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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 was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. [2] CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. [2]
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.
The Z-cohomology of RP 2a has an element y of degree 2 such that the whole cohomology is the direct sum of a copy of Z spanned by the element 1 in degree 0 together with copies of Z/2 spanned by the elements y i for i=1,...,a. The Z-cohomology of RP 2a+1 is the same together with an extra copy of Z in degree 2a+1. [10]
0, 1, 0, 0. This does work out, predicting the complement's reduced Betti numbers. The prototype here is the Jordan curve theorem, which topologically concerns the complement of a circle in the Riemann sphere. It also tells the same story. We have the honest Betti numbers 1, 1, 0. of the circle, and therefore 0, 1, 1. by flipping over and 1, 1, 0
The Whitehead conjecture is equivalent to the conjecture that every sub-presentation of an aspherical presentation is aspherical. In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture , or there must be a counterexample to the Whitehead conjecture; in other words ...
In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology. [1]