Subject: Answer to the question of Gabriele Mondello From: Andre Henriques Date: Wed, 14 Dec 2005 15:46:42 +0100 (CET) >> c. Question on orbi-CW complexes >> __________________________________________________________________________ >> >> Subject: Question on orbi-CW-complexes >> From: Gabriele Mondello >> Date: Fri, 9 Dec 2005 22:58:52 -0500 (EST) >> >> Hi, >> >> I am having some trouble in giving a reasonable >> definition of an orbi-CW-complex. >> The main problem is that an orbifold is described >> by local conditions, while a CW-complex structure >> is not a local fact. >> Also, I can define an orbifold a little more "globally" as a >> topological groupoid X=(X_0,X_1) with X_0,X_1 manifolds, >> and smooth maps s,t:X_1--->X_0 source and target, >> u:X_0--->X_1 unit, i:X_1--->X_1 inverse and m=composition; >> requiring that s,t are local diffeomorphisms and >> (s,t):X_1--->X_0xX_0 is proper (I want finite stabilizers). >> I suspect that for an orbi-CW-complex I should drop >> the condition "s,t local homeomorphisms" and ask something >> like "X_0,X_1 are CW-complexes and s,t are cellular maps, >> composition of a local homeomorphism and a closed embedding". >> Still I think someone should have already given such a definition. >> >> Does anyone know any reference? >> >> Gabriele Mondello Dear Gabrielle, I have two answers to your question. The first one assumes that we are modeling orbispaces by stacks (i.e. functors from the category of topological spaces to the category of groupoids). The second one assumes that we are modeling orbispaces by topological groupoids. 1) The usual Grothendieck topology on the category of topological spaces is not very compatible with the operation of glueing cells. Indeed, if X' is the pushout of a diagram X <- S^n -> D^{n+1}, then the sheaf Y(X') represented by the space X', is not the pushout of the sheaves Y(X) <- Y(S^n) -> Y(D^{n+1}). One possible solution is to go to a finer Grothendieck topology. I would suggest the following one: The underlying category is the category of compact spaces; a morphism A -> B is a cover if it is surjective (on the underlying sets). If you work with stacks in the above topology, the technical difficulty about Y(X') not being the pushout of Y(X) <- Y(S^n) -> Y(D^{n+1}) goes away. You can then define a CW orbispace to be a stack F which is the colimit of stacks F^(n), where F^(n+1) is obtained from F^(n) by taking a pushout F^(n) <- \coprod BG_i x Y(S^n) -> \coprod BG_i x Y(D^{n+1}). Depending on the applications that you have in mind, it might also be good idea to require that the attaching maps BG_i x Y(S^n) -> F^(n) be injective on stabilizer groups. Some notes about this approach to CW-orbispaces, are now accessible on my webpage http://wwwmath.uni-muenster.de/u/henrique/, but this is still work in progress. 2) If you want to model CW-orbispaces by topological groupoids, then it might indeed be a good idea to drop the condition "s,t local homeomorphisms". The question is then: what models does one take for the cells BG x D^n? It is not enough to just take groupoids of the form (objects = D^n ; morphisms = G x D^n). Indeed, the only CW-orbispaces that one can build using these cells are the ones where the map to the coarse moduli space is split. What you are suggesting amounts to taking cells of the form (objects = S x D^n ; morphisms = G x S2 x D^n), where S is a discrete set. It is possible to use this approach. However, if you also want to talk about maps between CW-orbispaces, you will soon want to refine your groupoids. And then you'll again note that the usual topology (with open covers) is not fine enough. Andre Henriques