Conditions for Solution to Pullback Equation on Forms?

[WWGD]

Hi, All:

I have a quotient map given by the mapping torus (S,h) , where S is a compact surface with nonempty boundary, and h: S→S is a homeomorphism. Let I=[0,1].
The mapping torus $S_h$ of the pair (S,h) is defined as the quotient q: $$ q:S \times I/~$$ , where (x,0)~(h(x),1), i.e., we glue $S \times I$ along h.

Now, let ω be a 1-form on S . I'm trying to see under what conditions we can solve the pullback equation:
$$ q^* β =ω $$ ,

i.e., I want to find a form β on $S_f$ which pulls-back to ω . Now, the problem is that the induced map ## q^*: H^m( S \times I/~)→ S \times I##on cohomology is not necessarily onto -- q is not a homeomorphism, for one, tho the gluing map is almost as nice as can be, since it is a homeomorphism.

Can anyone think of conditions on ω under which there is a form β with ω= q^* β ?

Thanks.

 

[WWGD]

Never mind, thanks, I got it.