# Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with duals upon a single object.

The proof sketch by Lurie uses Morse theory extensively, and has since been formalised and extended by several authors. Morse theory is a technique from real differential topology, and it is related to handle decompositions.

Piecewise linear (PL) manifolds are not equivalent to smooth manifolds from dimension 5 upwards. Topological manifolds famously already depart from PL and smooth manifolds in dimension 4. I don't know whether PL manifolds or topological have a well-developed analogue of Morse theory, but they have handle decompositions (except for 4d topological manifolds), so I would expect a lot of the technical procedures from the bordism hypothesis proof to work. I do not understand whether it is possible to define a $(\infty,n)$-category of bordisms in these cases, as opposed to a mere 1-category.

Then the awkward question is: How do the $(\infty,n)$-categories of PL and topological bordisms look like? They cannot possibly be equivalent to the smooth one, or otherwise we would have trivially proven that extended TQFTs do not detect PL or smooth structures? But then, which part of the construction goes awry? What's so special about the smooth categories that the bordism hypothesis works there?

• Just a trivial observation: There's a sense in which the Cobordism Hypothesis is a (far reaching) generalization of the Pontryagin Thom theorem about bordism groups. However, the latter relies very heavily on transverality (and is in fact false for topological manifolds). So a topological analog would have to somehow exclude this part of the story. – Saal Hardali Jul 17 '18 at 13:48
• @SaalHardali that is a good point, but there are generalizations of Pontrjagin-Thom theory to PL and topological manifolds: see here for some references. – Arun Debray Jul 17 '18 at 14:13

This is addressed in Remark 2.4.30 of Jacob's paper. The PL case has a very nice description but the topological case does not. In particular, there's no difference between framed bordisms in the PL and smooth case. So the framed part of the cobordism hypothesis goes through with no changes. This means that not only does $\mathrm{O}(n)$ act on the core of the bordism category, the larger group $\mathrm{PL}(n)$ does. So say unoriented PL TFTs valued in $\mathscr{S}$ are classified by $\mathrm{PL}(n)$-homotopy fixed points in the core $\mathscr{S}^\times$. In the topological case, Jacob argues that one should not expect a good similar description. Specifically, he says that the cobordism hypothesis is about handle decompositions of bordisms, but there are topological manifolds (such as Freedman's $E_8$ manifold) which do not have any handle decomposition at all.
• Handle decompositions do only not exist in 4 dimensions (although in some other low dimensions you have to prove hard theorems). If we look at nonsmoothable 4-manifolds like $E_8$ as some kind of pathological phenomenon, can't we ask about the topological bordism hypothesis in high dimensions? – Manuel Bärenz Jul 17 '18 at 13:53
• @ManuelBärenz Yes, exotic $S^7$s admit framings; see this MO post for a proof. – Arun Debray Jul 17 '18 at 14:34
• @ManuelBärenz A PL-framing of S^7 includes (up to contractible choices) the data of a smooth structure and a smooth framing of the resulting (exotic) smooth S^7. You can think of it this way: a PL-framing is a lift of the tangent microbundle map from BPL(7) to a contractible space (e.g. EPL(7)). A smooth structure is a lift to from BPL(7) to BO(7), and a smooth framing is a further lift to a contractible space. So they are the same! PL-framed $S^7$s corresponding to distinct framed exotic $S^7$s are already distinct in the PL-framed bordism category (since the cats are equivalent). – Chris Schommer-Pries Aug 15 '18 at 15:51