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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 ...
This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers b i of X and the Betti numbers b i,F with coefficients in a field F. These can differ, but only when the characteristic of F is a prime number p for which there is some p-torsion ...
The category of topological spaces Top has topological spaces as objects and as morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps.
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.
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.
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 .
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
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