Direct proof that the wedge product preserves integral cohomology classes?

Question

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$.

There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients in $\mathbb R$, which is isomorphic to the De Rham cohomology. As a previous question already revealed, the images of this map are precisely the classes of differential $k$-forms $[\omega]$ that yield integers when integrated over a $k$-cycle $\sigma$,

$$ \int_{\sigma} \omega \in \mathbb{Z} \quad\text{ whenever } d\sigma = 0$$

Let us call them "integral forms".

Motivated by the cup product on cohomology, my question/request is the following:

Give a direct proof that the wedge product $[\omega\wedge\eta]\in H^{k+l}(M,\mathbb R)$ of two integral forms $\omega\in \Omega^k(M)$ and $\eta\in \Omega^l(M)$ is again an integral form.

This should be true because the cup product is mapped to the wedge product, but the point of the exercise is to prove this statement directly, without constructing the singular cohomology $H^k(M,\mathbb Z)$ or homology first.

Maybe I also have to make sure that the condition of being an integral form is something that can be "checked effectively" without singular homology; this might be subject to a new question.

Answer 1

I mostly use cohomology as a blackbox so I may be overlooking technical difficulties but here's how I'd try to prove your statement.

First I would note that being integral is a local property: the class $[\omega]$ of a closed form is integral iff its restrictions $[\omega|_{U_i}]$ to a covering $M = \bigcup U_i$ is integral. One way is obvious: if $\sigma \subset U_i$, then $\int_\sigma \omega|_{U_i} = \int_\sigma \omega$. For the other, we can chose a (finite) triangulation $\sigma = \bigcup \sigma_\alpha$ such that each $\sigma_\alpha$ is included in one of the $U_i$ so that we can write $\int_\sigma \omega = \sum_\alpha \int_{\sigma_{\alpha}} \omega|_{U_{i(\alpha)}}$.

The point is that you can check integrality by using the triangulation you like.

Then I would write the cup product as $H^i_{dR}(X) \otimes H^j_{dR}(X) \to H^{i+j}_{dR}(X\times X) \to H^{i+j}_{dR}(X)$ the external product followed by restriction to the diagonal. And I'd check that each map sends integral integral classes to integral classes using triangulations.

Answer 2

My cohomology is a bit rusty, so I'm sorry for any mistakes or for being sketchy. It may also be that my suggestion is too close to the singular cohomology solution you want to avoid, although I'll be using Cech cohomology instead.

Let's first make a covering $\mathcal{U}=(U_j)_{j\in I}$ of $M$ where all $U_j$, as well as $U_J=\bigcap_{j\in J} U_j$ for $J\subset I$, are contractible (or empty). The basic idea is to use the double complex $$ \begin{array}{ccccccccc} &&0&\rightarrow&C^0(\mathbb{R})&\rightarrow&C^1(\mathbb{R}) &\rightarrow&\cdots\\ &&\downarrow&&\downarrow&&\downarrow&&\downarrow\\ 0&\rightarrow&\Omega^0&\rightarrow&C^0(\Omega^0)&\rightarrow&C^1(\Omega^0) &\rightarrow&\cdots\\ &&\downarrow&&\downarrow&&\downarrow&&\downarrow\\ 0&\rightarrow&\Omega^1&\rightarrow&C^0(\Omega^1)&\rightarrow&C^1(\Omega^1) &\rightarrow&\cdots\\ &&\downarrow&&\downarrow&&\downarrow&&\downarrow\\ \end{array} $$ where $C^q(\Omega^p)=C^q(\mathcal{U},\Omega^p)$ consists of $(\omega_J)_{J\in I_p}$ where $I_p$ is the set of size $p+1$ subsets of $I$ and $\omega_J\in\Omega^p(U_J)$: i.e. the setup for defining the Cech cohomology. We then have maps $\delta:C^q(\Omega^p)\rightarrow C^{q+1}(\Omega^p)$, and $d:C^q(\Omega^p)\rightarrow C^q(\Omega^q)$.

For convenience, I'll pretend $C^q(\Omega^{-1})=C^q(\mathbb{R})$ and $C^{-1}(\Omega^p)=\Omega^p$ (i.e. global $p$-form which I should probably have denoted $\Gamma(\Omega^p)$ instead of just $\Omega^p$).

Note that $H^p(C^q(\Omega^\bullet))=0$ for $p\ge 0$ since closed forms are exact on all the contractible sets $U_J$.

The proof that $H^k_\text{dR}(M)=H^k(\Omega^\bullet)$ is equal to $H^k(M;\mathbb{R})=H^k(C^\bullet(\mathbb{R}))$ is then done by diagram chasing. Start with $\omega\in\Omega^k$ with $d\omega=0$. This maps to $(\omega^{(0)}_J)_{J\in I_0}\in C^0(\Omega^k)$ by restriction of $\omega$ to each $U_J$, $J\in I_0$. Since $d\omega^{(0)}_J=0$, we have $\omega^{(0)}_J=d\upsilon^{(0)}_J$ for some $\upsilon^{(0)}_J\in\Omega^{k-1}(U_J)$. This $(\upsilon^{(0)}_J)$ is then mapped to $\omega^{(1)}\in C^1(\omega^{k-1})$. And so we go, alternatingly moving up and right, until we get an element of $C^k(\mathbb{R})$.

Similar diagram chasing is used to show that this mapping is one-to-one module the images $d\Omega^{k-1}$ and $\delta C^{k-1}(\mathbb{R})$. (Assuming I haven't mixed things up here.)

A $k$-form $\omega$ is integral in $H^k_\text{dR}(M)\equiv H^k(M;\mathbb{R})$ if it can be mapped by the above procedure to an element in $C^k(\mathbb{Z})$.

So, here's the trick. Let $\omega$ and $\nu$ be closed $p$- and $q$-forms resp. corresponding to integral cohomology classes. We can then replicate the entire diagram chasing for $\omega$, but apply it to $\omega\wedge\nu$ where through each step we leave $\nu$ unchanged, finally mapping $\omega$ to an element $\omega^{(p)}\in C^p(\mathbb{Z})$. In this way, we chase $\omega\wedge\nu$ to an element in $C^p(\Omega^q)$ which looks like the restriction of $\nu$ to $U_J$ sets, multiplied by the integer values of $\omega^{(p)}_J$.

Afterwards, we continue to do the same with $\nu$: the $\omega^{(p)}=(\omega^{(p)}_J)_{J\in I_p}$ part now consist of just constants by which the forms are multiplied.

As I warned, my cohomoloogy is somewhat rusty, so I may quite likely have missed a number of technical issues. However, I think this basic approach should work. And it should be a good excercise in diagram chasing.