Difference between revisions of "Cellular mapping"
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+ | $#C+1 = 26 : ~/encyclopedia/old_files/data/C021/C.0201140 Cellular mapping | ||
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− | + | A mapping | |
+ | from one relative [[CW-complex|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|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|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 |
Cellular mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cellular_mapping&oldid=46293