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

• Cells are disjoint: for any distinct ${\displaystyle e,e'\in {\mathcal {E}}}$, ${\displaystyle e\cap e'=\varnothing }$.
• No set gets mapped to a negative number: ${\displaystyle {\text{deg}}^{-1}(\{j\in \mathbb {Z} \mid j\leq -1\})=\varnothing }$.
• Cells look like balls: For any ${\displaystyle n\in \mathbb {N} _{0}}$ and for any ${\displaystyle e\in \deg ^{-1}(n)}$ there exists a continuous map ${\displaystyle \phi :B^{n}\to X}$ that is an isomorphism ${\displaystyle {\text{int}}B^{n}\cong e}$ and also ${\displaystyle \phi (\partial B^{n})\subseteq \cup {\text{deg}}^{-1}(n-1)}$.

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

See also

• CW complex

References

• 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