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Philip Engel
Philip Engel
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Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ of 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.

Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ of 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.

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.

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YCor
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Philip Engel
Philip Engel
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Structures between PL and smooth

Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ of 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.