anak

So perhaps orientable 3-manifolds.

But seems useful to have a volume form on the 3-manifold.

As of right now, I was just thinking surfaces in arbitrary 3-manifolds.

Another candidate might be $\int_\Sigma\iota_X\mu$ if we instead have $\mu$ as a volume form on the ambient manifold (instead of just the surface).

Does defining flux on a general manifold need a Riemannian metric? Strangely can't find much about it, but I was thinking the flux of a vector field $X$ would be defined as $\int_\Sigma g(X,n)\,\mu$ where $\mu$ is a volume form on $\Sigma$, $n$ a unit normal vector field to $\Sigma$, and $g$ a Riemannian metric.

Oh how I love drawing Seifert surfaces.

The ducks still frequent it, I hope?

Any more adventures at the fowl bowl (duck pond)

LESLIE

It's okay, apparently dumb question. :D

@Thorgott well the "dual" depends on what type of dual. For example, that's the dual as C^\infty(M)-modules, but the continuous dual (of either 1-forms or vector fields!) is the space of 1-currents. They aren't the same thing.

My experience is mostly limited to topological and module duals which I considered in this case but did not work for my purpose.

The point is indeed not present, it's rather complicated. I am just mining for any notion of dual that is natural in this case.

(in particular, the $C^\infty(M)$-dual of $\mathfrak X(M)$ is $\Omega^1(M)$)

Soft question for more algebra-oriented fellows: given a smooth manifold $M$, you get the $C^\infty(M)$-modules $\mathfrak X(M)$ and $\Omega^k(M)$ of vector fields and $k$-forms. You can find the $C^\infty(M)$-dual to these, but I was wondering if there is a more descriptive notion of "dual" that can be ascribed to this special case. Like they are both sheaves, for example, but I don't think this corresponds to a dual notion for sheaf since it heavily relies on the $C^\infty(M)$ part.

If it's one thing I've learned while doing math, it's not to trust grown-ups.

@XanderHenderson I can't imagine how there would be that much overhead. It's mathjax (or katex, if you want something faster), which I'd imagine is more efficient than even link previews.

Though they do have an unrelated page on elliptic chain complexes, also interesting.

"Remark Notice that there is no notion of cochain in this general setup. What are called cochains are specifically components of certain specific models for H(X,A) . More on this in the section on abelian cohomology below."

Very intriguing.

Yeah, the nlab page led me eventually back to the cohomology page where they define a very very general type of cohomology on (\infty,1) categories.

@BenSteffan Interesting, I didn't know that there were things called a cohomology theory that couldn't be phrased (in some manner) as arising from a cochain complex!

@BenSteffan Okay, so the "elliptic" part comes up only in the construction of the associated cochain complex?

@BenSteffan Like just a topological space?

Elliptic cohomology, huh. What's the object being assigned the cohomology groups in this case?

same

How have you been, @BenSteffan? What have you been working on?

lol stackexchange notifies me of them making improvements to the chat yet we still have yet to get native mathjax and have to run it manually ourselves

Hello world.

Anyone handy with relative de Rham cohomology?

For some reason I was taking it for granted that $N\cong (-\epsilon,\epsilon)\times H$

Thanks!

Oh it's implicitly through taking the neighbourhood as trivialized!

But now I wonder what's wrong/missing with my previous suggestion via tubular neighbourhoods? Am I implicitly appealing to this normal vector field?

Seems doable.

I guess you can do that by making sure that I guess the normal vector field defines an "in" and "out direction, and so as long as you pick $\rho_\alpha$ so that they agree on sign on the in an out sides, then you get that?

It would have to guarantee that all my local defining functions $\rho_\alpha$ have the same sign at $d\rho_\alpha|_p$ for $p\in H$, I guess?

@Thorgott I have trouble showing that 0 is a regular value in this case. Maybe tthe normal vector field can be used for this, but I guess I will have to think about it longer.

With the tubular neighbourhood idea, I would want to say that since regular points are generic, we can perturb our function $C^\infty$-small (?) to get a function, but this isn't to clean. I am also wondering if the normal vector field condition is even necessary?

Feels kind of stupid, but suppose I have a non-vanishing normal vector field to a hypersurface $H\subset M$. I want to define a function $\rho\colon N\to\mathbb R$ for a neighbourhood $N\supset H$ such that $\rho^{-1}(0) = H$ and $d\rho_p\neq 0$ for each $p\in H$. This is easy with the implicit function theorem around some point in $H$, and it's easy to come up with functions without the $d\rho$ constraint using tubular neighbourhoods, but I can't get anything satisfying everything.

Chat him up if he ever pops in. Anyway, I gotta go. Nice chatting. :D

A guy who used to (?) frequent here. Prodigious in all things topological and geometric.

You sound like you'd be a pal to Balarka.

Stolz's paper is a great geometric read though, maybe you can read it at some point and identify the more general theory being used.

I will have to check that out. Sounds more explicitly homotopy theory than Stolz's paper.

Interesting... do we have any good examples of people who purposefully used the language of stable homotopy theory to solve a geometric problem like Stolz?

Okay, so here they are just saying because it was with the spin cobordism, it's stable homotopy theory.

I am familiar with Stolz's paper, but I don't remember it being relevant to stable homotopy theory...

Out of curiosity I was just browsing the nlab page on stable homotopy theory and they have a note "The reduction of geometric phenomena to solvable problems in stable homotopy theory has remained an important mathematical theme, the most recent major success being Stolz’s use of Spin cobordism to study the classication of manifolds with positive scalar curvature."