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^{} $\mathrm{\Gamma}:\text{\bm{h}\bm{U}}\u27f6\text{\bm{h}\bm{U}}$ where hU is the homotopy category for unbased spaces , and a natural transformation $\gamma :\mathrm{\Gamma}\u27f6Id$ that asssigns a $CW$-complex $\mathrm{\Gamma}X$ and a weak equivalence^{} ${\gamma}_{e}:\mathrm{\Gamma}X\u27f6X$ to an arbitrary space $X$, such that the following diagram commutes:
$$\begin{array}{ccc}\hfill \mathrm{\Gamma}X\hfill & \hfill \stackrel{\mathrm{\Gamma}f}{\to}\hfill & \hfill \mathrm{\Gamma}Y\hfill \\ \hfill \mathit{\text{}}\gamma (X)\downarrow \hfill & & \hfill \downarrow \gamma (Y)\hfill & & \\ \hfill X\text{@}f\gg Y\hfill \end{array}$$ 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:
$\gamma *:[\mathrm{\Gamma}X,\mathrm{\Gamma}Y]\u27f6[\mathrm{\Gamma}X,Y]$ 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 ${\gamma}_{n+1}:{\pi}_{q}({X}_{n+1})\u27fc{\pi}_{q}(X)$ is a group epimorphism.
References
- 1 May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago
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 |