This question again might be silly, like the last post(deleted). Let me know I will delete it.
Problem: Let $\Sigma$ be a surface without boundary and $f:\Sigma\to \Sigma$ be a proper homotopy equivalence. Suppose for all closed curves $\alpha,\beta:\Bbb S^1\to\Sigma$ we have $i\big([f\circ\alpha],[f\circ\beta]\big)=i\big([\alpha],[\beta]\big)$, i.e. $f$ preserves all geometric intersection numbers. Is it true that $f$ is properly homotopic to a homeomorphism?
Note that if $\Sigma$ is a closed surface, then any homotopy equivalence is homotopic to a homeomorphism. So the problem is clear without the extra assumption: "geometric intersection number is preserved."
Also, note that any homeomorphism of $\Sigma$ preserves the geometric intersection number.
I am not sure about the term "proper," i.e., the problem might be well-posed if one replaces proper homotopy equivalence with ordinary homotopy equivalence and proper homotopy with ordinary homotopy. I used the term proper keeping in mind the open surface.
Even in the closed surface case, is it possible to prove the above problem a priori not assuming "every homotopy equivalence is homotopic to a homeomorphism"?
Any help will be appreciated. Thanks in advance.