nLab cellular approximation theorem



Homotopy theory

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Paths and cylinders

Homotopy groups

Basic facts


Algebraic topology



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 nn-skeleta for all nn.

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).



Given a continuous map f:(X,A)(X,A)f \colon (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 \colon (X, A) \to (X', A') that is homotopic to ff relative to YY.

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)


Finite-dimensional universal bundles

For GG a suitable topological group, consider the universal principal bundle EGBGE 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 dGE_d G of the universal principal bundle EGE G from BGB G to its d+1d+1-skeleton

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

Beware here the required skeletal degree: On the one hand, the cellular approximation theorem gives that every isomorphism class of a GG-principal bundle on X dX^d is hit already by pullback from just the dd-skeleton sk dBGsk_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)(d+1)-skeleton: Because the cylinder X d×[0,1]X^d \times [0,1] on which left homotopy of maps is defined, is d+1d+1-skeletal of X dX^d is dd-skeletal.

Such finite-dimensional GG-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).



See also

In equivariant homotopy theory

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

  • Takao Matumoto, Theorem 4.4 in: On GG-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

Textbook account:

Last revised on April 13, 2024 at 08:31:14. See the history of this page for a list of all contributions to it.