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 is an open cover with a function for which:

• Cells are disjoint: for any distinct , .
• No set gets mapped to a negative number: .
• Cells look like balls: For any and for any there exists a continuous map that is an isomorphism and also .

A cell complex is a pair where is a topological space and is a cellular decomposition of .

See also

• CW complex

References