# 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. 