# nLab branched manifold

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

Smooth branched $n$-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 $n$-plane.

Branched $n$-manifolds arise for instance as quotients of foliations.

## Definition

Let $X$ be a metrizable space with

1) a collection $\left\{{U}_{i}\right\}$ of closed subsets of $X$

2) for each ${U}_{i}$ a finite collection $\left\{{D}_{ij}\right\}$ of closed subsets of ${U}_{i}$

3). for each $i$ a map ${p}_{i}:{U}_{i}\to {D}_{i}^{n}$ to a closed $n$-disk of class ${C}^{k}$ in ${ℝ}^{n}$

such that

a) ${\cup }_{i}{D}_{ij}={U}_{i}$ and ${\cup }_{i}{U}_{i}^{\circ }=X$

b) ${p}_{i}{\mid }_{{D}_{ij}}$ is a homeomorphism onto its image which is a closed ${C}^{k}$ n-disk relative to $\partial {D}_{i}^{n}$.

c) There is a “cocycle” of diffeomorphisms $\left\{{\alpha }_{{i}^{\prime }i}\right\}$ of class ${C}^{k}$ such that ${p}_{{i}^{\prime }}={\alpha }_{{i}^{\prime },i}\circ {p}_{i}$ when defined. The domain of ${\alpha }_{{i}^{\prime }i}$ is ${p}_{i}\left({U}_{i}\cap {U}_{{i}^{\prime }}\right)$.

Then $X$ is called branched n-manifold of class ${C}^{k}$.

This appears as Williams, def. 1.0 ns.

If $X$ satisfies this set of axioms with b) replaced by

${b}_{\mathrm{ns}}\right)$ ${p}_{i}{\mid }_{{D}_{ij}}$ is a homeomorphisms onto ${D}_{i}^{n}$

$X$ is called nonsingular branched $n$-manifold of class ${C}^{k}$.

## 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)