### Fahrenberg and Raussen on Continuous Paths

#### Posted by Urs Schreiber

Suppose you want to transport something along some path through a space $X$. Before you do so, you need to know what a path in $X$ *is*.

If $X$ is a smooth space, we tend to demand a path to be a smooth map

up to reparameterization. ($I$ is the standard interval.)

What exactly is the analog of dividing out by reparameterization of paths in the case that $X$ is just a topological space?

Jim Stasheff, being interested in topological notions #, wondered why I kept going on about smooth paths # without ever talking about continuous paths. He was so kind to point me to the work

Ulrich Fahrenberg & Martin Raussen
*Reparametrizations of Continuous Paths*

Dept. of Mathematical Sciences, Aalborg University

Technical Report R-2006-22

(pdf)

where exactly this issue is investigated.

A central point of the discussion is the emphasis of the notion of a *directed* topological space, which is a space equipped with a collection of preferred paths that single out certain directions.

Under the term *$d$-space* this has been proposed and developed in

Marco Grandis
*Directed homotopy theory, I. The fundamental category*

math.AT/0111048 .

In this context it is natural to restrict to those reparameterizations

of the standard interval which are surjective (obviously), but also *increasing* (or rather: not decreasing).

Fahrenberg and Raussen study in detail which notion of path is obtained if we divide out maps

by this kind of continuous reparameterization.

A main result is that the space of
general paths modulo these reparametrizations is homeomorphic to the space of regular paths
modulo increasing auto-homeomorphisms of the interval. Here a *regular path* is one that never rests, i.e. one which is strictly increasing, in the obvious sense.

This is useful, because the space of regular paths in usually easier to deal with. Based on this, Martin Raussen develops a detailed theory of directed spaces

Martin Raussen
*Invariants of directed spaces*

Dept. of Mathematical Sciences, Aalborg University

Technical Report R-2006-28

(pdf).

Using this technology, one can go further and study *directed $n$-paths*

modulo continuous reparameterization, in a way analogous to what one would do for the smooth setup.

For 2-paths #, this will apparently be investigated in

Ulrich Fahrenberg
*Homotopy of Squares in a Directed Hausdorff Space*

(in preparation).

## Re: Fahrenberg and Raussen on Continuous Paths

One reason for being scared of parallel transport along continuous paths is that the differential equation describing parallel transport makes no obvious sense when the path fails to be differentiable:

$d g(t)/d t = A(\gamma'(t)) g(t)$

where $g$ is a $G$-valued function of $t$, $A$ is a $\mathrm{Lie}(G)$-valued 1-form on a manifold, and $\gamma$ is a path in this manifold.

One can take any ordinary differential equation and integrate both sides to get an integral equation which has a better chance of making sense. In the present case this doesn’t make the derivative in $\gamma'(t)$ go away, but it means we don’t need $\gamma'(t)$ to be defined pointwise - something weaker will suffice to make the integral well-defined.

Still, there are serious issues to deal with, and I’ve never heard of anyone showing that a smooth connection allows one to define parallel transport for arbitrary continuous paths.

The closest thing I’ve heard is Bismut’s work on the heat equation for vector bundles, which got used in Getzler’s famous heat equation proof of the Atiyah-Singer index theorem.

Bismut showed how the solutions of this heat equation could be described using Brownian motion of particles having an internal degree of freedom that gets parallel transported as they wiggle around. Since Brownian motion is almost surely everywhere nondifferentiable, one really needs to think hard about how this parallel transport is defined. And, he managed to succeed.

However, I think the stochastic aspect was crucial here: he wasn’t trying to define parallel transport for a

singlenondifferentiable path, merely its average over a bunch of them!