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The Whitehead theorem does not hold for general topological spaces or even for all subspaces of R n. For example, the Warsaw circle , a compact subset of the plane, has all homotopy groups zero, but the map from the Warsaw circle to a single point is not a homotopy equivalence.
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.
This is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary). If M {\displaystyle M} is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2 {\displaystyle 2} nd homology group
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 ...
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.
Through further reductions, it is possible to identify the homology of with the cohomology of . This is useful in algebraic geometry for computing the cohomology groups of projective varieties , and is exploited for constructing a basis of the Hodge structure of hypersurfaces of degree d {\displaystyle d} using the Jacobian ring .
Define W, the Whitehead continuum, to be =, or more precisely the intersection of all the for =,,, …. The Whitehead manifold is defined as X = S 3 ∖ W , {\displaystyle X=S^{3}\setminus W,} which is a non-compact manifold without boundary.
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]