Primer año de posgradoGeometry & TopologyPlanificado
Algebraic topology
Singular and cellular homology, CW complexes, exact sequences and Mayer–Vietoris, homotopy invariance, cohomology and cup products, with applications such as invariance of dimension and the Lefschetz fixed-point theorem.
math.geom.algebraic-topology
La lección de este tema todavía no se ha escrito. El esquema siguiente forma parte del mapa del plan de estudios y no ha sido revisado.
Se basa en
Requisitos previos obligatorios, hasta el fondo: 43
Aprenderás a
- Compute homology of spheres, projective spaces and surfaces using cellular homology and Mayer–Vietoris.
- Prove that homology is a homotopy-invariant functor and deduce that ℝᵐ and ℝⁿ are not homeomorphic for m ≠ n.
- Compute cohomology rings of simple spaces and use cup products to distinguish spaces with the same homology.
Definición
- The singular chain complex of X has as n-chains the formal integer combinations of continuous maps from the standard n-simplex into X.
- A CW complex is a space built by successively attaching cells of increasing dimension.
- Singular cohomology Hⁿ(X; R) is the cohomology of the dual cochain complex with coefficients in R.
Teorema
- Hₙ(Sⁿ) ≅ ℤ for n ≥ 1, and the reduced homology of Sⁿ vanishes in all other degrees.
- Mayer–Vietoris gives a long exact sequence relating the homology of a union to that of its pieces.
- ℝᵐ and ℝⁿ are homeomorphic only if m = n.
- Cup products distinguish ℂP² from S² ∨ S⁴, although the two spaces have isomorphic homology groups.
Errores frecuentes
- Thinking homology determines homotopy type.
- Confusing homology groups with homotopy groups; the higher homotopy groups of spheres are far more complicated.
Conduce a
Descubrimientos opcionales
Itinerarios de aprendizaje
Lecturas complementarias
- Allen Hatcher, Algebraic Topology (2002)