Branched surface

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In mathematics, a branched manifold is type of topological space. A small piece of an n-manifold looks topologically (i.e., up to homeomorphism) like $\mathbb {R} ^{n}$. A small piece of a branched n-manifold, on the other hand, might look like either of the following:

• $\mathbb {R} ^{n}$;
• the quotient space of two copies of $\mathbb {R} ^{n}$ modulo the identification of a closed half-space of each with a closed half-space of the other.

A neighborhood in a branched surface which cannot appear in a surface

A branched 1-manifold is called a train track; a two-dimensional branched manifold is called a branched surface.

A branched manifold can have a weight assigned to various of its subspaces; if this is done, the space is often called a weighted branched manifold. Weights are non-negative real numbers and are assigned to subspaces N that satisfy the following:

• N is open.
• N does not include any points whose only neighborhoods are the quotient space described above.
• N is maximal with respect to the above two conditions.

That is, N is the space from one branching to the next. Weights are assigned so that any if a neighborhood of a point is the quotient space described above, then the sum of the weights of the two unidentified hyperplanes of that neighorhood is the weight of the identified hyperplane space. (That must be reworded!)

File:Piece-of-train-tracks.png

A piece of a weighted train track

If a lamination or foliation of a manifold is looked at from a distance by a myopic person, the leaves might look like a branched manifold. Once this notion is made more precise (and it has been), if this occurs for a specific lamination or foliation, one says that the branched manifold carries the lamination or foliation.