Cellular filtration: Difference between revisions

Revision as of 22:51, 24 October 2007

Definition

A filtration of a topological space (say $X^{n}$ the filtration for $X$ ) is said to be a cellular filtration if it satisfes the following conditions:

$H_{p}(X_{n},X_{n-1})=0$ for all $p\neq n$
For any singular simplex, we can find a $n$ such that the image of the simplex sits inside $X^{n}$

A topological space equipped with the additional structure of a cellular filtration is termed a cellular space.

Relation with other structures

Stronger structures

CW complex: The structure of a CW complex on a topological space automatically gives it a cellular filtration

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 A [[filtration of a topological space]] (say <math>X^n</math> the filtration for <math>X</math>) is said to be a '''cellular filtration''' if it satisfes the following conditions:
-* <math>H^p(X_n,X_{n-1}) = 0</math>for all <math>p \ne n</math>
+* <math>H_p(X_n,X_{n-1}) = 0</math>for all <math>p \ne n</math>
-* For any singular simplex, we can find a <math>n</math> such that the simplex sits inside <math>X^n</math>
+* For any singular simplex, we can find a <math>n</math> such that the image of the simplex sits inside <math>X^n</math>
-A topological space equipped with the additional structure of a cellular gilstration is termed a '''cellular space'''.
+A topological space equipped with the additional structure of a cellular filtration is termed a '''cellular space'''.
+==Relation with other structures==
+===Stronger structures===
+* [[CW complex]]: The structure of a CW complex on a topological space automatically gives it a cellular filtration