nLab
thin homotopy

Contents

Idea

A thin homotopy between paths f,g:IXf,g: I \to X in a topological space XX (with I=[0,1]I = [0,1] the standard interval) is a homotopy I×III\times I \to I which, roughly speaking, has zero area.

Definition

Thin homotopies in smooth spaces

A (smooth) homotopy F:I×IXF: I \times I \to X between smooth paths in a smooth space XX is called thin if the rank of its differential dF(s,t):T s,tI×IT F(s),F(t)Xd F(s,t): T_{s,t} I \times I \to T_{F(s),F(t)} X is less than 2 for all s,tI×Is,t \in I \times I.

(More here…)

Thin homotopies in topological spaces

The following is taken from

We define a finite tree to be a one-dimensional finite polyhedron.

A homotopy F:I×IXF:I\times I \to X between paths F(,0)F(-,0) and F(,1)F(-,1) in the topological space XX is called thin if FF factors through a finite tree,

I×IF 0TF 1X I\times I \stackrel{F_0}{\to} T \stackrel{F_1}{\to} X

such that the paths F 0(,0):ITF_0(-,0):I\to T, F 0(,1):ITF_0(-,1):I\to T are piecewise-linear.

When XX is Hausdorff, points, paths and thin homotopies in XX form a bigroupoid.

Applications

Path n-groupoids

The definition of path groupoids and path n-groupoids as strict or semi-strict n-groupoids typically involves taking morphisms to be thin homotopy classes of paths. See there for more details.

Parallel transport

The parallel transport of a connection on a bundle is an assignment of fiber-homomorphisms to paths in a manifold that is invariant under thin homotopy.

Revised on April 9, 2014 06:21:33 by Tim Porter (2.26.27.237)