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 In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...

. A fundamental question is whether ''M'' is homeomorphic to ''M''/''G''. Bing's

dogbone space In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space \R^3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to \R^3. The name "dogbone spac ...

is an example with ''M'' (equal to R³) not homeomorphic to ''M''/''G''.

Definition

Cellular decomposition of

X

is an open cover

\mathcal

with a function

\text:\mathcal\to \mathbb

for which: * Cells are disjoint: for any distinct

e,e'\in\mathcal

e\cap e' = \varnothing

. * No set gets mapped to a negative number:

\text^(\) = \varnothing

. * Cells look like balls: For any

n\in\mathbb N_0

and for any

e\in \text^(n)

there exists a continuous map

\phi:B^n\to X

that is an isomorphism

\textB^n\cong e

and also

\phi(\partial B^n) \subseteq \cup \text^(n-1)

. A cell complex is a pair

(X,\mathcal E)

where

X

is a topological space and

\mathcal E

is a cellular decomposition of

X

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 Geometric topology

Definition

See also

References