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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211401.png" /> from one relative [[CW-complex|CW-complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211402.png" /> into another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211403.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211404.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211406.png" /> are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211407.png" />-skeletons of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c0211409.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114011.png" />, respectively. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114012.png" />, one obtains a cellular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114013.png" /> from the CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114014.png" /> into the CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114015.png" />.
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A homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114017.png" />, is called cellular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114019.png" />. The following cellular approximation theorem is the analogue of the simplicial approximation theorem (see [[Simplicial mapping|Simplicial mapping]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114020.png" /> be a mapping from one relative CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114021.png" /> into another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114022.png" /> the restriction of which to some subcomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114023.png" /> is cellular. Then there exists a cellular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114024.png" /> that is homotopic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114025.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021140/c02114026.png" />.
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For references see also [[CW-complex|CW-complex]].
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A mapping  $  f:  ( X, A) \rightarrow ( Y, B) $
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from one relative [[CW-complex|CW-complex]] $  ( X, A) $
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into another  $  ( Y, B) $
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such that  $  f (( X, A) ^ { p } ) \subset  ( Y, B) ^ { p } $,
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where  $  ( X, A) ^ { p } $
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and  $  ( Y, B) ^ { p } $
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are the  $  p $-
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skeletons of  $  X $
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and  $  Y $
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relative to  $  A $
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and  $  B $,
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respectively. In the case when  $  A, B = \emptyset $,
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one obtains a cellular mapping  $  f $
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from the CW-complex  $  X $
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into the CW-complex  $  Y $.
  
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A homotopy  $  F:  ( X, A) \rightarrow I \times ( Y, B) $,
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where  $  I = [ 0, 1] $,
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is called cellular if  $  F (( X, A) \times I) ^ { p } \subset  ( Y, B) ^ { p } $
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for all  $  p $.
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The following cellular approximation theorem is the analogue of the simplicial approximation theorem (see [[Simplicial mapping|Simplicial mapping]]): Let  $  f:  ( X, A) \rightarrow ( Y, B) $
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be a mapping from one relative CW-complex  $  ( X, A) $
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into another  $  ( Y, B) $
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the restriction of which to some subcomplex  $  ( L, N) \subset  ( X, A) $
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is cellular. Then there exists a cellular mapping  $  g:  ( X, A) \rightarrow ( Y, B) $
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that is homotopic to  $  f $
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relative to  $  L $.
  
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For references see also [[CW-complex|CW-complex]].
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Brown,  "Elements of modern topology" , McGraw-Hill  (1968)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.R.F. Maunder,  "Algebraic topology" , v. Nostrand  (1970)  pp. Section 7.5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Brown,  "Elements of modern topology" , McGraw-Hill  (1968)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.R.F. Maunder,  "Algebraic topology" , v. Nostrand  (1970)  pp. Section 7.5</TD></TR></table>

Revision as of 16:43, 4 June 2020


A mapping $ f: ( X, A) \rightarrow ( Y, B) $ from one relative CW-complex $ ( X, A) $ into another $ ( Y, B) $ such that $ f (( X, A) ^ { p } ) \subset ( Y, B) ^ { p } $, where $ ( X, A) ^ { p } $ and $ ( Y, B) ^ { p } $ are the $ p $- skeletons of $ X $ and $ Y $ relative to $ A $ and $ B $, respectively. In the case when $ A, B = \emptyset $, one obtains a cellular mapping $ f $ from the CW-complex $ X $ into the CW-complex $ Y $.

A homotopy $ F: ( X, A) \rightarrow I \times ( Y, B) $, where $ I = [ 0, 1] $, is called cellular if $ F (( X, A) \times I) ^ { p } \subset ( Y, B) ^ { p } $ for all $ p $. The following cellular approximation theorem is the analogue of the simplicial approximation theorem (see Simplicial mapping): Let $ f: ( X, A) \rightarrow ( Y, B) $ be a mapping from one relative CW-complex $ ( X, A) $ into another $ ( Y, B) $ the restriction of which to some subcomplex $ ( L, N) \subset ( X, A) $ is cellular. Then there exists a cellular mapping $ g: ( X, A) \rightarrow ( Y, B) $ that is homotopic to $ f $ relative to $ L $.

For references see also CW-complex.

Comments

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[a2] R. Brown, "Elements of modern topology" , McGraw-Hill (1968)
[a3] C.R.F. Maunder, "Algebraic topology" , v. Nostrand (1970) pp. Section 7.5
How to Cite This Entry:
Cellular mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cellular_mapping&oldid=46293
This article was adapted from an original article by D.O. Baladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article