PlanetPhysics/Approximation Theorem for an Arbitrary Space

\begin{theorem}(Approximation \htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html} for an arbitrary topological space in terms of the colimit of a sequence of cellular inclusions of -complexes}):

"There is a functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \Gamma: "'hU''' \longrightarrow '''hU''' } where is the homotopy category for unbased spaces , and a natural transformation that asssigns a -complex and a weak equivalence X</math>, such that the following diagram commutes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \begin{CD} \Gamma X @> \Gamma f >> \Gamma Y \\ @V <math>~~~~~~~} \gamma (X) VV @VV \gamma (Y) V \\ X @ > f >> Y \end{CD} </math> and is unique up to homotopy equivalence.

(viz. p. 75 in ref. [1]). \end{theorem}

The -complex specified in the approximation theorem for an arbitrary space is constructed as the colimit of a sequence of cellular inclusions of -complexes , so that one obtains . As a consequence of J.H.C. Whitehead's Theorem, one also has that:

is an isomorphism.

Furthermore, the homotopy groups of the -complex are the colimits of the homotopy groups of and is a group epimorphism.

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</ref>[1]</references>

  1. 1.0 1.1 May, J.P. 1999, A Concise Course in Algebraic Topology. , The University of Chicago Press: Chicago