Branched manifold

In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched n-manifold is covered by n-dimensional "coordinate charts", each of which involves one or several "branches" homeomorphically projecting into the same differentiable n-disk in Rn. Branched manifolds first appeared in the dynamical systems theory, in connection with one-dimensional hyperbolic attractors constructed by Smale and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train tracks (n = 1) and branched surfaces (n = 2) and play prominent role in the geometry of three-manifolds after Thurston.

Definition

Let K be a metrizable space, together with:

  1. a collection {Ui} of closed subsets of K;
  2. for each Ui, a finite collection {Dij} of closed subsets of Ui;
  3. for each i, a map πi: UiDin to a closed n-disk of class Ck in Rn.

These data must satisfy the following requirements:

  1. j Dij = Ui and ∪i Int Ui = K;
  2. the restriction of πi to Dij is a homeomorphism onto its image πi(Dij) which is a closed class Ck n-disk relative to the boundary of Din;
  3. there is a cocycle of diffeomorphisms {αlm} of class Ck (k ≥ 1) such that πl = αlm · πm when defined. The domain of αlm is πm(UlUm).

Then the space K is a branched n-manifold of class Ck.

The standard machinery of differential topology can be adapted to the case of branched manifolds. This leads to the definition of the tangent space TpK to a branched n-manifold K at a given point p, which is an n-dimensional real vector space; a natural notion of a Ck differentiable map f: KL between branched manifolds, its differential df: TpKTf(p)L, the germ of f at p, jet spaces, and other related notions.

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.

References

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