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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.
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).
A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an extraordinary homology theory (dually, extraordinary cohomology theory). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory , which are extraordinary co homology theories, and ...
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups: H i (X; Z) completely determine its homology groups with coefficients in A, for any abelian group A: H i (X; A)
The prototype here is the Jordan curve theorem, which topologically concerns the complement of a circle in the Riemann sphere. It also tells the same story. We have the honest Betti numbers 1, 1, 0. of the circle, and therefore 0, 1, 1. by flipping over and 1, 1, 0. by shifting to the left.
The Whitehead group of a connected CW-complex or a manifold M is equal to the Whitehead group (()) of the fundamental group of M.. If G is a group, the Whitehead group is defined to be the cokernel of the map {} ([]) which sends (g, ±1) to the invertible (1,1)-matrix (±g).
The complement to any connected knot or graph in a 3-dimensional sphere is of type (,); this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos. [ 1 ] Any compact , connected, non-positively curved manifold M is a K ( Γ , 1 ) {\displaystyle K(\Gamma ,1)} , where Γ = π 1 ( M ) {\displaystyle \Gamma =\pi ...