Branched manifold
Encyclopedia
In mathematics
, a branched manifold is a generalization of a differentiable manifold
which may have singularities of very restriced 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
attractor
s 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 track
s (n = 1) and branched surface
s (n = 2) and play prominent role in the geometry of three-manifolds after Thurston
.
These data must satisfy the following requirements:
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.
such that each point has a well-defined n-dimensional tangent space.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a branched manifold is a generalization of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
which may have singularities of very restriced type and admits a well-defined tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
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
Dynamical systems theory
Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference...
, in connection with one-dimensional hyperbolic
Hyperbolic set
In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding under f, with respect to some...
attractor
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
s constructed by Smale
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train track
Train track
In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions:#The curves meet at a finite set of vertices called switches....
s (n = 1) and branched surface
Branched surface
In mathematics, a branched surface is type of topological space. A small piece of an surface looks topologically like ℝ². A small piece of a branched surface, on the other hand, might look like either of the following:*ℝ²;...
s (n = 2) and play prominent role in the geometry of three-manifolds after Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
.
Definition
Let K be a metrizable space, together with:- a collection {Ui} of closed subsets of K;
- for each Ui, a finite collection {Dij} of closed subsets of Ui;
- 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:
- ∪j Dij = Ui and ∪i Int Ui = K;
- the restriction of πi to Dij is a homeomorphismHomeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
onto its image πi(Dij) which is a closed class Ck n-disk relative to the boundary of Din; - there is a cocycle of diffeomorphismDiffeomorphismIn mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
s {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
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
can be adapted to the case of branched manifolds. This leads to the definition of the tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
TpK to a branched n-manifold K at a given point p, which is an n-dimensional real vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
; a natural notion of a Ck differentiable map f: K → L between branched manifolds, its differential df: TpK → Tf(p)L, the germ
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...
of f at p, jet spaces, and other related notions.
Examples
Extrinsically, branched n-manifolds are n-dimensional complexes embedded into some Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
such that each point has a well-defined n-dimensional tangent space.
- A finite graphGraph (mathematics)In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
whose edges are smoothly embedded arcs in a surfaceSurfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
, such that all edges incident to a given vertex v have the same tangent line at v, is a branched one-manifold, or train trackTrain trackIn the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions:#The curves meet at a finite set of vertices called switches....
(there are several variants of the notion of a train track — here no restriction is placed on the valencies of the vertices). As a specific example, consider the "figure eight" formed by two externally tangent circles in the plane.
- A two-complex in R3 consisting of several leaves that may tangentially come together in pairs along certain double curves, or come together in triples at isolated singular points where these double curves intersect transversally, is a branched two-manifold, or branched surfaceBranched surfaceIn mathematics, a branched surface is type of topological space. A small piece of an surface looks topologically like ℝ². A small piece of a branched surface, on the other hand, might look like either of the following:*ℝ²;...
. For example, consider the space K obtained from 3 copies of the Euclidean plane, labelled T (top), M (middle) and B (bottom) by identifying the half-planes y ≤ 0 in T and M and the half-planes x ≤ 0 in M and B. One can imagine M being the flat coordinate plane z=0 in R3, T a leaf curling upward from M along the x-axis to the right (positive y-direction) and B another leaf curling downward from M along the y-axis in front (positive x-direction). The coordinate axes in the M plane are the double curves of K, which intersect transversally at a unique triple point (0,0).