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Difference between revisions of "Cellular mapping"

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For references see also [[CW-complex|CW-complex]].
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For references see also [[CW-complex]].
 
 
====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>
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<table>
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<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>
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</table>

Latest revision as of 18:02, 29 July 2024


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.

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=55921
This article was adapted from an original article by D.O. Baladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article