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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.
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.
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
For example it is common to take A to be Z/2Z, so that coefficients are modulo 2. 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.
The basic theorem is that the resulting homology is an invariant of the manifold (that is, independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular Betti numbers agree and gives an immediate proof of the Morse inequalities.
The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality (when those apply), and the universal coefficient theorem of homology theory. There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of harmonic forms .
As noted already, π 1 (S 1) = Z, and π 2 (S 2) contains a copy of Z generated by the identity map, so the fact that there is a surjective homomorphism from π 1 (S 1) to π 2 (S 2) implies that π 2 (S 2) = Z. The rest of the homomorphisms in the sequence are isomorphisms, so π n (S n) = Z for all n. [20] The homology groups H i (S n), with ...