Branched Manifold - Definition

Definition

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 Ck 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 Ck n-disk relative to the boundary of D_in;
3. there is a cocycle of diffeomorphisms {α_lm} of class Ck (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 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 T_pK 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: T_pK → T_f(p)L, the germ of f at p, jet spaces, and other related notions.