Cell complex

A cell complex or CW complex is a mathematical object from the field of algebraic topology . It is a generalization of the Simplizial Complex and was introduced by John Henry Constantine Whitehead in 1949 .

definition

A cell is a topological space that is too homeomorphic . An open cell is a topological space that is homeomorphic to the interior . is called the dimension of the cell. ${\displaystyle k}$ ${\displaystyle B^{k}:=[0,1]^{k}}$ ${\displaystyle k}$ ${\displaystyle B^{k}}$ ${\displaystyle k}$

A cell complex or CW complex ( closure-finite weak-topology ) is a Hausdorff space that breaks down into open cells , where: ${\displaystyle X}$ ${\displaystyle (c_{i})_{i\in I}}$

1. for every cell there is a continuous mapping so that the interior is mapped from homeomorphic to and the edge into a union of finitely many cells of the dimension . ( is called the characteristic image of the cell .) ${\displaystyle k}$ ${\displaystyle c_{i}\subseteq X}$ ${\displaystyle f_{i}:B^{k}\rightarrow X}$ ${\displaystyle B^{k}}$ ${\displaystyle c_{i}}$ ${\displaystyle <k}$ ${\displaystyle f_{i}}$ ${\displaystyle c_{i}}$
2. ${\displaystyle M\subseteq X}$is complete when is complete for all . ${\displaystyle M\cap f_{i}(B^{k})}$ ${\displaystyle i\in I}$

The skeleton of a CW complex is the union of all its cells of dimensions . ${\displaystyle k}$ ${\displaystyle \leq k}$

A finite CW complex is a CW complex made up of a finite number of cells.

properties

Every CW complex is normal , but does not necessarily satisfy the first countability axiom , so it is not necessarily metrizable . Each CW complex is locally contractible .

In connected CW complexes, Whitehead's theorem about homotopy equivalence applies .

A CW complex is the Kolimes of its finite sub -complexes .

Examples

• Every simplicial complex is a CW complex.
• ${\displaystyle \mathbb {R} }$is a CW complex. Look at the cells and the characteristic pictures . ${\displaystyle c_{i}=(i,i+1)}$ ${\displaystyle f_{i}:[0,1]\to \mathbb {R} ,x\mapsto i+x}$

Cellular images

The skeleton of a CW complex is the union of all its cells of dimension . ${\displaystyle n}$ ${\displaystyle K_{n}}$ ${\displaystyle K}$ ${\displaystyle \leq n}$

A CW map (or cellular map) is a continuous map that maps each cell of into the skeleton of . ( Cells do not necessarily have to be mapped to cells.) ${\displaystyle f\colon K\to L}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle L}$ ${\displaystyle n}$ ${\displaystyle n}$

See also

• Abstract cell complex

literature

• Allen Hatcher: Algebraic Topology . Cambridge University Press 2010, ISBN 978-0-521-79540-1 , p. 5ff., P. 102ff., P. 106ff
• DO Baladze: CW-complex . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ^ JHC Whitehead: Combinatorial homotopy , Bull. Amer. Math. Soc., Volume 55, 1949, 213-245 (part 1), pp. 453-496 (part 2)