nLab cellular approximation theorem

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

• CW-approximation theorem

• Whitehead's theorem

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)

See also

• Wikipedia, Cellular approximation

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)

See also

• Sören Illman, p. 90 (5 of 6) in: Equivariant algebraic topology, Annales de l’Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 (doi:10.5802/aif.458)

Textbook account:

• Tammo tom Dieck, Theorem II.2.1 in: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)