A topological application of the monodromy groupoid on principal bundles. (English) Zbl 1007.18006

A groupoid is a small category such that every morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all maps of the groupoid structure are continuous. Let \(G\) be a topological groupoid and \(W\) an open subset of \(G\) which contains all the identities. Let \(F(W)\) be the free groupoid on \(W\) and \(N\) the normal subgroupoid of \(F(W)\) generated by the elements in the form \([ba]^{-1}[b][a]\) for \(a, b\in W\) such that \(ba\) is defined and \(ba\in W\). Then the quotient groupoid \(M(G, W)\) of \(F(W)\) by \(N\) is called the monodromy groupoid of \(G\) for \(W\). Suppose \(p: E\rightarrow X\) is a principal bundle in the sense of C. Ehresmann [“Catégories topologiques et catégories différentiables”, in: Colloque Géom. Différ. Globale, Bruxelles 1958, 137-150 (1959; Zbl 0205.28202)] with certain conditions, it is proved that an open subset of the groupoid \(G = X\times X\) can be chosen such that the monodromy groupoid \(M(G, W)\) is a topological groupoid and acts topologically on the topological space \(E\) via \(p\).


18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
54B30 Categorical methods in general topology
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)


Zbl 0205.28202
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