Cellular mapping

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