everyone:
I was reading this question : What do we mean when we say a differential form "descends to the quotient"?
which is related to mine. But the reply given did not answer my question about the conditions for the form to descend, and I have not found any other posts that answers it, it did not give the conditions under which the form descends to the quotient (other than the obvious one about the form being constant on each equivalence class). Specifically, we have a quotient map $q$ given as a mapping torus $(S,f)$ ; $S$ a compact surface with boundary, $f$ a self-homeomorphism of $S$ that equals the identity near the boundary , and $I$ is ( I am?) the unit interval $[0,1]$ i.e., the quotient given is : $$ q:S \times I/== $$ ,
where $(x,0) == (f(x),1)$ . Now, ** my question: ** Let $\gamma$ be a 1-form on $S \times I$ Under what condition does there exist a form $\omega$ on $S \times I/==$ , so that $\gamma=q^*\omega$ ? We know that the contravariant map in cohomology is not necessarily onto (for one, $q$ is not an isomorphism.). Any Ideas?
Thanks.
This has nothing to do with 1-forms or surfaces. You can take any smooth manifold $N$, a smooth map $f: N\to N$, a differential form $\omega$ on the product $N\times I$ and ask when does $\omega$ descend to the mapping torus $M=N\times I/f$ where $f: M\times 0\to M\times 1$ is a diffeomorphism. The necessary condition for $\omega$ to descend to $M$ is that $f^*\omega|M\times 1=\omega|M\times 0$. However, this condition is not quite sufficient since there might be problems with behavior of $\omega$ in the direction normal to the boundary. The trick is to take instead $M\times (-\epsilon, 1+\epsilon)$ for small $\epsilon>0$, let $\omega$ be defined there and let $f$ be a diffeomorphism between $N_0=M\times (-\epsilon, \epsilon)$ and $N_1=M\times (1-\epsilon, 1+\epsilon)$. Now, the condition is that $f^*\omega|N_1=\omega|N_0$. Without this, you will/may have to deal with various unpleasant issues of analytical nature. (E.g., you can project the prom but projection can fail to be sufficiently smooth etc.)