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(LbEx1) The substantive significance of the use of labels may be seen in the
following fundamental example: One considers the (“line segment”) graph
Γseg
・ − ・
given by two vertices joined via a single edge. Write Γloop for the (“loop”)
graph obtained from Γseg by identifying the two distinct vertices of Γseg −
i.e., which may be thought of as distinct labels − to a single vertex. Thus,
Γseg is structurally different from Γloop in the sense that Γseg is simply
connected, while Γloop is not. This difference gives rise to very substantive consequences in many situations, e.g., situations where some sort of
“parallel transport” or “analytic continuation” along the loop of
Γloop gives rise to some sort of nontrivial monodromy operator. This
sort of nontrivial monodromy may be resolved precisely by distinguishing the situations that arise prior to and subsequent to the application of
the monodromy operator, i.e., by working over Γseg as opposed to Γloop.
When, moreover, one wishes to distinguish situations that arise prior to
and subsequent to multiple applications of the monodromy operator, it is
natural to consider the universal covering Γuni
... ・ − ・ − ・ − ・ ...
of Γloop, which may be thought of as the result of concatenating distinct
copies of Γseg labeled by elements ∈ Z. One fundamental example of this
sort of situation, i.e., of nontrivial monodromy around a loop, is the (angular portion of the) logarithm function in one-variable complex analysis.

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