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 $\text{deg}:ℰ\to \mathbb{Z}$ 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: ${\text{deg}}^{-1}(\{j\in \mathbb{Z}\mid j\le -1\})=\varnothing$.
• Cells look like balls: For any $n\in {\mathbb{N}}_0$ and for any $e\in {\mathrm{deg}}^{-1}(n)$ 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

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