Contents

# Contents

## 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 (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 $f \colon (X, A) \to (X', A')$ 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.

## 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).

## References

### 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)