The Whitehead Theorem

Algebraic Topology · Axiom Academy

A fundamental result connecting homotopy groups and homotopy equivalences for CW complexes 1. Statement of the Whitehead Theorem Let's begin with the precise statement of this fundamental result. This means that if f induces an isomorphism on π 0 (components), π 1 (fundamental group), π 2 , π 3 , and so on for all higher homotopy groups, then there exists a map g : Y → X such that g ∘ f ≃ id X and f ∘ g ≃ id Y . 2. Why CW Complexes Are Essential The Whitehead Theorem specifically requires CW complexes. Without this assumption, the theorem fails spectacularly! Built from cells: 0-cells (points), 1-cells (intervals), 2-cells (disks), etc. Each n -cell is attached to the ( n -1)-skeleton by a continuous map Weak topology: a set is closed iff its intersection with each cell is closed Examples: spheres S n , tori, compact manifolds, simplicial complexes Counterexample without CW assumption: Consider the Warsaw circle W , a pathological space that is not a CW complex. There exists a map from W to a point that induces isomorphisms on all homotopy groups (since W is simply connected with trivial higher homotopy groups), but W is not contractible, so this map is not a homotopy equivalence. 3. Weak Homotopy Equivalence vs Homotopy Equivalence The Whitehead Theorem bridges two important concepts in homotopy theory. Every homotopy equivalence is automatically a weak homotopy equivalence (follows from functoriality of homotopy groups)

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