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 ${ℰ}$ with a function $\text{deg}:{ℰ}\to \mathbb{\mathbb{Z}}$ for which:

• Cells are disjoint: for any distinct $e,e^{\prime }\in {ℰ}$, $e\cap e^{\prime }=\varnothing$.
• No set gets mapped to a negative number: ${\text{deg}}^{-1}(\{j\in \mathbb{\mathbb{Z}}\mid j\le -1\})=\varnothing$.
• Cells look like balls: For any $n\in {\mathbb{\mathbb{N}}}_0$ and for any $e\in {\text{deg}}^{-1}(n)$ there exists a continuous map $\varphi :B^n\to X$ that is an isomorphism $\text{int}{B}^n\cong e$ and also $\varphi (\partial B^n)\subseteq \cup {\text{deg}}^{-1}(n-1)$.

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

See also

• CW complex

References

• Daverman, Robert J. (2007), Decompositions of manifolds, AMS Chelsea Publishing, Providence, RI, p. 22, ISBN 978-0-8218-4372-7, https://www.ams.org/bookstore-getitem/item=chel-362.h