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approximation theorem for an arbitrary space (Theorem)
Theorem 0.1 (Approximation theorem for an arbitrary topological space in terms of the colimit of a sequence of cellular inclusions of $CW$-complexes)   :
“There is a functor $\Gamma: \textbf{hU} \longrightarrow \textbf{hU}$ where $\textbf{hU}$ is the homotopy category for unbased spaces , and a natural transformation $\gamma: \Gamma \longrightarrow Id$ that asssigns a $CW$-complex $\Gamma X$ and a weak equivalence $\gamma _e:\Gamma X \longrightarrow X$ to an arbitrary space $X$, such that the following diagram commutes:

$\displaystyle \begin{CD} \Gamma X @> \Gamma f >> \Gamma Y \\ @V $~~~~~~~$\gamma (X) VV @VV \gamma (Y) V \\ X @ > f >> Y \end{CD}$
and $\Gamma f: \Gamma X\rightarrow \Gamma Y$ is unique up to homotopy equivalence.”
(viz. p. 75 in ref. [1]).
Remark 0.1   The $CW$-complex specified in the approximation theorem for an arbitrary space is constructed as the colimit $\Gamma X$ of a sequence of cellular inclusions of $CW$-complexes $X_1, ..., X_n$ , so that one obtains $X \equiv colim [X_i]$. As a consequence of J.H.C. Whitehead's Theorem, one also has that:

$\gamma* : [\Gamma X,\Gamma Y] \longrightarrow[\Gamma X, Y]$ is an isomorphism.

Furthermore, the homotopy groups of the $CW$-complex $\Gamma X$ are the colimits of the homotopy groups of $X_n$ and $\gamma_{n+1} : \pi_q(X_{n+1})\longmapsto\pi_q (X)$ is a group epimorphism.

Bibliography

1
May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago



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Keywords:  theorem for an arbitrary space

Cross-references: epimorphism, group, homotopy groups, isomorphism, topological, theorem
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This is version 1 of approximation theorem for an arbitrary space, born on 2009-02-04.
Object id is 479, canonical name is ApproximationTheoremForAnArbitrarySpace.
Accessed 351 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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