Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a mapping:
$$
hol: L\mapsto hol(L)\in Aut(V),
$$


the holonomy map, which gives us the (linear) transformation of vectors after parallel transport around the loop $L$. Its image $Hol(\nabla)$ (dropping the reference to the base point) is a subgroup of $Aut(V)$.


If $X$ is simply connected, then it is known that $Hol(\nabla)$ is a path-connected Lie subgroup of $Aut(V)$. (See for example "[Riemannian Holonomy Groups and Calibrated Geometry](http://rads.stackoverflow.com/amzn/click/0199215596)" by D. Joyce, chapter 2.)


This can be seen in the following way: a continuous homotopy $L\_s, s\in[0,1]$ between two loops $L\_0$ and $L\_1$ defines a continuous curve in $Hol(\nabla)$, 
given by:
$$
\gamma:s\mapsto \gamma\_s= hol(L\_s).
$$


Therefore, homotopic loops have path-connected images in $Hol(\nabla)$.


My question is: how further can we go?


Between points in $Hol(\nabla)$, having a continuous curve that joins them is an equivalence relation.
If we denote by $\pi\_0(Hol(\nabla))$ the equivalence classes of such relation, we have a well defined map $hol^1:\pi\_1(X)\to \pi\_0(Hol(\nabla))$.
In fact, if two loops are homotopic, their images lie in the same class.


$\pi\_0$ has no group structure defined (would it be possible to define it?), so our map $hol^1$ is not a morphism of groups.


Now, a 2-dimensional homotopy $S^2\to X$ can be thought of a homotopy $L\_s$ of loops, like before, but where both $L\_1$ and $L\_0$ are trivial (while for generic $s$, $L\_s$ may not be trivial).
The curve $s\mapsto hol(L\_s)$ now defines a *closed* loop in $Hol(\nabla)$, based at the identity. A homotopy between two 2-dimensional homotopies in $X$ defines a homotopy of loops in $Hol(\nabla)$.
In particular, we can define a mapping $hol^2:\pi\_2(X)\to \pi\_1(Hol(\nabla))$. It is well defined, because as we saw $hol$ preserves homotopy of different orders.


So in general, I would expect that $hol$ defines mappings:
$$
hol^k:\pi\_k(X)\to \pi\_{k-1}(Hol(\nabla)).
$$


Moreover, for $k>0$, $\pi\_k$ are groups, and since composing homotopies of order $k$ in $X$ corresponds to composing homotopies of order $k-1$ in $Hol(\nabla)$ (and the image of trivial homotopies are trivial), the maps $hol^k$ are morphisms at least for $k>1$.
So I would say:



> 
> The holonomy map defines morphisms $hol^k:\pi\_k(X)\to \pi\_{k-1}(Hol(\nabla))$.
> 
> 
> 


This would imply:



> 
> A bundle $V\to E\to X$ with holonomy $Hol(\nabla)$ can exist only if every $\pi\_k(Hol(\nabla))$ admits as a subgroup a homomorphic image (i.e. a quotient) of $\pi\_{k+1}(M)$.
> 
> 
> 


Is this reasoning correct? Is it helpful? Has it been done anywhere?


EDIT: For $TS^2$, the tangent bundle to the 2-sphere, and the Levi-Civita connection, the Gauss-Bonnet theorem can be stated in this (beautiful!) way: since $Hol(\nabla)=SO(2)\cong S^1$,



> 
> The morphism $hol^2: \pi\_2(S^2) \to \pi\_1(S^1)$ induces an isomorphism $\Bbb{Z}\to 2\Bbb{Z}$.
> 
> 
> 


Does this result generalize?
Thank you!



Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?
I can try to answer to myself, using the very helpful comments you wrote. (You are all still free to write an answer yourselves!)


*Notice:* For convenience, here I will denote by $hol^k$ what in the question I denoted by $hol^{k+1}$. So we look for 
group morphisms $hol^k:\pi\_{k+1}(X)\to \pi\_k(Hol(\nabla))$.


First of all, there is a very natural answer to:



> 
> Is it possible to define a group structure on $\pi\_0(Hol(\nabla))$?
> 
> 
> 


Yes. Since $Hol\_0(\nabla)$ is a *normal* subgroup of $Hol(\nabla)$, we simply define:
$$
\pi\_0(Hol(\nabla)) := \frac{Hol(\nabla)}{Hol\_0(\nabla)}.
$$


Holonomies of loops of the same homotopy class differ by an element of $Hol\_0(\nabla)$. This is easy to see, if loops $L,L'$ are homotopic,
take their composition with reversed $L$: $L'L^{-1}$ is homotopic to the trivial loop. 
Therefore $hol^0$ maps homotopic loops into the same element of $\pi\_0(Hol(\nabla))$.


It is naturally a morphism, because if $L,L'$ are loops:
$$
hol(LL') = hol(L)hol(L'),
$$


where the composition on the left member is loop concatenation, and the composition in the right member is group multiplication on $Hol(\nabla)$.
The quotients $\pi\_1(X)$ and $\pi\_0(Hol(\nabla))$ inherit the multiplication law, which for $\pi\_1(X)$ becomes a group law, and 
therefore we have a morphism. (Of course, trivial loops are mapped into the identity.)



> 
> Does $hol$ define morphisms $\pi\_{k+1}(X)\to\pi\_k(hol(\nabla))$?
> 
> 
> 


Yes. As abx pointed out, $hol:\Omega X\to Hol(\nabla)$ defines maps:
$$
 hol^k:\pi\_k(\Omega X)\to \pi\_k(Hol(\nabla)),
$$


where $\Omega X$ is the loop space of $X$, and $\pi\_k(\Omega X) = \pi\_{k+1}(X)$.



> 
> Does this impose restrictions on the bundles and connections?
> 
> 
> 


Hardly, as was pointed out by abx, Ulrich Pennig, and Konrad Waldorf, because the image of $hol^k$ can be trivial.


This, nevertheless, and the remark on $TS^2$, raises the following question:



> 
> What happens when $Im(hol^k)$ is *not* trivial? What properties does it have?
> 
> 
> 


Or, following the example of $TS^2$:



> 
> Are characteristic classes linked to the "size" of $Im(hol^k)$?
> 
> 
> 


When I gather enough details of the problem, I will ask a new question, and post the link here.


**EDIT:** I asked the question, [here it is](https://mathoverflow.net/questions/161603/possible-homotopy-theoretical-approach-to-gauss-bonnet).
**EDIT 2:** It doesn't generalize in the way I (wrongly) thought. I'll ask more when I find something *actually* interesting.


