Cellular decomposition explained

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

l{E}

with a function

deg:l{E}\toZ

for which:

Cells are disjoint: for any distinct

e,e'\inl{E}

e\cape'=\varnothing

No set gets mapped to a negative number:

deg^-1(\{j\inZ\midj\leq-1\})=\varnothing

Cells look like balls: For any

n\inN₀

and for any

e\in\deg^-1(n)

there exists a continuous map

\phi:B^n\toX

that is an isomorphism

intB^n\conge

and also

\phi(\partialBⁿ⁾\subseteq\cupdeg^-1(n-1)

A cell complex is a pair

(X,lE)

where

is a topological space and

is a cellular decomposition of

Cellular decomposition explained

Definition

See also