Since $M$ is a closed manifold and $gf$ is homotopic to the identity, it must be surjective, otherwise it wouldn't preserve the fundamental class mod 2. In particular $g$ is surjective. Your isotopy condition implies that $fg$ is a homeomorphism onto the image, which coincides with the image of $f$ since $g$ is onto. The image of $f$ is compact since $M$ is compact, so $N$ should be compact too.

The isotopy condition also implies that $gf$ is injective, so $f$ is injective too. Since $M$ is compact and $N$ is Hausdorff, $f$ is closed and hence a homeomorphism onto the image. we have seen that the image of $f$ coincides with the image of $fg$. Therefore we conclude that $N$ and $M$ are homeomorphic. Your isotopy equivalence notion is the same as the relation of being homeomorphic.

Since $M$ is a closed manifold and $gf$ is homotopic to the identity, it must be surjective, otherwise it wouldn't preserve the fundamental class mod 2. In particular $g$ is surjective. Your isotopy condition implies that $fg$ is a homeomorphism onto the image, which coincides with the image of $f$ since $g$ is onto. The image of $f$ is compact since $M$ is compact, so $N$ should be compact too.

The isotopy condition also implies that $gf$ is injective, so $f$ is injective too. Since $M$ is compact and $N$ is Hausdorff, $f$ is closed and hence a homeomorphism onto the image. we have seen that the image of $f$ coincides with the image of $fg$. Therefore we conclude that $N$ and $M$ are homeomorphic.

Since $M$ is a closed manifold and $gf$ is homotopic to the identity, it must be surjective, otherwise it wouldn't preserve the fundamental class mod 2. In particular $g$ is surjective. Your isotopy condition implies that $fg$ is a homeomorphism onto the image, which coincides with the image of $f$ since $g$ is onto. The image of $f$ is compact since $M$ is compact, so $N$ should be compact too.

Since $M$ is a closed manifold and $gf$ is homotopic to the identity, it must be surjective, otherwise it wouldn't preserve the fundamental class mod 2. In particular $g$ is surjective. Your isotopy condition implies that $fg$ is a homeomorphism onto the image, which coincides with the image of $f$ since $g$ is onto. The image of $f$ is compact since $M$ is compact, so $N$ should be compact too.

The isotopy condition also implies that $gf$ is injective, so $f$ is injective too. Since $M$ is compact and $N$ is Hausdorff, $f$ is closed and hence a homeomorphism onto the image. we have seen that the image of $f$ coincides with the image of $fg$. Therefore we conclude that $N$ and $M$ are homeomorphic. Your isotopy equivalence notion is the same as the relation of being homeomorphic.