Branched Manifold - Definition

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: Ui → Din 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(Ul ∩ Um).

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: K → L between branched manifolds, its differential df: TpK → Tf(p)L, the germ of f at p, jet spaces, and other related notions.

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