derivation of cohomology group theorem
Let
be a general CW-complex and consider the set
of basepoint preserving homotopy classes of maps from
to Eilenberg-MacLane spaces
for
, with
being an Abelian group.
Theorem 0.1 (
Fundamental, [or reduced] cohomology theorem, [1])
.
There exists a natural group isomorphism:
 |
(0.1) |
for all CW-complexes
, with
any Abelian group and all
. Such a group isomorphism
has the form
for a certain distinguished class in the cohomology group
, (called a “fundamental class”).
For connected CW-complexes,
, the set
of basepoint preserving homotopy classes maps from
to Eilenberg-MacLane spaces
is replaced by the set of non-basepointed homotopy classes
, for an Abelian group
and all
, because every map
can be homotoped to take basepoint to basepoint, and also every homotopy between basepoint -preserving maps can be homotoped to be basepoint-preserving when the image space
is simply-connected.
Therefore, the natural group isomorphism in Eq. (0.1) becomes:
![$\displaystyle \iota : [X, K(\pi,n)] \cong \overline{H}^n (X;\pi)$](img22.png) |
(0.2) |
When
the above group isomorphism results immediately from the condition that
is an Abelian group. QED
Remarks.
- A direct but very tedious proof of the (reduced) cohomology theorem can be obtained by constructing maps and homotopies cell-by-cell.
- An alternative, categorical derivation via duality and generalization of the proof of the cohomology group theorem ([2]) is possible by employing the categorical definitions of a limit, colimit/cocone, the definition of Eilenberg-MacLane spaces (as specified under related), and by verification of the axioms for reduced cohomology groups (pp. 142-143 in Ch.19 and p. 172 of ref. [2]).
This also raises the interesting question of the propositions that hold for non-Abelian groups G, and generalized cohomology theories.
-
- 1
-
Hatcher, A. 2001. Algebraic Topology., Cambridge University Press; Cambridge, UK., (Theorem 4.57, pp.393-405).
- 2
-
May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
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