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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.
Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with.
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.
Morse homology can also be formulated for Morse–Bott functions; the differential in Morse–Bott homology is computed by a spectral sequence. Frederic Bourgeois sketched an approach in the course of his work on a Morse–Bott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.
Two Weeks Notice opened at number two domestically, behind The Lord of the Rings: The Two Towers, and spent its first five weeks in the Top 10 at the box office. [10] It grossed $93.3 million in the United States and Canada, and $105.7 million in other territories, for a worldwide total of $199 million, against a budget of $60 million. [11]
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.
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]