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algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
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).
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)
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).
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
Cellular approximation for G-CW complexes in equivariant homotopy theory is due to:
Textbook account:
Review:
Last revised on April 6, 2021 at 15:50:10. See the history of this page for a list of all contributions to it.