@VítTuček Yes, I was thinking more along the lines of $M$ and $N$ being compact manifolds.

@VítTuček Would you know of any such book titles I could look up, or any keywords that could point me in this direction? Thank you.

@JoséFigueroa-O'Farrill I meant to indicate smooth throughout, and I have edited the question now to correct this, although if any of the listed questions have answers for continuous replacing smooth, that would also be of equal interest. Thank you very much.

Thank you very much, this answer is very interesting and helpful. I have also posted a related question mathoverflow.net/questions/186668/inverse-problem-for-pullba‌​ck.

Thank you very much. I have also posted the related question mathoverflow.net/questions/186668/inverse-problem-for-pullba‌​ck