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The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map f: X → Y has a homotopy inverse g: Y → X, which is not at all clear from the assumptions.) This implies the same conclusion for spaces X and Y that are homotopy equivalent to CW complexes.
CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups. A covering space of a CW complex is also a CW complex.
An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map h takes a homotopy class of maps from X to K(G, i) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor. [1]
on homology is an isomorphism for all integers n. (Here H n (X) is the object of A defined as the kernel of X n → X n−1 modulo the image of X n+1 → X n.) The resulting homotopy category is called the derived category D(A).
Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing. Another important result is the approximation theorem. First, the homotopy category of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point. The free rank of the nth homology group of a simplicial complex is the nth Betti number, which allows one to calculate the Euler–Poincaré characteristic.
Typically these only hold for larger dimensions. The first such result was Hans Freudenthal's suspension theorem, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results. [19] In 1953 George W. Whitehead showed that there is a metastable range for the homotopy groups of spheres.
Well-ordering theorem (mathematical logic) Whitehead theorem (homotopy theory) Whitney embedding theorem (differential manifolds) Whitney extension theorem (mathematical analysis) Whitney immersion theorem (differential topology) Whitney–Graustein Theorem (algebraic topology) Wick's theorem ; Wiener's tauberian theorem (real analysis)