# When do fibre products of smooth manifolds exist?

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 $M \times_L N$.

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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 $B$.