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 $C\mathrm{}W$-complexes):

“There is a functor $\mathrm{\Gamma }:\text{𝒉𝑼}⟶\text{𝒉𝑼}$ where hU is the homotopy category for unbased spaces , and a natural transformation $\gamma :\mathrm{\Gamma }⟶Id$ that asssigns a $CW$-complex $\mathrm{\Gamma }X$ and a weak equivalence ${\gamma }_e:\mathrm{\Gamma }X⟶X$ to an arbitrary space $X$, such that the following diagram commutes:

$$\begin{array}{ccc}\hfill \mathrm{\Gamma }X\hfill & \hfill \stackrel{\mathrm{\Gamma }f}{\to }\hfill & \hfill \mathrm{\Gamma }Y\hfill \\ \hfill \gamma (X)\downarrow \hfill & & \hfill \downarrow \gamma (Y)\hfill & & \\ \hfill X>f\gg Y\hfill \end{array}$$

and $\mathrm{\Gamma }f:\mathrm{\Gamma }X\to \mathrm{\Gamma }Y$ 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 (http://planetmath.org/ApproximationTheoremForAnArbitrarySpace) is constructed as the colimit $\mathrm{\Gamma }X$ of a sequence of cellular inclusions of $CW$-complexes $X_1,\mathrm{\dots },X_n$ , so that one obtains $X\equiv colim[X_i]$. As a consequence of J.H.C. Whitehead’s Theorem, one also has that:

$\gamma *:[\mathrm{\Gamma }X,\mathrm{\Gamma }Y]⟶[\mathrm{\Gamma }X,Y]$ is an isomorphism.

Furthermore, the homotopy groups of the $CW$-complex $\mathrm{\Gamma }X$ are the colimits of the homotopy groups of $X_n$ and ${\gamma }_{n+1}:{\pi }_q(X_{n+1})⟼{\pi }_q(X)$ is a group epimorphism.

Title	approximation theorem for an arbitrary space
Canonical name	ApproximationTheoremForAnArbitrarySpace
Date of creation	2013-03-22 18:14:40
Last modified on	2013-03-22 18:14:40
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	43
Author	bci1 (20947)
Entry type	Theorem
Classification	msc 81T25
Classification	msc 81T05
Classification	msc 81T10
Classification	msc 55U15
Classification	msc 57Q05
Classification	msc 57Q55
Classification	msc 55U05
Classification	msc 55U10
Synonym	approximation theorems for topological spaces
Related topic	TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT
Related topic	CWComplex
Related topic	SpinNetworksAndSpinFoams
Related topic	HomotopyCategory
Related topic	WeakHomotopyEquivalence
Related topic	GroupHomomorphism
Related topic	ApproximationTheoremAppliedToWhitneyCrMNSpaces
Defines	unique colimit of a sequence of cellular inclusions of $CW$-complexes