Is integration on manifolds unique?

Question

I asked this question on math.stackexchange.com, but haven't received an answer, so I thought I'd brave the waters here.

Suppose $M$ is a smooth oriented compact connected $m$-dimensional manifold and let $A^m(M)$ denote the set of smooth exterior differential $m$-forms on $M$. We can integrate members of $A^m(M)$, and the mapping $\phi\mapsto\int_M\phi$ has several properties:

• It's not identically $0$.
• It's linear.
• It's symmetric.

By "it's symmetric", I mean that if $f:M\to M$ is an orientation-preserving diffeomorphism, then $\int_M \phi = \int_M f^*\circ \phi$. I think that expresses the idea of symmetry I have in mind. Basically, all points on $M$ are equivalent as far as integration is concerned.

My question is: Do these properties uniquely determine the space $A^m(M)$ and the usual definition of integration? More precisely, is there some smooth compact connected $m$-manifold $M$, a space $S$ of smooth sections of the tensor bundle of $M$, a large family $F$ of diffeomorphisms of $M$, and a non-trivial linear functional $\lambda:S\to\mathbb{R}$ such that $\lambda( \phi) = \lambda( f^*\circ \phi)$ for all $\phi\in S$ and $f\in F$, but is not just integration of exterior $m$-forms, up to constant scale factor?

My motivation for asking this question: I suspect almost everyone who's ever learned about integration of differential forms on manifolds has wondered why those particular definitions were chosen. The answer is probably some variant of, "Because those definitions work." But must it be this way? Are they the only definitions that make integration work?

Answer 1

Here's a somewhat abstract but pretty satisfying way to see that integration of compactly supported densities on a not-necessarily orientable and not-necessarily compact $n$-manifold $M$ is unique.

This point of view is explained in Remark 3.3.10 in Kashiwara & Schapira's Sheaves on Manifolds.

Let $t: M \to 1$ be the terminal map to a point, and recall that the category of sheaves of $\mathbb{R}$-vector spaces on a point is just the familiar category of vector spaces.

Recall also that the pushforward with proper supports along the terminal map $t$ is the compactly supported global sections functor; i.e. $$t_!=\Gamma_c(M; -).$$

The counit $\epsilon: Rt_! t^! \Rightarrow 1$ of the Poincare-Verdier duality adjunction $Rt_! \dashv t^!$ is a generalization of integration.

Since adjoints, like other universal objects, are unique up to unique isomorphism, it follows that integration is unique in this sense.

To see how the counit here generalizes integration, apply it to the real numbers (or complex numbers) and take the $0$th cohomology to recover integration. In a bit more detail, notice that

$$H^0 Rt_! t^! \mathbb{R}=H^0 Rt_! \mathrm{or}_M[n] =H^0 R \Gamma_c(M; \mathrm{or}_M[n]) = H^n_c(M; \mathrm{or}_M).$$

So the counit mentioned above induces the integration map in compactly supported sheaf cohomology: $$\int_M: H^n_c(M; \mathrm{or}_M) \to \mathbb{R}.$$

The densities (or $n$-forms if you pick an orientation for an orientable manifold) appear when you take the usual de Rham resolution of the orientation sheaf $\mathrm{or}_M$, thus using de Rham cohomology to compute sheaf cohomology.

For $p: E \to M$ a topological submersion, we can use Poincare-Verdier duality, the relative orientation sheaf $\mathrm{or}_{E/M}$, and a somewhat similar approach to define fibre integration of "relative" densities.