Locally topological groupoids and extendibility
by
Osman Mucuk
Erciyes University
Coauthors: Ilhan Içen (Inonu University)

A groupoid is a small category in which each morphism is an isomorphism. Let G be a groupoid and W a subset of G containing all the identities in G. Suppose that W has a topology. For certain conditions on W the pair (G, W) is called a locally topological groupoid. The topological structure of W does not in general extend to a topological groupoid structure on G which restricts to that on W, but there is a topological groupoid H called the holonomy groupoid with a morphism H --> G such that H contains W as a subspace and H has a universal property. The full details of this result are given by Aof and Brown in [1].

A locally topological groupoid (G, W) is called extendible if there is a topology on G such that G is a topological groupoid with this topology and W is open in G. A locally topological groupoid is not in general extendible. It is proven by Brown and Mucuk in [3] that the charts of a foliated manifold give rise to a locally topological groupoid which is not extendible. We have also examples of locally topological groupoids, due to Pradines, which are not extendible. A full account of the monodromy groupoids was given in [5] and published in [2].

Let G be a topological groupoid in which each star G_x has a universal cover. Then the monodromy groupoid GM is constructed by Mackenzie in [4] as the union of the universal coverings of G_x. In the locally trivial case in [4], MG is given a topology such that MG is a topological groupoid with this topology.

In this paper we prove that if G is a locally sectionable topological groupoid and W is an open subset containing all the identities, then using the criterion obtained from the holonomy, the monodromy groupoid MG gives rise to a locally topological groupoid (MG, W) which is extendible.

References

[1]. Aof, M.E.-S.A.-F. and Brown, R., The holonomy groupoid of a locally topological groupoid, Top. Appl., 47, 1992, 97-113.

[2]. Brown, R. and Mucuk, O., The monodromy groupoid of a Lie groupoid, Cah. Top. Géom. Diff. Cat., 36, 1995, 345-369.

[3]. Brown, R. and Mucuk, O., Foliations, locally Lie groupoids and holonomy, Cah. Top. Géom. Diff. Cat., 37, 1996, 61-71.

[4]. Mackenzie, K.C.H., Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Note Series 124, Cambridge University Press, 1987.

[5]. Mucuk O., Covering groups of non-connected topological groups and the monodromy groupoid of a topological groupoid, PhD Thesis, University of Wales, 1993.

Date received: July 17, 2001