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Topology II -- Spring 2014

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MTH 631 - Topology II. First course in algebraic topology. Homotopy, fundamental group, simplicial homology. PR: MTH550 and MTh630. 3 hours.

  • Time and Place: 12:30 pm - 1:45 pm TR 263 Smith Hall.
  • Instructor: Peter Saveliev (call me Peter)
  • Office: Smith Hall 325
  • Office Hours: MW 2:30 - 4:30, or by appointment
  • Office Phone: x4639
  • E-mail: saveliev@marshall.edu
  • Class Web-Page: math02.com
  • Prerequisites: Topology I -- Fall 2013
  • Text: Topology Illustrated (online draft), specific chapters linked below
  • Goals: Introduction to point-set and algebraic topology
  • Grade Breakdown:
    • homework and quizzes: 40%
    • midterm: 25%
    • final exam: 35%
  • Letter Grades: A: 90-100, B: 80-89, C: 70-79, D: 60-69, F: <60

See also Course policy.

1 Lectures

They will appear here as the course progresses.

A better organized version of the content of the lectures appears in one of the chapters below within a week. The most current material is marked with $\star$.

2 Chapters

This is the text that will be followed. The chapters will be updated or rewritten, sometimes significantly, as the course progresses. Read them.

Chapter 4. Topological spaces

  1. Compact spaces
  2. Quotients
  3. Cell complexes
  4. Triangulations
  5. Manifolds
  6. Products

Chapter 5. Maps

  1. Homotopy
  2. Cell maps
  3. Homology theory $\star$
  4. Euler and Lefschetz numbers
  5. Homology of parametric complexes

Chapter 6. Advanced topics

  1. Short overview
  2. Singular complexes
  3. Homology as a group
  4. Properties of homology groups
  5. Spaces vs subspaces: relative homology
  6. Exact sequences
  7. Cochain complexes and cohomology
  8. Homology vs homology of the complement: Alexander duality
  9. Homology vs cohomology in manifolds: Poincare duality
  10. Cup product and cap product
  • Appendices:
  1. Commutative diagrams
  2. Computing homology
  3. Fundamental group
  4. Homology and algebra
  5. Homology as a vector space
  6. More about manifolds
  7. Mobius band

3 Notes

Related texts: