2
$\begingroup$

I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. I'll phrase the question in terms of the definition which I like the most. Feel free to answer with a different definition if simplifies things.

Let $X$ be a manifold. Let $\mathcal{OR}$ be the orientation sheaf on $X$ which gives for every open set $U \subset X$ the abelian group $\mathcal{OR}(U) = H_n(X,X-U ; \mathbb{Z})$.

Definition: The sheaf of compactly supported $n$-forms twisted by orientation is called the sheaf of densities and is denoted by $\mathcal{S = \Omega^{n}_{c} \otimes_{\mathbb{Z}}\mathcal{OR} }$ .

Question 1: Let $\omega \in \Gamma (U,\mathcal{S})$ be a local section. How do I get/compute the integral $\int_U \omega $ out of this information?

$\endgroup$
7
  • 1
    $\begingroup$ I'm fuzzy on this material this so I won't try to give a precise answer, but you should be able to think of $\mathcal{OR}(U)$ as equivalence classes of $n$-cycles that "agree on" $U$ and then simply define $\int (\mu \otimes [a]) = \int_a \mu$, and since $\mu$ is "supported on" $U$ the representative cycle $a$ you choose won't matter. $\endgroup$ – Anthony Carapetis Feb 4 '16 at 1:11
  • $\begingroup$ @AnthonyCarapetis This is precisely what I tried to do but I didn't manage to prove the invariance with respect to the representing class... $\endgroup$ – Saal Hardali Feb 4 '16 at 8:25
  • 1
    $\begingroup$ What's the issue? If $[a] = [b]$ then $a-b \in C_n(X-U)$, so $\int_a \mu - \int_b \mu = \int_{a-b} \mu$ vanishes so long as $\mu$ is compactly supported inside $U$. I'm not sure on the exact definition of a local section of $\Omega^n_c(U)$ but from an analytic standpoint this would surely be enough with a sufficiently fine cover and a partition of unity argument. $\endgroup$ – Anthony Carapetis Feb 4 '16 at 9:08
  • $\begingroup$ @AnthonyCarapetis Got it thanks! $\endgroup$ – Saal Hardali Feb 4 '16 at 9:12
  • 1
    $\begingroup$ The double cover of orientation of $M$ is canonically oriented (even if $M$ is not). A section of your sheaf yields an ordinary $n$ form on this double cover. Integrate it and divide by 2. $\endgroup$ – Thomas Feb 4 '16 at 11:07
2
$\begingroup$

The key to answering your questions in my opinion is to get a nice description of densities in local coordinates. (For this purpose, the definition of densities that you have chosen does not seem very convenient to me). The point is that, as for $n$-forms, one can describe the restriction of a density to the domain of a coordinate chart by a single smooth function. However, under a coordinate change, the function corresponding to a density transforms by multiplication with the absolute value of the Jacobian of the chart change (in constrast to $n$-forms, where just the Jacobian of the chart change enters). Once you have this description, you can define integration exactly as for $n$-forms in the oriented chase.

The simplest way to get to this description is to define the bundle of densities as an associated bundle to the linear frame bundle corresponding to the represenation defined by the absolute value of the determinant. Then the above description holds by definition. To get the description in your picture, you probably just have to check that the action of a diffeomorphism on the orientation shaef you define is given by multiplication by the signature of the Jacobian (which perfectly fits the intuitive meaning of this sheaf).

$\endgroup$
2
  • $\begingroup$ The problem I have with this definition is it's not so clear to me why the integral is invariant under a transformation. under a transformation the integral trasforms corresponding to the jacoiban of the transformation. And then I think about it and it really ought to change since the transformation did something to the measure. Why is your definition good? Beyonf the fact that it's local and it's the most common one. $\endgroup$ – Saal Hardali Feb 4 '16 at 8:23
  • $\begingroup$ The definition via the frame bundle is not local, it is a result that sections of the associated bundle admit a local description as indicated. Second, the reason why one has to deal with orientations classically is that the integral transforms by the abosolute value of the Jacobian rather than by the Jacobian. Hence the same proof that shows that the integral over $n$-forms is well defined when using an oriented atlas shows that the integral of densities is well defined in general. (It seems that I did not use the term "Jacobian" well, I've edited accordingly.) $\endgroup$ – Andreas Cap Feb 4 '16 at 19:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.