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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.
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.
By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Finding techniques to robustly calculate homology using various triangulation strategies over multiple length scales is the topic of persistent homology. [11]
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 .
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]
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
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 ...
In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered as dual to its complement in the n-sphere, where n is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifolds.