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 $e\in {\text{deg}}^{-1}(n)$ there exists a continuous map $\phi :B^{n}\to X$ that is an isomorphism ${\text{int}}B^{n}\cong e$ and also $\phi (\partial B^{n})\subseteq \cup {\text{deg}}^{-1}(n-1)$ .