Abstract Dorette Pronk

Title: Atlases for Nonreduced Orbifolds

 Abstract: Satake introduced the notion of orbifold (also called, V-manifold) in terms of an underlying space with an atlas of orbifold charts consisting of open subsets of Euclidean space with a finite group of diffeomorphisms such that the quotient is an open subset of the underlying space. Chart embeddings are embeddings between charts that send orbits to orbits. Satake observes that any such chart embedding induces a unique group monomorphism between the structure groups which makes the embedding equivariant. In my work with Moerdijk I have shown that orbifolds defined in this way can be represented by effective topological groupoids with a proper diagonal and structure maps that are local homeomorphisms. This gave rise to a new way of defining smooth maps between orbifolds. Motivated by examples in mathematical physics there has been an interest to extend the notion of orbifold to include nonreduced orbifolds. For a nonreduced orbifold each chart has an action of a finite group, but we do not require that the group is a subgroup of the group of diffeomorphisms, so the action need not be effective. From a groupoid perspective it is fairly clear how this should be done: one simply removes the requirement that the groupoid be effective from the definition of orbifold groupoid. This does not affect the notion of map between orbifolds, so it seems fairly straightforward to do this. However, for the atlas definition it is not as straightforward to adjust the notion of chart embedding. When the group actions are not effective there is in general not a unique homomorphisms that makes the embedding equivariant. Some authors have therefore required that chart embeddings be pairs of an embedding of chart spaces and a group monomorphism together with a a requirement that the group homomorphism induce an isomorphism on between the kernels of the actions. We will show that orbifold atlases defined with this type of embeddings do not correspond to noneffective orbifold groupoids. Then we will propose a new definition of atlas for noneffective orbifolds which is expressed in terms of profuctors between the structure groups. We will show that this generalizes Satake's definitiion of orbifold atlas, and that our new atlases do correspond to not necessarily effective orbifold groupoids. If there is time I will discuss what notions of map between orbifolds we obtain from our new definition of orbifold atlas.

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