研究生一年级Geometry & Topology计划中

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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前置知识

Homotopy and the fundamental group Quotient groups and quotient rings

43 项必需的前置知识（一直追溯到底）

你将学会

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.

定义

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.

定理

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.

常见误解

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