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*: Hm( 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.