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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.
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.
That is, the correct answer in honest Betti numbers is 2, 0, 0. Once more, it is the reduced Betti numbers that work out. With those, we begin with 0, 1, 0. to finish with 1, 0, 0. From these two examples, therefore, Alexander's formulation can be inferred: reduced Betti numbers ~ are related in complements by
Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra , homological algebra , algebraic topology and algebraic number theory , as well as in applications to group theory ...
William Thurston calls this "combing all the tangles to one point". In the original 2-page paper, J. W. Alexander explains that for each t > 0 {\displaystyle t>0} the transformation J t {\displaystyle J_{t}} replicates f {\displaystyle f} at a different scale, on the disk of radius t {\displaystyle t} , thus as t → 0 {\displaystyle t ...
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.
The basic theorem is that the resulting homology is an invariant of the manifold (that is, independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular Betti numbers agree and gives an immediate proof of the Morse inequalities.