# Approximation of classifying space BG of compact Lie group G by finite CW complexes

The classifying space of a topological group $$G$$ is usually constructed as follows: one constructs a sequence of spaces $$E_1G$$, $$E_2G$$, $$E_3G$$, … with a free $$G$$-action such that the (homotopy) colimit $$EG$$ of this sequence is contractible, and obtains $$BG$$ as the (homotopy) colimit of the sequence of spaces $$B_iG:=E_iG/G$$. I would like to work with a version of this construction that has the following property:

If $$G$$ is a compact Lie group, each of the approximating spaces $$B_iG$$ has the homotopy type of a finite CW complex.

I do care about the spaces $$B_iG$$ being of the form $$E_iG/G$$ as in the construction outlined – I am not just looking for an arbitrary approximation. For example, for $$BU(1)$$ we can take $$E_iG = S^{2i+1}$$ and then $$B_iG = \mathbb{C}P^i$$ is a finite CW complex. Similarly explicit descriptions can be given more generally for $$U(n)$$ and presumably also for other classical compact Lie groups.

I know of two general constructions: Milnor’s construction using joins of $$G$$, and a simplicial construction.

Milnor’s construction (see [Mil56]): I have no idea whether this construction has the desired property. Milnor argues that we can give $$BG$$ itself the structure of countable CW complex provided that $$G$$ is a “countable CW group” in the strong sense that multiplication and inversion on $$G$$ are cellular maps [Mil56, §5]. If compact Lie groups were “finite CW groups” in this strong sense, i.e. if they had a finite CW structure in which multiplication and inversion are cellular, then I could see a way of adapting Milnor’s arguments to verify that it has the above property. But according to [Rov18, Example 1.13], it’s not even clear whether the classical groups $$O(n)$$ and $$U(n)$$ satisfy this strong assumption – it’s not even clear whether they are “countable CW groups” in Milnor’s sense.

Simplicial construction (see e.g. [May99, §16.5]): It seems to me that using this construction in combination with Segal’s fat geometric realization (see [Seg72, Appendix A] or ncatlab), one obtains models of $$EG$$ and $$BG$$ which more generally satisfy the following property: If the inclusion of the neutral element in $$G$$ is a cofibration, and if $$G$$ has the homotopy type of a finite/countable/finite-dimensional CW complex, then each of the spaces $$E_iG$$ and $$B_iG = E_iG/G$$ has the homotopy type of a finite/countable/finite-dimensional CW complex. (The assumption on the neutral element is necessary for the simplicial classifying space to be “good” in Segal’s sense, so that the fat geometric realization is homotopy equivalent to the usual geometric realization.)

Could this be correct? I can provide more details, but for now I’d rather ask whether anyone knows either a reference that any known construction has the desired property, or a reason for this to be impossible.

[May99] May, A concise course in algebraic topology (1999)
[Mil56]$$~~$$ Milnor, Construction of Universal Bundles, II (1956)
[Rov18] Rovelli, Characteristic classes as complete obstructions (preprint)
[Seg72] Segal, Categories and Cohomology Theories (1972)

• If you're looking at Lie groups, then you might as well use ordinary (not fat) geometric realisation to start with. – David Roberts Feb 20 '19 at 10:18
• I agree with David, and it should be clear that the simplicial construction works, using the standard simplicial filtration. – Peter May Oct 31 '19 at 1:50
• The reason I was looking at Segal's fat realization was that I thought the usual geometric realisation of a simplicial topological space could take me outside of the category of CW complexes altogether. Are you saying that if I start with a simplicial space that is termwise an actual CW complex (and not just a space of this homotopy type), then it is evident that the successive stages of the geometric realization are still (homotopy equivalent to?) CW complexes? – Communicative Algebra Nov 1 '19 at 12:33

Yes, but non-functorially in the compact Lie group $$G$$.

In 1951 Norman Steenrod produced the very first model [4, Theorem 19.6] of a $$n$$-universal space [4, Definition 19.2]. See the equivalence [4, Theorem 19.4]. His $$E_n O_k = V_k(\mathbb{R}^{n+k})$$, a Stiefel manifold.

Strangely, this was not acknowledged by his Princeton colleague John Milnor in his 1956 article [5], which actually has no references! (Was this rush due to the Cold War, though the Space Race started with Sputnik in 1957?)

Namely, since any compact Lie group $$G$$ is known to embed into some orthogonal group $$O_k$$, Steenrod's $$B_n G := O_{n+k}/(O_n \times G)$$ is a closed real-analytic manifold by the classical slice theorem. Hence it's finitely triangulable by opaquely Cairns's thesis [1], first using Whitney's embedding [2, Theorem 1] of $$B_n G$$ smoothly into some euclidean space.

This triangulation process was elucidated by Whitehead's followup article [3, Theorem 7] and later by Whitney's book [6, Theorem 12A]. (Is this why people nowadays call it the "Whitney triangulation"?)

[1] Stewart Cairns, On the triangulation of regular loci, Annals Math 35(3):579–587, 1934

[2] Hassler Whitney, Differentiable manifolds, Annals Math 37(3):645–680, 1936

[3] John Whitehead, On $$C^1$$-complexes, Annals Math 41(4):809–824, 1940

[4] Norman Steenrod, The Topology of Fibre Bundles, 1951

[5] John Milnor, Construction of universal bundles II, Annals Math 63(3):430–436, 1956

[6] Hassler Whitney, Geometric Integration Theory, 1957