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| In [[algebraic topology]], in the '''cellular approximation theorem''', a [[Map (mathematics)|map]] between [[CW-complex]]es can always be taken to be of a specific type. Concretely, if ''X'' and ''Y'' are CW-complexes, and ''f'' : ''X'' → ''Y'' is a continuous map, then ''f'' is said to be ''cellular'', if ''f'' takes the [[n-skeleton|''n''-skeleton]] of ''X'' to the ''n''-skeleton of ''Y'' for all ''n'', i.e. if <math>f(X^n)\subseteq Y^n</math> for all ''n''. The content of the cellular approximation theorem is then that any continuous map ''f'' : ''X'' → ''Y'' between CW-complexes ''X'' and ''Y'' is [[Homotopy|homotopic]] to a cellular map, and if ''f'' is already cellular on a subcomplex ''A'' of ''X'', then we can furthermore choose the homotopy to be stationary on ''A''. From an algebraic topological viewpoint, any map between CW-complexes can thus be taken to be cellular.
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| == Idea of proof ==
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| The proof can be given by [[Mathematical induction|induction]] after ''n'', with the statement that ''f'' is cellular on the skeleton ''X''<sup>''n''</sup>. For the base case n=0, notice that every [[Path-connected#Path_connectedness|path-component]] of ''Y'' must contain a 0-cell. The [[Image (mathematics)|image]] under ''f'' of a 0-cell of ''X'' can thus be connected to a 0-cell of ''Y'' by a path, but this gives a homotopy from ''f'' to a map which is cellular on the 0-skeleton of X.
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| Assume inductively that ''f'' is cellular on the (''n'' − 1)-skeleton of ''X'', and let ''e''<sup>''n'' </sup> be an ''n''-cell of ''X''. The [[Closure (topology)|closure]] of ''e''<sup>''n''</sup> is [[compact set|compact]] in ''X'', being the image of the characteristic map of the cell, and hence the image of the closure of ''e''<sup>''n''</sup> under ''f'' is also compact in ''Y''. Then it is a general result of CW-complexes that any compact subspace of a CW-complex meets (that is, [[Intersection (set theory)|intersects]] [[non-trivial]]ly) only finitely many cells of the complex. Thus ''f''(''e''<sup>''n''</sup>) meets at most finitely many cells of ''Y'', so we can take <math>e^k\subseteq Y</math> to be a cell of highest dimension meeting ''f''(''e''<sup>''n''</sup>). If <math>k\leq n</math>, the map ''f'' is already cellular on ''e''<sup>''n''</sup>, since in this case only cells of the ''n''-skeleton of ''Y'' meets ''f''(''e''<sup>''n''</sup>), so we may assume that ''k'' > ''n''. It is then a technical, non-trivial result (see Hatcher) that the [[Function_(mathematics)#Restrictions_and_extensions|restriction]] of ''f'' to <math>X^{n-1}\cup e^n</math> can be [[Homotopy#Relative_homotopy|homotoped relative]] to ''X''<sup>''n-1''</sup> to a map missing a point ''p'' ∈ ''e''<sup>''k''</sup>. Since ''Y''<sup>''k''</sup> − {''p''} [[Deformation retraction|deformation retracts]] onto the subspace ''Y''<sup>''k''</sup>-''e''<sup>''k''</sup>, we can further homotope the restriction of ''f'' to <math>X^{n-1}\cup e^n</math> to a map, say, ''g'', with the property that ''g''(''e''<sup>''n''</sup>) misses the cell ''e''<sup>''k''</sup> of ''Y'', still relative to ''X''<sup>''n-1''</sup>. Since ''f''(''e''<sup>''n''</sup>) met only finitely many cells of ''Y'' to begin with, we can repeat this process finitely many times to make <math>f(e^n)</math> miss all cells of ''Y'' of dimension larger than ''n''.
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| We repeat this process for every ''n''-cell of ''X'', fixing cells of the subcomplex ''A'' on which ''f'' is already cellular, and we thus obtain a homotopy (relative to the (''n'' − 1)-skeleton of ''X'' and the ''n''-cells of ''A'') of the restriction of ''f'' to ''X''<sup>''n''</sup> to a map cellular on all cells of ''X'' of dimension at most ''n''. Using then the [[homotopy extension property]] to extend this to a homotopy on all of ''X'', and patching these homotopies together, will finish the proof. For details, consult Hatcher.
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| == Applications ==
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| === Some homotopy groups ===
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| The cellular approximation theorem can be used to immediately calculate some [[homotopy group]]s. In particular, if <math>n<k</math>, then <math>\pi_n(S^k)=0 \,</math>: Give <math>S^n \,</math> and <math>S^k \,</math> their [[canonical form|canonical]] CW-structure, with one 0-cell each, and with one ''n''-cell for <math>S^n \,</math> and one ''k''-cell for <math>S^k \,</math>. Any [[Base-point|base-point preserving]] map ''f'':<math>S^n \,</math>→<math>S^k \,</math> is by the cellular approximation theorem homotopic to a constant map, whence <math>\pi_n(S^k)=0 \,</math>.
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| === Cellular approximation for pairs ===
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| Let ''f'':''(X,A)''→''(Y,B)'' be a map of [[CW-pair]]s, that is, ''f'' is a map from ''X'' to ''Y'', and the image of <math>A\subseteq X \,</math> under ''f'' sits inside ''B''. Then ''f'' is homotopic to a cellular map ''(X,A)''→''(Y,B)''. To see this, restrict ''f'' to ''A'' and use cellular approximation to obtain a homotopy of ''f'' to a cellular map on ''A''. Use the homotopy extension property to extend this homotopy to all of ''X'', and apply cellular approximation again to obtain a map cellular on ''X'', but without violating the cellular property on ''A''.
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| As a consequence, we have that a CW-pair ''(X,A)'' is [[n-connected]], if all cells of <math>X-A</math> have dimension strictly greater than ''n'': If <math>i\leq n \,</math>, then any map <math>(D^i,\partial D^i) \,</math>→''(X,A)'' is homotopic to a cellular map of pairs, and since the ''n''-skeleton of ''X'' sits inside ''A'', any such map is homotopic to a map whose image is in ''A'', and hence it is 0 in the relative homotopy group <math>\pi_i(X,A) \,</math>.<br />
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| We have in particular that <math>(X,X^n)\,</math> is ''n''-connected, so it follows from the long exact sequence of homotopy groups for the pair <math>(X,X^n) \,</math> that we have isomorphisms <math>\pi_i(X^n) \,</math>→<math>\pi_i(X) \,</math> for all <math>i<n \,</math> and a surjection <math>\pi_n(X^n) \,</math>→<math>\pi_n(X) \,</math>.
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| == References ==
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| * {{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2005}}
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| [[Category:Theorems in algebraic topology]]
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