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It's often a great technical convenience to know that you can "do homotopy theory" with simplicial sets. But if you really get down to it geometrically, it can be awkward.

In general, if I have a CW complex that I want to study simplicially, I have to build up a simplicial structure cell-by-cell, and I need to keep applying barycentric subdivision to simplicially approximate my attaching maps.

There are a few interesting spaces with economical simplicial structures. For example spheres, or $K(G,1)$'s (for which one can take the nerve of $G$ as a model), and suspensions thereof. I suppose Moore spaces aren't too bad either. One trick suggested by this is to take classifying spaces of other categories, but when it comes down to it there really aren't that many categories whose classifying spaces I know.

As far as I can tell, that might be about it! For example, even to write down a simplicial structure on $\mathbb C \mathbb P^\infty$ that I could actually work with, I'm not sure what I'd do. I could imagine that if there were an "economical" model of $S^1$ which was a simplicial topological group, then one could perform a bar construction to get $\mathbb C \mathbb P^\infty$, but I don't even really know a good model for $S^1$ as a simplicial group, short of geometrically realizing and taking the singular complex, which I think is very far from economical. It's not even clear to me whether $Ex^\infty S^1$ is a simplicial topological group!

Questions:

  1. Is there a general method to write down a simplicial model of a homotopy type which is more economical than just subdividing to simplicially aproximate each attaching map?

  2. Are there at least some examples of homotopy types which are not suspensions of $K(G,1)$'s or Moore spaces which admit "simple" simplicial models? By "simple", I roughly mean that one might really think about the space "geometrically", rather than just on some formal level.

  3. As a test case, what would be a good simplicial model for $\mathbb C \mathbb P^\infty$?

EDIT: I should clarify that when I talk about a "simplicial model" of a homotopy type, I'm asking for a simplical set whose geometric realization has that homotopy type. I'm not requiring a simplicial complex or anything.

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    $\begingroup$ Concerning the comment about CP^∞: why not just take Γ(Z[2]), where Γ:Ch→sAb is the Dold–Kan functor? $\endgroup$
    – Dmitri Pavlov
    Commented Aug 31, 2019 at 22:09
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    $\begingroup$ @DmitriPavlov Actually, that sounds like a great idea! I'll have to ponder what comes out, but it seems about as economical as one could like! $\endgroup$
    – Tim Campion
    Commented Aug 31, 2019 at 22:11
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    $\begingroup$ There are some reasons to expect that there might be an efficient and explicit algebraic/combinatorial simplicial model for the space $\Omega^\infty K(n)$, but I do not believe that anyone has ever produced one. This is potentially a rather interesting problem. $\endgroup$
    – Neil Strickland
    Commented Aug 31, 2019 at 23:43
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    $\begingroup$ @skd The Postnikov tower for $\Omega^\infty K(n)$ is very thin and quite regular. The corresponding minimal Kan complex is a twisted form of $\prod_kK(\mathbb{Z}/p,2(p^n-1)k)$, and the first nontrivial $k$-invariant is the Milnor Bockstein operation $Q_n$. It seems plausible that there should be an explicit simplicial map realising $Q_n$, which might be describable in terms of the McClure-Smith operad. One might hope that this would involve congruences of binomial coefficients and so make contact with formal group theory. $\endgroup$
    – Neil Strickland
    Commented Sep 1, 2019 at 9:03
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    $\begingroup$ If we interpret your question a little more broadly, it's well known that delta complexes give efficient (and very useful) descriptions for surfaces and "most" cusped hyperbolic 3-manifolds. There's some beautiful delta complex structures on $\mathbb CP^2$ and I imagine many of them fit into a natural family. Ben Burton and I found a delta complex structure on $\mathbb CP^2$ with just four $4$-dimensional simplices. $\endgroup$
    – Ryan Budney
    Commented Sep 1, 2019 at 15:46

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