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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.
Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing. Another important result is the approximation theorem. First, the homotopy category of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
Here might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups . The form of the result is that other coefficients A may be used, at the cost of using a Tor functor .
Cellular homology can also be used to calculate the homology of the genus g surface. The fundamental polygon of Σ g {\displaystyle \Sigma _{g}} is a 4 n {\displaystyle 4n} -gon which gives Σ g {\displaystyle \Sigma _{g}} a CW-structure with one 2-cell, 2 n {\displaystyle 2n} 1-cells, and one 0-cell.
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.
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences.The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology.
There are two basic invariants of a space X in the rational homotopy category: the rational cohomology ring (,) and the homotopy Lie algebra ().The rational cohomology is a graded-commutative algebra over , and the homotopy groups form a graded Lie algebra via the Whitehead product.