When do fibre products of smooth manifolds exist?

Question

Harold asks what conditions on $f:M\to L$ and $g:N\to L$, both smooth maps of smooth manifolds, ensures the existence of the fibre product $M \times_L N$.

Answer 1

It suffices for one of the two maps to be a submersion for the fibre product to exist. In fact, if $f$ and $g$ are maps from $X$ to $Y$, the fibre product is the inverse image of the diagonal in $Y \times Y$ under the map $f \times g : X\times X \to Y \times Y$. So a sufficient condition to have a nice fibre product is that $f \times g$ be transverse to the diagonal.

The next best thing to transverse intersection is clean intersection, as Ben has pointed out. Another definition of clean intersection of $A$ and $B$ in $X$ (more easily checked than the one about the local normal form) is that the intersection of $A$ and $B$ is a submanifold $C$, and that the tangent bundle to $C$ is the intersection of the tangent bundles to $A$ and $B$.

Answer 2

The most popular is that both be submersions (maps whose differentials are surjective at each point). More generally, you could be submersions onto cleanly intersecting submanifold.

Two submanifolds are said to intersect cleanly if their intersection is locally isomorphic to two intersecting vector subspaces at each point of intersection. This is a generalization of transverse.