Is there a PL, or topological, bordism hypothesis? - MathOverflowmost recent 30 from mathoverflow.net2019-10-20T18:07:58Zhttps://mathoverflow.net/feeds/question/306197https://creativecommons.org/licenses/by-sa/4.0/rdfhttps://mathoverflow.net/q/30619711Is there a PL, or topological, bordism hypothesis?Manuel Bärenzhttps://mathoverflow.net/users/137672018-07-17T11:27:23Z2018-07-17T13:45:15Z
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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?</p>
https://mathoverflow.net/questions/306197/-/306215#3062159Answer by Noah Snyder for Is there a PL, or topological, bordism hypothesis?Noah Snyderhttps://mathoverflow.net/users/222018-07-17T13:45:15Z2018-07-17T13:45:15Z<p>This is addressed in Remark 2.4.30 of <a href="http://www.math.harvard.edu/~lurie/papers/cobordism.pdf" rel="noreferrer">Jacob's paper</a>. The PL case has a very nice description but the topological case does not. In particular, there's no difference between <em>framed</em> 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.</p>
<p>It's not obvious to me why there couldn't be some description of topological bordisms by starting with the smooth bordism category and then adding some new generators (for example, a new generator corresponding to the fake 4-ball). But at any rate there's no good result known in this direction and it certainly seems like it would be very difficult. </p>