nLab cellular approximation theorem

Contents

Context

Homotopy theory

Algebraic topology

Idea
Statement
Applications

Finite-dimensional universal bundles

Related concepts
References

General
In equivariant homotopy theory

Idea

The cellular approximation theorem states that every continuous map between CW complexes (with chosen CW presentations) is homotopic to a cellular map, hence a map that respects the cell complex-structure, mapping n-skeleta to $n$ -skeleta for all $n$ .

This is the analogue for CW-complexes of the simplicial approximation theorem: 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 $f \colon (X, A) \to (X', A')$ between relative CW-complexes that is cellular on a subcomplex $(Y, B)$ of $(X, A)$ , there is a cellular map $g \colon (X, A) \to (X', A')$ that is homotopic to $f$ relative to $Y$ .

It follows that if two cellular maps between CW-complexes are homotopic, then they are so by a cellular homotopy.

(Spanier 66, p. 404, review in May 99, Section 10.4, Hatcher 02, Thm. 4.8)

Applications

Finite-dimensional universal bundles

For $G$ a suitable topological group, consider the universal principal bundle $E G \to B G$ over the classifying space equipped with some CW-complex-structure (as typically comes with its construction as a sequential colimit of Grassmannians).

Then the cellular approximation theorem (Thm. ) implies at once that the pullback $E_d G$ of the universal principal bundle $E G$ from $B G$ to its $d+1$ -skeleton

is universal for $G$ -principal bundles over $d$ -dimensional cell complexes $X^d$ (in particular: over $d$ -dimensional smooth manifolds, via any of their smooth triangulations) – in that forming pullback of $E_d G$ identifies homotopy classes of maps $X^d \to sk_{d+1} B G$ with isomorphism classes of $G$ -principal bundles over $X^d$ .

Beware here the required skeletal degree: On the one hand, the cellular approximation theorem gives that every isomorphism class of a $G$ -principal bundle on $X^d$ is hit already by pullback from just the $d$ -skeleton $sk_d B G$ . But in order for the isomorphism relation of bundles to be reflected in the homotopy relation of their classifying maps one needs the $(d+1)$ -skeleton: Because the cylinder $X^d \times [0,1]$ on which left homotopy of maps is defined, is $d+1$ -skeletal of $X^d$ is $d$ -skeletal.

Such finite-dimensional $G$ -principal bundles, universal for base spaces of fixed bounded dimension, have the advantage that they carry an ordinary smooth manifold-structure (instead of just a generalized smooth space-structure) and as such an ordinary principal connection, which is a universal connection for bundles over fixed bounded-dimensional base spaces. In this way these finite-dimensional universal bundles serve as a foundation for Chern-Weil theory (Chern 51 – p. 45 and 67, Narasimhan-Ramanan 61, Narasimhan-Ramanan 63, Sclafly 80).

Related concepts

References

General

Edwin Spanier, ch. 7. sec. 6, cor. 18 (p. 404) of: Algebraic topology, Springer 1966 (doi:10.1007/978-1-4684-9322-1)
Peter May, sec. 10.4 in: A Concise Course in Algebraic Topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)
Allen Hatcher, Theorem 4.8, p. 349 in: Algebraic topology, Cambridge Univ. Press 2002 (web)

In equivariant homotopy theory

Cellular approximation for G-CW complexes in equivariant homotopy theory is due to:

Takao Matumoto, Theorem 4.4 in: On $G$ -CW complexes and a theorem of JHC Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA 18, 363-374, 1971 (PDF)

and, independently, due to:

Sören Illman, Prop. 2.4 of: Equivariant singular homology and cohomology for actions of compact lie groups (doi:10.1007/BFb0070055) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), Proceedings of the Second Conference on Compact Transformation Groups Lecture Notes in Mathematics, vol 298. Springer 1972 (doi:10.1007/BFb0070029)