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 ''B<sup>n</sup>'').
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 '''R'''<sup>3</sup>) not homeomorphic to ''M''/''G''.
==Definition== Cellular decomposition of <math>X</math> is an open cover <math>\mathcal{E}</math> with a function <math>\text{deg}:\mathcal{E}\to \mathbb{Z}</math> for which: * Cells are disjoint: for any distinct <math>e,e'\in\mathcal{E}</math>, <math>e\cap e' = \varnothing</math>. * No set gets mapped to a negative number: <math>\text{deg}^{-1}(\{j\in\mathbb Z\mid j\leq -1\}) = \varnothing</math>. * Cells look like balls: For any <math>n\in\mathbb N_0</math> and for any <math>e\in \deg^{-1}(n)</math> there exists a continuous map <math>\phi:B^n\to X</math> that is an isomorphism <math>\text{int}B^n\cong e</math> and also <math>\phi(\partial B^n) \subseteq \cup \text{deg}^{-1}(n-1)</math>.
A cell complex is a pair <math>(X,\mathcal E)</math> where <math>X</math> is a topological space and <math>\mathcal E</math> is a cellular decomposition of <math>X</math>. ==See also== *CW complex
==References== *{{Citation|author1-link=Robert Daverman | last1=Daverman | first1=Robert J. | title=Decompositions of manifolds | url=https://www.ams.org/bookstore-getitem/item=chel-362.h | publisher=AMS Chelsea Publishing, Providence, RI | isbn=978-0-8218-4372-7 | mr=2341468 | year=2007 |page=22| arxiv=0903.3055 }}
Category:Geometric topology