TITLE: Classification of mapping tori
QUESTION [5 upvotes]: Assume $M$ is a topological space and $f\in \operatorname{Homeo}(M)$, then $f$ obviously plays a significant role in the torus bundle 
$$M_f = M\times I/\{(x,0)\sim (f(x),1)\mid x\in M\}.$$
Hence there should be some general result of the following type: $M_f$ and $M_g$ are bundle isomorphic (resp. diffeomorphic) if and only if "W", where W is a relation between $f$ and $g$.
Can someone help give W and explain? Thank you!

REPLY [5 votes]: Mapping tori come equipped with a projection to the circle: $p\colon M_f\to S^1$. Bundle isomorphism is isomorphism of the total space that commutes with the projection. A weaker notion of isomorphism for mapping tori is an isomorphism that commutes with the projection only up to homotopy. Pseudoisotopy is the name for the relation on isomorphisms of $M$ that replaces isotopy in this coarser notion of equivalence of mapping tori. 
If $M$ is a simply connected manifold of high dimension, then pseudoisotopy implies isotopy and preserving the homotopy class of the map to the circle isn't a big deal, so isomorphism of mapping tori is pretty much isotopy of mapping classes. But if $M$ is not simply connected, there are two ways that mapping tori may be isomorphic without being bundle isomorphic: there may be more pseudoisotopies than isotopies and there may be isomorphisms that do not preserve the homotopy class of the map to the circle.