The language of topology. Continuity,
homeomorphism, compactness for subsets of Rn
(from the epsilon-delta language to the language
of neighborhoods and coverings).
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).
Examples of surfaces (2-manifolds),
orientability, Euler characteristic.
Classification of surfaces (geometric proof for
Homotopy and homotopy equivalence, the
homotopy groups πn (.) for n > 1 and their
Vector fields on the plane. Generic singular
points. The index of a plane vector field. Vector
fields on surfaces. The Poincaré index theorem.
Infinite constructions (counterexamples to
“obvious” statements): the Cantor set, the Peano
curve, the Brouwer continuum (as a strange
attractor related to the equation z3 – 1 = 0),
Antoine’s necklace, and Alexander’s horned
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
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.
Knots and links in 3-space. Reidemeister
moves. The Alexander-Conway polynomial.
W.Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, Springer-Verlag, New York, 1991.