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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.
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz , and generalizes earlier results of Henri Poincaré .
The free rank of the nth homology group of a simplicial complex is the nth Betti number, which allows one to calculate the Euler–Poincaré characteristic. One can use the differential structure of smooth manifolds via de Rham cohomology , or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the ...
In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology. [1]
As noted already, π 1 (S 1) = Z, and π 2 (S 2) contains a copy of Z generated by the identity map, so the fact that there is a surjective homomorphism from π 1 (S 1) to π 2 (S 2) implies that π 2 (S 2) = Z. The rest of the homomorphisms in the sequence are isomorphisms, so π n (S n) = Z for all n. [20] The homology groups H i (S n), with ...
Morse homology can also be formulated for Morse–Bott functions; the differential in Morse–Bott homology is computed by a spectral sequence. Frederic Bourgeois sketched an approach in the course of his work on a Morse–Bott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.
The two curves of the Whitehead link have linking number zero. Any two unlinked curves have linking number zero. However, two curves with linking number zero may still be linked (e.g. the Whitehead link). Reversing the orientation of either of the curves negates the linking number, while reversing the orientation of both curves leaves it unchanged.