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

finiteCW 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)