Cellular decomposition

In geometric topology, a cellular decomposition G of a manifold M is a decomposition of M as the disjoint union of cells (spaces homeomorphic to n-balls Bn).

The quotient space M/G has points that correspond to the cells of the decomposition. There is a natural map from M to M/G, which is given the quotient topology. A fundamental question is whether M is homeomorphic to M/G. Bing's dogbone space is an example with M (equal to R3) not homeomorphic to M/G.

## Definition[edit]

Cellular decomposition of  X {\displaystyle X} is an open cover  E {\displaystyle {\mathcal {E}}} with a function  deg : E -> Z {\displaystyle {\text{deg}}:{\mathcal {E}}\to \mathbb {Z} } for which:

* Cells are disjoint: for any distinct  e , e ′ ∈ E {\displaystyle e,e'\in {\mathcal {E}}} ,  e ∩ e ′ = ∅ {\displaystyle e\cap e'=\varnothing } .
* No set gets mapped to a negative number:  deg − 1 ( { j ∈ Z ∣ j ≤ − 1 } ) = ∅ {\displaystyle {\text{deg}}^{-1}(\\{j\in \mathbb {Z} \mid j\leq -1\\})=\varnothing } .
* Cells look like balls: For any  n ∈ N 0 {\displaystyle n\in \mathbb {N} _{0}} and for any  e ∈ deg − 1 ⁡ ( n ) {\displaystyle e\in \deg ^{-1}(n)} there exists a continuous map  ϕ : B n -> X {\displaystyle \phi :B^{n}\to X} that is an isomorphism  int B n ≅ e {\displaystyle {\text{int}}B^{n}\cong e} and also  ϕ ( ∂ B n ) ⊆ ∪ deg − 1 ( n − 1 ) {\displaystyle \phi (\partial B^{n})\subseteq \cup {\text{deg}}^{-1}(n-1)} .

A cell complex is a pair  ( X , E ) {\displaystyle (X,{\mathcal {E}})} where  X {\displaystyle X} is a topological space and  E {\displaystyle {\mathcal {E}}} is a cellular decomposition of  X {\displaystyle X} .

## See also[edit]

* CW complex

## References[edit]

* Daverman, Robert J. (2007), Decompositions of manifolds, AMS Chelsea Publishing, Providence, RI, p. 22, arXiv:0903.3055, ISBN 978-0-8218-4372-7, MR 2341468