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For instance, take X= S 2 × RP 3 and Y= RP 2 × S 3. Then X and Y have the same fundamental group, namely the cyclic group Z/2, and the same universal cover, namely S 2 × S 3; thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the Künneth formula); thus, X and Y are not ...
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. [13] The product of two CW complexes can be made into a CW complex.
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 Hairy ball theorem: any continuous vector field on the 2-sphere (or more generally, the 2k-sphere for any ) vanishes at some point. The Borsuk–Ulam theorem : any continuous function from an n -sphere into Euclidean n -space maps some pair of antipodal points to the same point.
A pointed space means a pair (X,x) with X a topological space and x a point in X, called the base point. The category Top * of pointed spaces has objects the pointed spaces, and a morphism f : X → Y is a continuous map that takes the base point of X to the base point of Y. The naive homotopy category of pointed spaces has the same objects ...
The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point. The free rank of the n th homology group of a simplicial complex is the n th Betti number , which allows one to calculate the Euler–Poincaré characteristic .
(For X and Y path-connected, the first condition is automatic, and it suffices to state the second condition for a single point x in X.) For simply connected topological spaces X and Y, a map f: X → Y is a weak homotopy equivalence if and only if the induced homomorphism f *: H n (X,Z) → H n (Y,Z) on singular homology groups is bijective ...
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they ...