Cornell Math - Math 651 (SP01)

Math 651 — Spring 2001
Introduction to Algebraic Topology


Instructor: Marshall Cohen
Time: MW 3:10–4:25
Room: MT 207 (M); MT 206 (W)

Course Outline:

  1. INTRODUCTION: Categories and Functors
    (wherein we learn to trade hard problems for easier ones)
  2. Homotopy and the fundamental group
    (the first functor in topology)
  3. Covering spaces
    (the geometric expression of fundamental groups; natural objects in complex analysis and combinatorial group theory)
  4. CW and simplicial complexes
    (spaces which are built of cells; ideal for a combinatorial analysis)
  5. Simplicial homology
    (a functor which associates to each simplicial complex a graded abelian group)
  6. The homology process
    (the algebra motivated by Part 5)
  7. Singular homology
    (a homology functor for arbitrary spaces; application of this to get cellular homology - the homology of CW complexes)
  8. Applications
    (Lefschetz fixed point theorem, Jordan-Brouwer separation theorem, invariance of domain, etc.)

Prerequisite: Mathematics 453 (Introduction to Topology) and Mathematics 434 (Honors Algebra — in particular the basics of group theory).