6
$\begingroup$

Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\beta(0)$ to $E$. According to the nlab, a Hurewicz connection is a continuous section $\lambda$ of the induced map $E^I \to Cocyl(p)$, i.e. a continuous choice of lifts for each path in $B$ once an endpoint is specified.

Let's say that a Hurewicz connection $\lambda$ is functorial if

  1. For any order-preserving homeomorphism $\phi : I \to I$ of the interval, $\lambda$ commutes with reparameterization by $\phi$, in the sense that $\lambda(e,\beta)(\phi(t)) = \lambda(e,\beta\circ \phi)(t)$ for $t \in I$.

  2. $\lambda$ commutes with gluing of paths: if $\beta_1,\beta_2$ are paths from $b_0$ to $b_1$ and $b_1$ to $b_2$ respectively, and if $\beta_1 \ast \beta_2$ is the glued path from $b_0$ to $b_2$, then for any order-preserving homeomorphism $\phi : I \to I \vee I$, we have $\lambda(e,(\beta_1 \ast \beta_2) \circ \phi)(t) = (\lambda(e,\beta_1) \ast \lambda(e',\beta_2))(\phi(t))$ for $t \in I$, where $e'= \lambda(e,\beta_1)(1)$.

  3. (Maybe say something about inverses when reversing paths?)

Question 1: Is there a standard name for "functorial Hurewicz connections" (or something similar)?

Question 2: What can one say about classifying functorial Hurewicz connections in more "local" terms, more analogous to Ehresmann connections?

Question 3: If we impose appropriate smoothness conditions on a functorial Hurewicz connection on a fiber bundle, do we recover Ehresmann connections exactly, or do we get something more general?

Improve this question
$\endgroup$
3
  • $\begingroup$ For Q3, I'm wondering if uniqueness theorems for ODEs would tell us something useful. Let me do some thinking out loud. Since it's a local question, you can assume the base $B$ is a Euclidean space (assuming you are considering fin.dim. at least to start), and if you are working with a smoothly locally trivial fibre bundle, then you might as well assume it's a product $B\times F$, the cocylinder becomes the product $B^I \times F$, and the map you need a section of is basically $id\times ev\colon B^I\times F^I \to B^I\times F$. Hence the section is really a smooth map $B^I\times F\to F^I$. $\endgroup$
    – David Roberts ♦
    Commented Aug 27, 2022 at 14:39
  • $\begingroup$ Then one uncurries to get $I\times B^I \times F \to F$. From this it seems to me one could define the horizontal distribution by looking at the derivatives of the lifted paths, and from there to other usual definitions of connections eg differential forms. $\endgroup$
    – David Roberts ♦
    Commented Aug 27, 2022 at 14:42
  • 2
    $\begingroup$ The answer to Question 3 is well known: you need to require λ to be a smooth map (meaning it sends smooth families of paths to smooth families of transport maps). See, for example, arxiv.org/abs/0705.0452v5 or arxiv.org/abs/1501.00967v2 for the cases of principal G-bundles or vector bundles with connection, and the general case of Ehresmann connections on submersions is treated in the same way. $\endgroup$
    – Dmitri Pavlov
    Commented Aug 27, 2022 at 21:26

0

Reset to default

You must log in to answer this question.