geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originat ...

, a cellular decomposition ''G'' of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

''M'' is a decomposition of ''M'' as the disjoint union of cells (spaces homeomorphic to ''n''-balls ''Bⁿ''). The

quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...

''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 t ...

. 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 space ...

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