approximation theorem for an arbitrary space

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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