nLab
cellular approximation theorem

Contents

Idea

The cellular approximation theorem states that every continuous map between CW complexes (with chosen CW presentations) is homotopic to a cellular map? (a map induced by a map of cell complexes).

This is a CW analogue of the simplicial approximation theorem (sometimes also called lemma): that every continuous map between the geometric realizations of simplicial complexes is homotopic to a map induced by a map of simplicial complexes (after subdivision).

Statement

Theorem

Given a continuous map f:(X,A)(X,A)f: (X, A) \to (X', A') between relative CW complexes that is cellular on a subcomplex (Y,B)(Y, B) of (X,A)(X, A), there is a cellular map g:(X,A)(X,A)g: (X, A) \to (X', A') that is homotopic to ff relative to YY.

  • wikipedia: cellular approximation

  • E. H. Spanier, Algebraic topology, Springer 1966, ch. 7. sec. 6

  • A. Hatcher, Algebraic topology, Cambridge Univ. Press 2002 , link

Revised on January 30, 2014 14:00:27 by Todd Trimble (67.80.194.40)