e-mail:`olga@math.sunysb.edu`**Office hours:**MW 11:00am-12:55 pm or by appointment.**Class meetings:**Monday and Friday, 1:00-2:20 pm, in Math 4-130.

This course forms a one-year sequence in algebraic topology together with MAT 566 that I will teach in Spring 2014. The goal of the one-year course is to cover the basics on singular homology, cohomology, homotopy theory and vector bundles; as the year progresses, the focus will be shifting from working with more general topological spaces to understanding topology of smooth manifolds. In particular, MAT 539 will mainly deal with singular homology and cohomology and a little bit of homotopy theory. MAT 566 will explore more of homotopy theory and will be concerned with vector bundles, characteristic classes, and smooth topology.

- 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
- 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

**Students with Disabilities: **If you have a physical,
psychological, medical, or learning disability that may impact on your
ability to carry out assigned course work, you are strongly urged to
contact the staff in the Disabled Student Services (DSS) office: Room
133 in the Humanities Building; 632-6748v/TDD. The DSS office will
review your concerns and determine, with you, what accommodations are
necessary and appropriate. A written DSS recommendation should be
brought to your lecturer who will make a decision on what special
arrangements will be made. All information and documentation of
disability is confidential. Arrangements should be made early in the
semester so that your needs can be accommodated.