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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 can be constructed from two points (x and y), and the 1-dimensional ball B (an interval), such that one endpoint of B is glued to x and the other is glued to y. The two points x and y are the 0-cells; the interior of B is the 1-cell. Alternatively, it can be constructed just from a single interval, with no 0-cells. A circle.
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 ...
In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by
In this table, the entries are either a) the trivial group 0, the infinite cyclic group Z, b) the finite cyclic groups of order n (written as Z n), or c) the direct products of such groups (written, for example, as Z 24 ×Z 3 or Z 2 2 = Z 2 ×Z 2). Extended tables of homotopy groups of spheres are given at the end of the article.
Two pairs (X 1, A) and (X 2, A) are said to be equivalent, if there is a simple homotopy equivalence between X 1 and X 2 relative to A. The set of such equivalence classes form a group where the addition is given by taking union of X 1 and X 2 with common subspace A. This group is natural isomorphic to the Whitehead group Wh(A) of the CW-complex A.
Theorem: The map : is a homomorphism. If is odd, is trivial (since () is torsion). If is even, the image of contains .Moreover, the image of the Whitehead product of identity maps equals 2, i. e. ([,]) =, where : is the identity map and [,] is the Whitehead product.
In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. [1] For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own advantages.