# Structures between PL and smooth

Let $$X$$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $$ks(X)\in H^4(X,\mathbb{Z}_2)$$ is an obstruction to the existence of a PL structure on $$X$$. If it vanishes, the set of PL structures up to concordance is $$H^3(X,\mathbb{Z}_2)$$. This can be phrased as the homotopy equivalence $$TOP/PL=K(\mathbb{Z}_2,3)$$.

Now, let $$X$$ be a PL manifold. It is known that $$X$$ admits a smooth structure if its dimension is seven or less, and that this smooth structure is unique if the dimension is six or less. But $$PL/O\neq K(\pi,7)$$ for some group $$\pi$$---the homotopy groups are complicated, and depend on homotopy groups of higher dimensional spheres.

But perhaps there is some natural class $$C$$ of manifolds sitting between PL and smooth for which $$PL/C=K(\pi,7)$$? Very simple ideas like taking piecewise quadratic functions don't work because such functions aren't invertible within the given class. I'm not really looking for a "formal solution" about the abstract existence of such a class $$C$$, but rather for some concretely defined geometric structure.

• The "formal solution" that you're not looking for might define $BC \to BPL$ as a Moore-Postnikov factorization of $BO \to BPL$ and then define a $C$-structure to be a lift of a map classifying the stable tangent microbundle. More geometrically, perhaps a $C$-structure can be thought of as a smooth structure outside a subset of codimension 9. (I'm not sure how to make that precise though.) Jan 12, 2021 at 21:37
• Another idea, if a triangulation of $M$ is specified. Cheeger defined explicit real cocycles $L_i \in C^{4i}(M;\mathbb{R})$ in the simplicial cochain complex, representing the Hirzebruch classes. Now $p_2 = 9(L_1^2 + 5L_2)/7 \in C^8(M;\mathbb{R})$ represents the Pontryagin class. If we define a $C$-structure on $M$ to be a simplicial cochain $c \in C^7(M;\mathbb{R}/\mathbb{Z})$ with $\delta(c) = p_2$ mod $\mathbb{Z}$, then $PL/C \simeq K(\mathbb{R}/\mathbb{Z},7)$ ... Jan 16, 2021 at 1:39
• ... This is "concrete" in that a $C$-structure is an assignment of $c(\sigma)$ to every 7-simplex $\sigma$ in the triagulation. Integrality of Pontryagin classes of vector bundles gives preferred $C$-structures on smooth manifolds. Jan 16, 2021 at 1:41