nLab
holonomy groupoid

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Higher Lie theory

          ∞-Lie theory (higher geometry)

          Background

          Smooth structure

          Higher groupoids

          Lie theory

          ∞-Lie groupoids

          ∞-Lie algebroids

          Formal Lie groupoids

          Cohomology

          Homotopy

          Examples

          \infty-Lie groupoids

          \infty-Lie groups

          \infty-Lie algebroids

          \infty-Lie algebras

          Contents

          Idea

          A holonomy groupoid is a (topological/Lie-) groupoid naturally associated with a foliation \mathcal{F} of a manifold X X . It is, in some sense, the smallest de-singularization of the leaf-space quotient X/ X/\mathcal{F} of the foliation, which is, in general, not a manifold itself. Every foliation groupoid of \mathcal{F} has this de-singularization property, but the holonomy groupoid is, in some sense, minimal with respect to this property.

          Explicitly, given a foliation \mathcal{F} on a manifold X X , the holonomy groupoid of \mathcal{F} has as objects the points of X X . Given points x,y x,y in the same leaf, a morphism between them is the equivalence class of a path in the leaf from x x to y y , where two paths are identified if they induce the same germ of a holonomy transformation between small transversal sections through x x and y y . If x x and y y are not in the same leaf, then there is no morphism between them.

          This is naturally a topological groupoid and a Lie groupoid if done right.

          The monodromy groupoid of the foliation is obtained from this by further dividing out the homotopy between paths in a leaf.

          Definition

          Foliation Holonomy

          Let (X,) (X,\mathcal{F}) be a foliated manifold with q=codim() q = codim(\mathcal{F}) . Let L L be a leaf of \mathcal{F} ; let x,yL x,y \in L be two points; and let S S and T T be transversal sections through x x and y y respectively (i.e., S S and T T are sub-manifolds transversal to the leaves of \mathcal{F} ).

          To a path γ:[0,1]L \gamma: [0,1] \to L from x x to y y , we assign the germ of a (partially defined) diffeomorphism

          hol S,T(γ):(S,x)(T,y), {hol^{S,T}}(\gamma): (S,x) \to (T,y),

          called the holonomy transformation of the path γ \gamma with respect to S S and T T , as follows:

          If there exists a single foliation chart U U of \mathcal{F} that contains the image of γ \gamma , then there exists a sufficiently small open neighborhood A A of x x in the space SU S \cap U for which there exists a unique smooth map f:AT f: A \to T satisfying the following conditions:

          • f(x)=y f(x) = y ;

          • For every aA a \in A , the point f(a) f(a) lies in the same plaque in U U as a a . Observe that f f is a diffeomorphism onto its image.

          Then define hol S,T(γ) {hol^{S,T}}(\gamma) to be the germ of this diffeomorphism at x x :

          hol S,T(γ)=defgerm x(f). {hol^{S,T}}(\gamma) \stackrel{def}{=} {germ_{x}}(f).

          In general, the image of γ \gamma is not contained inside any single foliation chart U U , but as it is a compact subspace of X X , there exist finitely many foliation charts U 1,,U n+1 U_{1},\ldots,U_{n+1} and numbers t 0,,t n+1 t_{0},\ldots,t_{n+1} such that

          • 0=t 0<t 1<<t n<t n+1=1 0 = t_{0} \lt t_{1} \lt \cdots \lt t_{n} \lt t_{n+1} = 1 and

          • γ([t i1,t i]) \gamma([t_{i-1},t_{i}]) is contained in U i U_{i} for i=1,,n+1 i = 1,\ldots,n+1 .

          Then arbitrarily choose transversal sections T i T_{i} through γ(t i) \gamma(t_{i}) for i=1,,n i = 1,\ldots,n and define

          hol S,T(γ)=defhol T n,T(γ| [t n,t n+1])hol T n1,T n(γ| [t n1,t n])hol T 1,T 2(γ| [t 1,t 2])hol S,T 1(γ| [0,t 1]). {hol^{S,T}}(\gamma) \stackrel{def}{=} {hol^{T_{n},T}} \left( \gamma|_{[t_{n},t_{n+1}]} \right) \circ {hol^{T_{n-1},T_{n}}} \left( \gamma|_{[t_{n-1},t_{n}]} \right) \circ \cdots \circ {hol^{T_{1},T_{2}}} \left( \gamma|_{[t_{1},t_{2}]} \right) \circ {hol^{S,T_{1}}} \left( \gamma|_{[0,t_{1}]} \right).

          This definition is independent of the choice of U i U_{i} ‘s, t i t_{i} ’s and T i T_{i} ’s. It only depends on the initial and final transversal sections S S and T T .

          Proposition

          • Two homotopic paths with the same endpoints induce the same holonomy. (Note, however, that the converse is not true. Two paths with the same endpoints inducing the same holonomy may not be homotopic.)

          • If S,S S,S' are two transversal sections through x x and T,T T,T' two transversal sections through y y , then

          hol S,T(γ)=hol T,T(const y)hol S,T(γ)hol S,S(const x). hol^{S',T'}(\gamma) = {hol^{T,T'}}(const_{y}) \circ {hol^{S,T}}(\gamma) \circ {hol^{S',S}}(const_{x}).

          Holonomy Groupoid

          Given a foliated manifold (X,) (X,\mathcal{F}) , the monodromy groupoid is the disjoint union of the fundamental groupoids of the leaves of \mathcal{F} , which is the groupoid having the following properties:

          1. Its objects are the points of X X .

          2. There are no morphisms between two points on different leaves.

          3. The morphisms between two points on the same leaf are homotopy-classes of paths lying in the leaf joining those points.

          The holonomy groupoid is defined analogously, where instead of identifying two paths if they are homotopic, they are identified if they induce the same holonomy as described above.

          References

          The holonomy groupoid appears in

          • Charles Ehresmann, Structures Feuilletées , Proc. 5th Canadian Math. Congress, Univ. of Toronto Press 1963, 1961, pp. 109–172.

          and was studied extensively in

          • H. E. Winkelnkemper, The graph of a foliation , Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75.

          See also

          Applications to singular foliations and to Poisson manifolds in particular are in

          Last revised on July 2, 2016 at 06:57:30. See the history of this page for a list of all contributions to it.