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Consider principal connections on the frame bundle of a compact, connected, smooth, orientable Riemannian surface embedded in $\mathbb{R}^3$. On a disk $D$, it is apparent that you can construct a connection $\omega$ with zero holonomy everywhere: for instance, map $D$ to the plane and use Euclidean translation to induce parallel transport. Further, suppose that $D$ is actually an embedding of $S^2$ with a single point $p$ removed. If we now compactify $D$ to get $S^2$ again, then we have a connection $\tilde{\omega}$ on the sphere which is well-defined for any loop that does not contain $p$, and exhibits zero holonomy around any such loop. In a similar way, we can construct a connection with a single "singular" point on a surface of any genus by removing a set of loops that generate the fundamental group rather than just a single point (though we can no longer rely on Euclidean translation to provide the connection). And more generally, we can imagine connections with zero holonomy except at a number of singularities (map a punctured disk to the plane, say).

Is there a more formal description of this type of construction, and does it have a name? Any pointers to literature?

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    $\begingroup$ If you remove a small disk from a higher-genus Riemann surface, the result can be immersed in the plane, and the trivial connection pulled back. $\endgroup$
    – S. Carnahan
    Commented Apr 9, 2010 at 18:03

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I think this concerns the moduli space of flat connections on Riemann surfaces with punctures (aka holes). If there is at least one puncture $\pi_1$ of the Riemann surface is a free group and the moduli space in question reduces to the moduli space of ($G$-valued) representations of the free group (in some letters). Hence you need to study so-called character varieties, see e.g.

Florentino, Carlos; Lawton, Sean, The topology of moduli spaces of free group representations, Math. Ann. 345, No. 2, 453-489 (2009). arXiv:0807.3317v2. ZBL1200.14093.

You might also have a look at:
Florentino, Carlos; Lawton, Sean, Singularities of free group character varieties, Pac. J. Math. 260, No. 1, 149-179 (2012). arXiv:0907.4720v2. ZBL1264.14064.

For some general stuff see also
http://www.springerlink.com/content/r57w32lhk6346157/

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  • $\begingroup$ The link to springerlink.com is broken. I'm also unable to find any copy saved on the Wayback Machine. $\endgroup$ Commented Jun 8, 2022 at 8:57

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