Cellular decomposition
Encyclopedia
In geometric topology
, a cellular decomposition G of a manifold
M is a decomposition of M as the disjoint union of cells.
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.
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
, a cellular decomposition G of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
M is a decomposition of M as the disjoint union of cells.
The quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
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
Dogbone space
In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space R3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to R3. The name "dogbone space" refers to a fanciful resemblance between some of the...
is an example with M (equal to R3) not homeomorphic to M/G.