Cell complex: Difference between revisions
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* Any space which can be given a cell complex structure, is a [[homotopy-CW-space]]; in other words, it has the homotopy type of a [[CW-complex]]. Moreover, the number of cells of each dimension in the associated CW-complex is the same. | * Any space which can be given a cell complex structure, is a [[homotopy-CW-space]]; in other words, it has the homotopy type of a [[CW-complex]]. Moreover, the number of cells of each dimension in the associated CW-complex is the same. | ||
* A product of cell complexes is a cell complex. More generally, given any fiber bundle with cell complex structures on the base and the fiber, we can combine those to give a cell complex structure to the total space. This is ''not'' possible for [[CW-complex]] structures since the order of attaching cells gets changed drastically. | |||
Revision as of 19:35, 24 December 2007
Definition
A cell complex is a topological space along with a cell structure, which is given as follows. We start off with the empty set, and at each stage, we attach a -cell (for arbitrary ) using an attaching map from to the space constructed so far. The dimensions of the cells could be in any order, and the sequence of attachments could be transfinite.
A special case is a CW-complex where the cells are attached in increasing order of dimension.
Facts
- Any space which can be given a cell complex structure, is a homotopy-CW-space; in other words, it has the homotopy type of a CW-complex. Moreover, the number of cells of each dimension in the associated CW-complex is the same.
- A product of cell complexes is a cell complex. More generally, given any fiber bundle with cell complex structures on the base and the fiber, we can combine those to give a cell complex structure to the total space. This is not possible for CW-complex structures since the order of attaching cells gets changed drastically.