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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 $K(\pi,n)$ be Eilenberg-MacLane spaces for connected CW complexes $X$, Abelian groups $\pi$ and integers $n {\geqslant}0$. Let us also consider the set of non-basepointed homotopy classes $[X, K(\pi,n)]$ of non-basepointed maps $\eta :X \to K(\pi,n)$ and the cohomolgy groups $\overline{H}^n(X;\pi)$. Then, there exist the following natural isomorphisms:

$\displaystyle [X, K(\pi,n)] \cong \overline{H}^n(X;\pi),$ (0.1)
Proof. For a complete proof of this theorem the reader is referred to ref. [1] $\qedsymbol$

Related remarks:

  1. In order to determine all cohomology operations one needs only to compute the cohomology of all Eilenberg-MacLane spaces $K(\pi,n)$; (source: ref [1]);
  2. When $n = 1$, and $\pi$ is non-Abelian, one still has that $[X,K(\pi ,1)] \cong Hom(\pi_1(X),\pi)/\pi$, that is, the conjugacy class or representation of $\pi_1$ into $\pi$;
  3. A derivation of this result based on the fundamental cohomology theorem is also attached.

Bibliography

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, Abelian groups, 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 943 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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