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Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the homotopy coherent nerve of the sing of the corresponding topological categories).

Let $\operatorname{BTop}(n)$ and $ \mathrm{B}O(n)$ be the classifying spaces of topological respectively orthogonal $\mathbb{R}^n$-bundles. Similarly let $\operatorname{BPL}(n)$ denote the classifying space of $\mathrm{PL}$-bundles of rank $n$ (it's not quite the classifying space of a topological group but is actually the simplicial set classifying $\mathrm{PL}$ bundles over polyhedra). Let $\mathcal{S}_{/X}$ denote the slice category of the $\infty$-category of spaces over a fixed space $X$. There's a canonical commutative diagram of $\infty$-categories (for the middle row in this diagram you have to work a bit but I'm pretty sure this is true): $\require{AMScd}$

\begin{CD} \mathrm{Diff}_n @>>> \mathrm{PL}_n @>>> \mathrm{Top}_n\\ @VVV @VVV @VVV \\ \mathcal{S}_{/\mathrm{B}O(n)} @>>> S_{ /\mathrm{B}\mathrm{PL}(n)} @>>> \mathcal{S}_{/\mathrm{BTop}(n)} \end{CD}

Question: (for $n \ne 4$) Are all the squares in this diagram pullback squares? If so where can I find this or at least the relevant pieces of the argument?

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    $\begingroup$ I think you need to replace BO, BPL and BTop by their unstable versions. Then it sounds like a reformulation of smoothing theory as in Kirby-Siebenmann (which would even work for dimension 4 when going from PL to Diff). $\endgroup$ – skupers Oct 5 at 16:09
  • $\begingroup$ @skupers Having the stable versions is part of the statement which I think shiuld be true due to Product Structure theorems. Could you perhaps point at the exact statements from Kirby and Siebenmann you have in mind? $\endgroup$ – Saal Hardali Oct 5 at 16:17
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    $\begingroup$ The product structure theorems give $n$-equivalences, but here you ask for full homotopy equivalences. It is true that $Top(n)/PL(n)=Top/PL=K(\mathbb Z/2,3)$, though, so that square is OK either way. Unstable PL bundles realize all Pontrjagin classes, while smooth ones don't. So there is a PL $S^3$ bundle on $S^{4n}$ with nontrivial $p_n$ that is not smoothable, but which your stable statement would imply smoothable. . . Also, $Top(n)$ probably isn't a topological group, either, and you should do something simplicial there, too. $\endgroup$ – Ben Wieland Oct 5 at 16:45
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    $\begingroup$ By Kisters theorem, BTop(n) defined as classifying n-dimensional topological microbundles is weakly equivalent to BHomeo(R^n). Anyway, it's Essay V.1 you want to look at. Another reference is Lashof's Embedding Spaxes $\endgroup$ – skupers Oct 5 at 17:52
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    $\begingroup$ Remark 3.29 of the paper arxiv.org/abs/1206.5522 claims that the answer is yes, and that this follows from Kirby-Siebenmann, but there is no precise reference. I was not able to find it in KS, but if you manage I would be quite interested to see it myself. $\endgroup$ – Yonatan Harpaz Oct 16 at 19:33

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