It is elementary that differential forms can be pulled back via a smooth map between manifolds. However, I was reading a paper and came across a construction about push forward of a differential form via a submersion which I didn't fully understand.
The paper pointed to Differential Forms in Algebraic Topology by Bott and Tu for reference. However, since I have little background in algebraic topology, I would like to know if anyone can show me a more detailed explanation, or point me to some references.
Below is the construction as described in the paper:
If $f: X\rightarrow Y$ is a submersion from an oriented manifold of dimension $n$ to an oriented manifold of dimension $m \leq n$. Then the fibers are manifolds of dimension $r=n-m$.
So far this is OK, and it continues:
Integration over the fibers gives a map $f_*: D^p(X)\rightarrow D^{p-r}(Y)$ defined as follows.
Any $p$-form $\phi$ on $X$ with compact support can be written $\phi = \psi \wedge f^*\omega$, where $\psi$ is an $r$-form with compact support on $X$ and $\omega$ is a $(p-r)$-form on $Y$. To see this, use a partition of unity to write $\phi$ as a sum of forms with support in a coordinate neighborhood, and in local coordinates the decomposition becomes obvious.
We can then consider $f_*\psi$ on $Y$ with compact support defined by $f_*\psi (y)=\int_{f^{-1}(y)} \psi$ and define $f_*\phi = f_* \psi \wedge \omega$.
What I didn't understand is how the $p$-form $\phi$ on $X$ can be decomposed as $\psi \wedge f^*\omega$. (Even though it says it's obvious in local coordinates...) Is this decomposition unique? If not, then the push forward $f_* \phi$ better not depend on the decomposition..?