cohomology group theorem

The following theorem involves Eilenberg-MacLane spaces in relationPlanetmathPlanetmath to cohomology groupsPlanetmathPlanetmath for connected CW-complexesMathworldPlanetmath.

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 groupsMathworldPlanetmath π and integers n0. Let us also consider the set of non-basepointed homotopy classes [X,K(π,n)] of non-basepointed maps η:XK(π,n) and the cohomolgy groups ( H¯n(X;π). Then, there exist the following natural isomorphisms:

[X,K(π,n)]H¯n(X;π), (0.1)

0.1 Related remarks:

  1. 1.

    In order to determine all cohomology operations one needs only to compute the cohomologyPlanetmathPlanetmath of all Eilenberg-MacLane spaces K(π,n); (source: ref [1]);

  2. 2.

    When n=1, and π is non-AbelianMathworldPlanetmathPlanetmath, one still has that [X,K(π,1)]Hom(π1(X),π)/π, that is, the conjugacy classMathworldPlanetmathPlanetmath or representationPlanetmathPlanetmath of π1 into π;

  3. 3.

    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.
Title cohomology group theorem
Canonical name CohomologyGroupTheorem
Date of creation 2013-03-22 18:14:43
Last modified on 2013-03-22 18:14:43
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 48
Author bci1 (20947)
Entry type Theorem
Classification msc 55N33
Classification msc 55N20
Synonym fundamental cohomology theorem
Related topic GroupCohomology
Related topic EilenbergMacLaneSpace
Related topic HomotopyGroups
Related topic HomotopyCategory
Related topic GroupCohomologyTopologicalDefinition
Related topic TangentialCauchyRiemannComplexOfCinftySmoothForms
Related topic HomologyTopologicalSpace
Related topic ProofOfCohomologyGroupTheorem
Related topic OmegaSpectrum
Related topic ACRcomplex
Defines conjugacy class or representation of π1 into π
Defines set of based homotopy classes of based maps