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Homederivation of cohomology group theorem for connected CW-complexes

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# derivation of cohomology group theorem for connected CW-complexes

# 0.1 Introduction

Let $X_{g}$ be a general CW-complex and consider the set

$\left\langle{X_{g},K(G,n)}\right\rangle$ |

of basepoint preserving homotopy classes of maps from $X_{g}$ to Eilenberg-MacLane spaces $K(G,n)$ for $n{\geqslant}0$, with $G$ being an Abelian group.

###### Theorem 0.1.

(Fundamental, [or reduced] Cohomology Theorem, [1]).

There exists a natural group isomorphism:

$\iota:\left\langle{X_{g},K(G,n)}\right\rangle\cong\overline{H}^{n}(X_{g};G)$ | (0.1) |

for all CW-complexes $X_{g}$ , with $G$ any Abelian group and all $n{\geqslant}0$. Such a group isomorphism
has the form $\iota([f])=f^{*}(\Phi)$ for a certain distinguished class in the cohomology group
$\Phi\in\overline{H}^{n}(X_{g};G)$, (called a *fundamental class*).

# 0.2 Derivation of the Cohomology Group Theorem for Connected CW-complexes.

For connected CW-complexes, $X$, the set $\left\langle{X_{g},K(G,n)}\right\rangle$ of basepoint preserving homotopy classes maps from $X_{g}$ to Eilenberg-MacLane spaces $K(G,n)$ is replaced by the set of non-basepointed homotopy classes $[X,K(\pi,n)]$, for an Abelian group $G=\pi$ and all $n{\geqslant}1$, because every map $X\to K(\pi,n)$ 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 $K(\pi,n)$ is simply-connected.

Therefore, the natural group isomorphism in Eq. (0.1) becomes:

$\iota:[X,K(\pi,n)]\cong\overline{H}^{n}(X;\pi)$ | (0.2) |

When $n=1$ the above group isomorphism results immediately from the condition that $\pi=G$ is an Abelian group. QED

# 0.3 Remarks

1. A direct but very tedious proof of the (reduced) cohomology theorem can be obtained by constructing maps and homotopies cell-by-cell.

2. 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.

# References

- 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

## Mathematics Subject Classification

18-00*no label found*55P20

*no label found*55N33

*no label found*55N20

*no label found*

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