MAT 541 Algebraic Topology, Fall 2019.

  • Instructor: Olga Plamenevskaya, office 2-112 Math Tower, e-mail: olga@math.stonybrook.edu
  • Office hours: Tue,We 1:00-2:30pm or by appointment.
  • Class meetings: Tuesday and Thursday, 11:30am-12:50pm, Math 4-130.
    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.
  • Prerequisites : MAT 530, preferably MAT 531 as well.
  • References :
    Allen Hatcher, Algebraic Topology. This book is available for free download from Hatcher's webpage.
    James R. Munkres, Elements of Algebraic Topology.

  • Homework and grading: there will be a few problem sets, as well as some general recommendations to solve problems from certain sections of Hatcher. If you have passed comps, you should still do at least some homework to learn how to compute and use (co)homology. If you have not passed the comps and are taking the course for a grade, the homework is mandatory. You must either submit the homeworks to me by the due dates, or make arrangements with me to discuss homework orally. Your course grade will be determined by homeworks. There will be no exams.

    Homework 1 (pdf), due October 3.

    Homework 2 due November 21.
    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: 2, 7, 8, 9, 10, 12, 15, 17, 28, 29 6, 22, 27, 41.
    p.184: 4
    section 3.1 p.204: 8, 9, 5
    section 3.2 p.228: 1, 2

    Homework 3 due anytime before December 18.
    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.

  • Syllabus: I hope to cover the topics listed below, perhaps in a slightly different order. Additional topics may be discussed if time permits.
    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
      • 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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