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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.
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 ...
A general paradigm in group theory is that a group G should be studied via its group representations.A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M.
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
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.
Whitehead group in mathematics may mean: A group W with Ext(W, Z)=0; see Whitehead problem; For a ring, the Whitehead group Wh(A) of a ring A, equal to () For a group, the Whitehead group Wh(G) of a group G, equal to K 1 (Z[G])/{±G}. Note that this is a quotient of the Whitehead group of the group ring.
The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their product space. In the simplest possible case the relationship is that of a tensor product , but for applications it is very often necessary to apply certain tools of homological algebra to express the answer.