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cohomology group theorem

(Theorem)

The following theorem involves Eilenberg-MacLane spaces in relation to cohomology groups for connected CW-complexes.

Theorem 0.1. Cohomology group theorem for connected CW-complexes ([1]):

Let K(π,n) be Eilenberg-MacLane spaces for connected CW complexes X, Abelian groups π and integers n≥0. Let us also consider the set of non-basepointed homotopy classes [X,K(π,n)] of non-basepointed maps η : X → K(π,n) and the cohomolgy groups Hn(X; π). Then, there exist the following natural isomorphisms:

--n [X, K (π,n )] ∼= H (X; π),
(0.1)

Proof. For a complete proof of this theorem the reader is referred to ref. [1] □

0.1 Related remarks:

1.
In order to determine all cohomology operations one needs only to compute the cohomology of all Eilenberg-MacLane spaces K(π,n); (source: ref [1]);
2.
When n = 1, and π is non-Abelian, one still has that [X,K(π, 1)]∼=Hom(π1(X),π)∕π, that is, the conjugacy class or representation of π1 into π;
3.
A derivation of this result based on the fundamental cohomology theorem is also attached.

References

[1]   May, J.P. 1999. A Concise Course in Algebraic Topology, The University of Chicago Press: Chicago.,p.173.


"cohomology group theorem" is owned by bci1.
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Other names:  fundamental cohomology theorem
Also defines:  conjugacy class or representation of into, set of based homotopy classes of based maps
Keywords:  Cohomology group theorem for connected CW-complexes, Abelian and non-Abelian groups, the cohomology group theorem, cohomology group, homotopy group, theorem on the equivalence of homology and homotopy groups, natural isomorphisms, fundamental cohomology theorem, reference to proof of theorem

Attachments:
derivation of cohomology group theorem (Derivation) by bci1

Cross-references: representation, non-Abelian, operations, natural isomorphisms, homotopy, cohomology groups, relation, theorem

This is version 5 of cohomology group theorem, born on 2009-01-26, modified 2009-01-27.
Object id is 436, canonical name is CohomologyGroupTheorem.
Accessed 4130 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )

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