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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.
0, 1, 0, 0. This does work out, predicting the complement's reduced Betti numbers. The prototype here is the Jordan curve theorem, which topologically concerns the complement of a circle in the Riemann sphere. It also tells the same story. We have the honest Betti numbers 1, 1, 0. of the circle, and therefore 0, 1, 1. by flipping over and 1, 1, 0
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 ...
It is a theorem, proved first by Frank Adams, and subsequently by Adams and Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1. Whitehead integral formula
A continuous map: of simply connected topological spaces is called a rational homotopy equivalence if it induces an isomorphism on homotopy groups tensored with the rational numbers . [1] Equivalently: f is a rational homotopy equivalence if and only if it induces an isomorphism on singular homology groups with rational coefficients. [ 3 ]
If r is 0 or 1 mod 8 and positive, the order of the image is 2 (so in this case the J-homomorphism is injective). If r is 3 or 7 mod 8, the image is a cyclic group of order equal to the denominator of B 2 n / 4 n {\displaystyle B_{2n}/4n} , where B 2 n {\displaystyle B_{2n}} is a Bernoulli number .
b 0 is the number of connected components; b 1 is the number of one-dimensional or "circular" holes; b 2 is the number of two-dimensional "voids" or "cavities". Thus, for example, a torus has one connected surface component so b 0 = 1, two "circular" holes (one equatorial and one meridional) so b 1 = 2, and a single cavity enclosed within the ...