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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 
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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
 +
and    Y
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relative to    A
 +
and    B ,
 +
respectively. In the case when    A, B = \emptyset ,
 +
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)
 +
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