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[parent] derivation of cohomology group theorem (Derivation)

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:

$\displaystyle \iota : \left\langle(X_g, K(G,n))\right\rangle \cong \overline{H}^n (X_g;G)$ (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”).

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:

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

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

Bibliography

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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Other names:  fundamental cohomology theorem
Also defines:  fundamental class, natural isomorphism, cohomology group, fundamental cohomology theorem
Keywords:  derivation of cohomology group theorem for connected CW-complexes

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Cross-references: cohomology theories, non-Abelian, propositions, cohomology groups, duality, QED, homotopy, isomorphism, group, theorem, Abelian group, homotopy classes of maps
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This is version 6 of derivation of cohomology group theorem, born on 2009-01-26, modified 2009-01-27.
Object id is 435, canonical name is DerivationOfCohomologyGroupTheorem.
Accessed 1334 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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