some $m\le
n$. All our bifolds will be equipped with Riemannian metrics, in the orbifold sense.

#### Webs and foams.

The underlying topological space of a bifold is a manifold $X$, and the set of points with non-trivial local stabilizer is a codimension-$2$ subcomplex of $X$. In the case of dimension $2$, this subcomplex is a set of points. In the case $n=3$, it is a trivalent graph, which we refer to as a web.

In the case $n=4$, the points with $H_{z}\ne 1$ form a $2$-complex which we call a foam. A foam can have tetrahedral points, where the local stabilizer is $H_{4}$. The set of points with $H_{z}\cong H_{3}$ is a union of arcs and circles: these are the seams, which together with the tetrahedral points comprise a $4$-valent graph. The remainder of the foam is a $2$-manifold whose components are the *faces*.

A pair $(Y,K)$ consisting of a smooth $3$-manifold and smoothly embedded web can be used to construct a corresponding bifold $\check Y$. The same is true for a pair $(X,\Phi)$ consisting of a $4$-manifold and an embedded foam: we may write the corresponding bifold as $\check X$. There is a cobordism category in which the 