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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

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

Cells are disjoint: for any distinct e,e'\in\mathcal{E}, e\cap e' = \varnothing.
No set gets mapped to a negative number: \text{deg}^{-1}({j\in\mathbb Z\mid j\leq -1}) = \varnothing.
Cells look like balls: For any n\in\mathbb N_0 and for any e\in \deg^{-1}(n) there exists a continuous map \phi:B^n\to X that is an isomorphism \text{int}B^n\cong e and also \phi(\partial B^n) \subseteq \cup \text{deg}^{-1}(n-1).

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

References

Info: Wikipedia Source

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