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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]
Since C −1 = 0, every 0-chain is a cycle (i.e. Z 0 = C 0); moreover, the group B 0 of the 0-boundaries is generated by the three elements on the right of these equations, creating a two-dimensional subgroup of C 0. So the 0th homology group H 0 (S) = Z 0 /B 0 is isomorphic to Z, with a basis given (for example) by the image of the 0-cycle (v 0).
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 connectivity of the empty space is, by convention, conn H (X) = -2. Some computations become simpler if the connectivity is defined with an offset of 2, that is, ():= +. [2] The eta of the empty space is 0, which is its smallest possible value. The eta of any disconnected space is 1.
This does not contradict Whitehead theorem since the Long Line does not have the homotopy type of a CW-complex. Another prominent example for this phenomenon is the Warsaw circle . References
A simply connected finite complex X is rationally elliptic if and only if the rational homology of the loop space grows at most polynomially. More generally, X is called integrally elliptic if the mod p homology of grows at most polynomially, for every prime number p. All known Riemannian manifolds with nonnegative sectional curvature are in ...
A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an extraordinary homology theory (dually, extraordinary cohomology theory). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory , which are extraordinary co homology theories, and ...