Prerequisites
Concepts studied in the Foundations of Analysis and Analysis I courses, Elementary Group Theory.
Curriculum
- The language of topology. Continuity, homeomorphism, compactness for subsets of \(\mathbb{R}^n\) (from the epsilon-delta language to the language of neighborhoods and coverings).
- The objects of topology: topological and metric spaces, simplicial 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 triangulated surfaces).
- Homotopy and homotopy equivalence, the homotopy groups \(\pi_n(\cdot)\) for \(n > 1\) and their main properties.
- 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 \(z^3 – 1 = 0\)), Antoine’s necklace, and Alexander’s horned sphere.
- 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.
- 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.
Textbooks
- W. Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag, New York, 1991.