Algebraic topology

Inajengwa juu ya

Masharti ya awali ya lazima 43 (hadi chini kabisa)

Utajifunza

• 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.

Fasili

• 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.

Nadharia

• 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.

Mawazo potofu ya kawaida

• Thinking homology determines homotopy type.
• Confusing homology groups with homotopy groups; the higher homotopy groups of spheres are far more complicated.

Inaelekea kwa

Ugunduzi wa hiari

• Low-dimensional topology: 3- and 4-manifolds Homology and the fundamental group are the first tools for classifying 3- and 4-manifolds, where many questions remain open.
• ← Functors
• ← Surfaces, polyhedra and the Euler characteristic
• ← Natural transformations
• ← Differential forms and Stokes' theorem
• ← Homological algebra

Njia za kujifunza

• Graduate preparation
• Geometry and topology

Usomaji zaidi

• Allen Hatcher, Algebraic Topology (2002)