MAT 539 Algebraic Topology, Fall 2013.

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.
  • Prerequisites : MAT 530.
  • References :
    Allen Hatcher, Algebraic Topology. This book is available for free download from Hatcher's webpage.
    James R. Munkres, Elements of Algebraic Topology.
    We will use other books as well but you are not required to buy anything. (Please download Hatcher, though.)

  • Homework: here are some problem sets from past years: Homework 1, Homework 2, Homework 3. They are not mandatory (I'm not going to grade), but strongly recommended. In addition, you can also do extra exercises from Hatcher. The homeworks, are mostly your own responsibility. (If, however, you missed a lot of lectures, I will ask you to submit some written homeworks to pass the course. There are plenty of useful questions you can think about; doing the homework will hugely improve your understanding of the material.
  • Syllabus: I hope to cover the topics listed below, perhaps in a slightly different order. Additional topics may be discussed if time permits. The end of the course will be somewhat flexible as the discussion will continue in the spring.
    1. 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
    2. 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
    3. 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.