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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 ...
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
For example, the two embedded circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus and 2-sphere represent 2-cycles. Cycles form a group under the operation of formal addition, which refers to adding cycles symbolically rather than combining them geometrically.
[2] Example: The circle S 1, the plane R 2 minus the origin, and the Möbius strip are all homotopy equivalent, although these topological spaces are not homeomorphic. The notation [X,Y ] is often used for the hom-set from a space X to a space Y in the naive homotopy category (but it is also used for the related categories discussed below).
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
Define f: X → Y by mapping 0 to 0 and n to 1/n for positive integers n. Then f is continuous, and in fact a weak homotopy equivalence, but it is not a homotopy equivalence . The homotopy category of topological spaces (obtained by inverting the weak homotopy equivalences) greatly simplifies the category of topological spaces.
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 .
Specifically, if X is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from a two-point space to X, : {,}. This map can be perturbed to be disjoint from the 0-skeleton of X if and only if f ( 0 ) {\displaystyle f(0)} and f ( 1 ) {\displaystyle f(1)} are not 0-valence vertices of X .