differential topology - When do two holonomy maps determine flat bundles that are isomorphic as just bundles (w/o regard to the flat connections)?

For an abelian structure group $A$, principal $A$-bundles over a smooth manifold $M$ can be described completely in terms of their holonomy functionals $f: LM to A$.

Here, $LM$ is the thin loop space whose elements are homotopy classes of loops, with the rank of the homotopies bounded by one. A map $f:LM to A$ is by definition smooth, if its pullback to the ordinary loop space is smooth (in the Fréchet sense).

Definition. A fusion map is a smooth map $f: LM to A$ such that for every triple $(gamma_1,gamma_2,gamma_3)$ of paths in $M$, with a common initial point and a common end point, and each path is smoothly composable with the inverse of each other, the map satisfies

$$f(overline{gamma_2} star gamma_1) cdot f(overline{gamma_3} star gamma_2) = f(overline{gamma_3} star gamma_1).$$

Theorem.

1. The group of isomorphism classes of principal $A$-bundles with connection over $M$ is isomorphic to the group of fusion maps.




2. The group of isomorphism classes of principal $A$-bundles with flat connection over $M$ is isomorphic to the group of locally constant fusion maps.




3. The group of isomorphism classes of principal $A$-bundles over $M$ is isomorphic to the group of homotopy classes of fusion maps (where the homotopies are smooth and go through fusion maps).

Parts 1 and 2 are a theorem of J. Barrett, proved in "Holonomy and Path Structures in General Relativity and Yand-Mills Theory". Part 3 is in my paper "Transgression to Loop Spaces and its Inverse I".

Corollary.
Two principal $A$-bundles with connection (no matter if flat or not) are isomorphic as bundles, if and only if their holonomy functionals are homotopic in the sense of 3.

The non-abelian case should be similiar. In fact, Barrett's result is valid for non-abelian groups.

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