Skip to content
The Lumenary
First-year graduateGeometry & TopologyPlanned

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

Show everything needed to reach this topic

The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.

Builds on

Homotopy and the fundamental groupQuotient groups and quotient rings

Required prerequisites, all the way down: 43

You will learn to

  • 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.

Definition

  • 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.

Theorem

  • 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.

Common misconceptions

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

Leads to

Homotopy type theory∞-categories

Optional discoveries

Learning pathways

Further reading

  • Allen Hatcher, Algebraic Topology (2002)