Branched manifold

In

Let K be a metrizable space, together with:

1. a collection {U_i} of
   closed subsets
   of K;
2. for each U_i, a finite collection {D_ij} of closed subsets of U_i;
3. for each i, a map π_i: U_i → D_iⁿ to a closed n-disk of class C^k in Rⁿ.

These data must satisfy the following requirements:

1. ∪_j D_ij = U_i and ∪_i Int U_i = K;
2. the restriction of π_i to D_ij is a homeomorphism onto its image π_i(D_ij) which is a closed class C^k n-disk relative to the boundary of D_iⁿ;
3. there is a cocycle of diffeomorphisms {α_lm} of class C^k (k ≥ 1) such that π_l = α_lm · π_m when defined. The domain of α_lm is π_m(U_l ∩ U_m).

Then the space K is a branched n-manifold of class C^k.

The standard machinery of

Examples

Extrinsically, branched n-manifolds are n-dimensional complexes embedded into some Euclidean space such that each point has a well-defined n-dimensional tangent space.

• A finite graph whose edges are smoothly embedded arcs in a surface, such that all edges incident to a given vertex v have the same tangent line at v, is a branched one-manifold, or train track (there are several variants of the notion of a train track — here no restriction is placed on the valencies of the vertices). As a specific example, consider the "figure eight" formed by two externally tangent circles in the plane.
• A two-complex in R³ consisting of several leaves that may tangentially come together in pairs along certain double curves, or come together in triples at isolated singular points where these double curves intersect transversally, is a branched two-manifold, or branched surface. For example, consider the space K obtained from 3 copies of the Euclidean plane, labelled T (top), M (middle) and B (bottom) by identifying the half-planes y ≤ 0 in T and M and the half-planes x ≤ 0 in M and B. One can imagine M being the flat coordinate plane z=0 in R³, T a leaf curling upward from M along the x-axis to the right (positive y-direction) and B another leaf curling downward from M along the y-axis in front (positive x-direction). The coordinate axes in the M plane are the double curves of K, which intersect transversally at a unique triple point (0,0).

References

• Robert F. Williams, Expanding attractors, Publ. Math. IHÉS, t. 43 (1974), p. 169–203