Topology I

Curriculum:

  1. The language of topology. Continuity, homeomorphism, compactness for subsets of Rn (from the epsilon-delta language to the language of neighborhoods and coverings).
  2. The objects of topology: topological and metric spaces, simplectic and cell spaces, manifolds. Topological constructions (product, disjoint union, wedge, cone, suspension, quotient spaces, cell spaces, examples of fiber bundles).
  3. Examples of surfaces (2-manifolds), orientability, Euler characteristic. Classification of surfaces (geometric proof for triangulated surfaces).
  4. Homotopy and homotopy equivalence, the homotopy groups πn (.) for n > 1 and their main properties.
  5. Vector fields on the plane. Generic singular points. The index of a plane vector field. Vector fields on surfaces. The Poincaré index theorem.
  6. Infinite constructions (counterexamples to “obvious” statements): the Cantor set, the Peano curve, the Brouwer continuum (as a strange attractor related to the equation z 3 – 1 = 0), Antoine’s necklace, and Alexander’s horned sphere.
  7. Curves in the plane, degree of a point with respect to a curve, Whitney index (winding number) of a curve, classification of immersions, the “fundamental theorem of algebra”. Degree of a map of a circle into itself. Brouwer fixed point theorem.
  8. Fundamental group (main properties, simplest computations), covering spaces. Algebraic classification of covering spaces (via subgroups of the fundamental group of the base). Branched coverings, Riemann-Hurwitz theorem.
  9. Knots and links in 3-space. Reidemeister moves. The Alexander-Conway polynomial.

Textbooks

  • W.Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, Springer-Verlag, New York, 1991.