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.