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The Lumenary
السنة الأولى من الدراسات العلياGeometry & Topologyمُخطَّط له

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

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Homotopy and the fundamental groupQuotient groups and quotient rings

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 43

ستتعلّم

  • 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.

تعريف

  • 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.

مبرهنة

  • 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.

مفاهيم خاطئة شائعة

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

يقود إلى

Homotopy type theory∞-categories

اكتشافات اختيارية

مسارات التعلّم

قراءات إضافية

  • Allen Hatcher, Algebraic Topology (2002)