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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 ...
It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. [2] CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. [2]
Here might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups . The form of the result is that other coefficients A may be used, at the cost of using a Tor functor .
Cellular homology can also be used to calculate the homology of the genus g surface. The fundamental polygon of Σ g {\displaystyle \Sigma _{g}} is a 4 n {\displaystyle 4n} -gon which gives Σ g {\displaystyle \Sigma _{g}} a CW-structure with one 2-cell, 2 n {\displaystyle 2n} 1-cells, and one 0-cell.
As a ring, it is generated by a class η in degree 1, a class x 4 in degree 4, and an invertible class v 1 4 in degree 8, subject to the relations that 2η = η 3 = ηx 4 = 0, and x 4 2 = 4v 1 4. KO 0 (X) is the ring of stable equivalence classes of real vector bundles over X. Bott periodicity implies that the K-groups have period 8.
A manifold is orientable when the top-dimensional integral homology group is the integers, and is non-orientable when it is 0. The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n = 2, this is sometimes called the "hairy ball theorem".)
The hyperbolic volume of the complement of the Whitehead link is 4 times Catalan's constant, approximately 3.66.The Whitehead link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the pretzel link with parameters (−2, 3, 8).
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.