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 [math]\displaystyle{ X }[/math] is an open cover [math]\displaystyle{ \mathcal{E} }[/math] with a function [math]\displaystyle{ \text{deg}:\mathcal{E}\to \mathbb{Z} }[/math] for which:
- Cells are disjoint: for any distinct [math]\displaystyle{ e,e'\in\mathcal{E} }[/math], [math]\displaystyle{ e\cap e' = \varnothing }[/math].
- No set gets mapped to a negative number: [math]\displaystyle{ \text{deg}^{-1}(\{j\in\mathbb Z\mid j\leq -1\}) = \varnothing }[/math].
- Cells look like balls: For any [math]\displaystyle{ n\in\mathbb N_0 }[/math] and for any [math]\displaystyle{ e\in \text{deg}^{-1}(n) }[/math] there exists a continuous map [math]\displaystyle{ \phi:B^n\to X }[/math] that is an isomorphism [math]\displaystyle{ \text{int}B^n\cong e }[/math] and also [math]\displaystyle{ \phi(\partial B^n) \subseteq \cup \text{deg}^{-1}(n-1) }[/math].
A cell complex is a pair [math]\displaystyle{ (X,\mathcal E) }[/math] where [math]\displaystyle{ X }[/math] is a topological space and [math]\displaystyle{ \mathcal E }[/math] is a cellular decomposition of [math]\displaystyle{ X }[/math].
See also
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
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