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cohomology group theorem
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(Theorem)
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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 $n \geq 0$ . 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:
\begin{equation} [X, K(\pi,n)] \cong \overline{H}^n(X;\pi), \end{equation}
- 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]);
- 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 $\pi$ ;
- A derivation of this result based on the fundamental cohomology theorem is also attached.
- 1
- May, J.P. 1999. A Concise Course in Algebraic Topology, The University of Chicago Press: Chicago.,p.173.
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"cohomology group theorem" is owned by bci1.
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See Also: group cohomology, Eilenberg-MacLane space, homotopy groups, homotopy category, group cohomology (topological definition), tangential Cauchy-Riemann complex of ![$C^{\infty)$]( "$C^{\infty)$") -smooth forms, homology, derivation of cohomology group theorem for connected CW-complexes,  -spectrum, tangential Cauchy-Riemann complex of smooth forms
| Other names: |
fundamental cohomology theorem |
| Also defines: |
conjugacy class or representation of  into  , set of based homotopy classes of based maps |
| Keywords: |
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 |
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Cross-references: derivation, representation, conjugacy class, non-Abelian, source, operations, cohomology, order, natural isomorphisms, maps, classes, homotopy, integers, abelian groups, CW-complexes, connected, cohomology groups, relation, Eilenberg-MacLane spaces, theorem
There is 1 reference to this entry.
This is version 44 of cohomology group theorem, born on 2008-07-20, modified 2009-01-26.
Object id is 10838, canonical name is CohomologyGroupTheorem.
Accessed 1727 times total.
Classification:
| AMS MSC: | 55N20 (Algebraic topology :: Homology and cohomology theories :: Generalized homology and cohomology theories) | | | 55N33 (Algebraic topology :: Homology and cohomology theories :: Intersection homology and cohomology) |
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Pending Errata and Addenda
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