nLab
branched manifold

Contents

Idea

Smooth branched nn-manifolds are a generalization of smooth manifolds which may have ill-defined tangent spaces at certain “branch loci”, where however they are required to have a well defined tangent nn-plane.

Branched nn-manifolds arise for instance as quotients of foliations.

Definition

Let XX be a metrizable space with

1) a collection {U i}\{U_i\} of closed subsets of XX

2) for each U iU_i a finite collection {D ij}\{D_{i j}\} of closed subsets of U iU_i

3). for each ii a map p i:U iD i n\p_i:U_i\to D_i^n to a closed nn-disk of class C kC^k in n\mathbb{R}^n

such that

a) iD ij=U i\cup_i D_{i j}=U_i and iU i =X\cup_i U^\circ_i =X

b) p i| D ijp_i |_{D_{i j}} is a homeomorphism onto its image which is a closed C kC^k n-disk relative to D i n\partial D_i^n.

c) There is a “cocycle” of diffeomorphisms {α i i}\{\alpha_{i^' i}\} of class C kC^k such that p i =α i ,ip ip_{i^'}=\alpha_{i^',i}\circ p_i when defined. The domain of α i i\alpha_{i^' i} is p i(U iU i )p_i(U_i\cap U_{i^'}).

Then XX is called branched n-manifold of class C kC^k.

This appears as Williams, def. 1.0 ns.

If XX satisfies this set of axioms with b) replaced by

b ns)b_{ns}) p i| D ijp_i |_{D_{i j}} is a homeomorphisms onto D i nD_i^n

XX is called nonsingular branched nn-manifold of class C kC^k.

References

  • Robert F. Williams, Expanding attractors, Publications mathématique d l’IHÉS, tome 43 (1974) (numdam)
  • Dusa McDuff, Groupoids, branched manifolds and multisections, J. Symplectic Geom. Volume 4, Number 3 (2006), 259-315 (project euclid)

Discussion relating to orbifolds is in

Revised on February 13, 2013 13:06:36 by Urs Schreiber (82.169.65.155)