The n-Category Café

It would be fun to try to define an EHQFT, to absorb the homotopy in a better way. Has anyone done this, do you know?)

Just heard the talk by Dan Freed following up on Mike Hopkins’ talk yesterday.

He started by “recalling” the theorem Mike Hopkins gave yesterday but actually stated it in more generality, which seems to be just the E-HQFT picture for the case $X=BG$:

namely, they talk about the $\mathrm{\infty }n$-category of manifolds “with $G$ structure” $${\mathrm{Bord}}_n^G$$ which is (as far as I understood and he seemed to have confirmed that when I talked to him after the talk) just ${\mathrm{Bord}}_n$ but with everything in sight equipped with homotopy classes of maps to $BG$.

So in particular ${\mathrm{Bord}}_n^{\mathrm{SO}(n)}$ is oriented manifolds, that’s the case he explicitly mentioned.

The Baez-Dolan-hypothesis theorem is supposed to say in this case:

Thm. For $C$ some $\mathrm{\infty }n$-category the space of monoidal $\mathrm{\infty }n$-functors $${\mathrm{Bord}}_n^G\to C$$ is that of very dualizable objects in $C^{\mathrm{fd}}$ which are “homotopy fixed points for the $G$-action on $C^{\mathrm{fd}}$.”

I suppose that just means that there is a weak action of $G$ on these objects.

I’ll try to confirm this.

Urs wrote:

I wish John would chime in and help me a bit here.

I’m at Berkeley busy talking to Alan Weinstein and Jenny Harrison — but more importantly, the computer here seems not to have Java enabled, and is thus unable to post comments to the blog. Actually, let me test that hypothesis once more…

Hmm, so it actually works! Weird — I thought yesterday it didn’t.

Okay…

they talk about the $(\mathrm{\infty },n)$-category of manifolds “with G structure” ${\mathrm{Bord}}_n^G$ which is (as far as I understood and he seemed to have confirmed that when I talked to him after the talk) just ${\mathrm{Bord}}_n$ but with everything in sight equipped with homotopy classes of maps to $BG$.

That sounds roughly right… let me start by reminding you of the usual story about cobordism theories and Thom spaces, which is lurking behind all this stuff.

Homotopy theorists usually study cobordisms where all manifolds and cobordisms between them have their ‘stable normal bundle’ equipped with the structure of a $G$-bundle. To intuit the concept of ‘stable normal bundle’, first imagine your $n$-manifolds being embedded in ${ℝ}^{n+k}$, and cobordisms embedded in ${ℝ}^{n+k}\times [\mathrm{0,1}]$. Then your manifolds and cobordisms will have a ‘normal bundle’ in the usual sense: the orthogonal complement of the tangent bundle. This normal bundle will be a vector bundle with $k$-dimensional fibers.

Then, take the limit $k\to \mathrm{\infty }$. You get the notion of ‘stable normal bundle’, which is an infinite-dimensional vector bundle.

Note that before we take the limit $k\to \mathrm{\infty }$, we are really considering higher-dimensional tangles. In the limit $k\to \mathrm{\infty }$, things ‘stabilize’ and we are considering cobordisms. Homotopy theorists often focus on the stable situation, but the generalized tangle hypothesis discusses the case of finite $k$ as well.

Homotopy theorists like to equip the stable normal bundle of their manifolds or cobordisms between these with the structure of a $G$-bundle. Here $G$ is some group that’s equipped with a homomorphism to $\mathrm{GL}(\mathrm{\infty })$ —or, with no loss of generality, $O(\mathrm{\infty })$ (which is homotopy equivalent).

Let’s call a manifold a ‘$G$-manifold’ if its stable normal bundle is equipped with the structure of a $G$-bundle, and similarly for ‘$G$-cobordism’.

Let’s say two $n$-dimensional $G$-manifolds are ‘equivalent’ if there’s a $G$-cobordism between them. Equivalence classes of $n$-dimensional $G$-manifolds form an abelian group under disjoint union.

The big theorem in cobordism theory is that this group is the $n$th homotopy group of a certain space, the ‘Thom space’ $MG$. This is an infinite loop space — that is, a space of loops in a space of loops in a space…

But, this big theorem is just a pale shadow of a bigger idea: the generalized cobordism hypothesis. This says, first, that we can make a stable $\mathrm{\infty }$-groupoid

$$\mathrm{\infty }{\mathrm{Cob}}^G$$

whose objects are $0$-dimensional $G$-manifolds, whose morphisms are $G$-cobordisms between these, whose 2-morphisms are $G$-cobordisms between these, and so on.

Note: $\mathrm{\infty }{\mathrm{Cob}}^G$ should a stable $\mathrm{\infty }$-category with duals, but now I believe that’s the same thing as a stable $\mathrm{\infty }$-groupoid. And that should be the same thing as an infinite loop space.

And, the generalized cobordism hypothesis says this infinite loop space is just the Thom space $MG$!

The generalized cobordism hypothesis is itself a pale shadow of an even bigger idea: the generalized tangle hypothesis. This bigger idea is actually simpler in some ways, since it doesn’t involve taking the $k\to \mathrm{\infty }$ limit. It deals with tangles whose normal bundle is equipped with the structure of a $G$-bundle where $G$ is equipped with a homomorphism to $O(k)$.

I explained the generalized tangle hypothesis here. If anyone forgets what a ‘Thom space’ is, I also defined that there.

There’s a lot more to say, but the main danger in this business is saying so much that people think you must be talking about something complicated and scary. Pictures would help a lot…