# Stable homotopy theory of orbifolds

Is there a notion of stable homotopy, spectrum, or a stable homotopy category which corresponds to orbifolds and orbispaces, in the same way that classical stable homotopy theory corresponds to ordinary manifolds and spaces?

I'm not so interested in speculations about what such a thing might look like; I can come up with my own speculations. I'm curious whether orbifold theorists have actually come up with such notions before and used them for things.

• Stefan Schwede gave a talk about "global spectra" in Edinburgh a little while ago. These consisted of a model structure on orthogonal spectra that produced equivariant spectra for any group $G$ in some kind of compatible way. I believe that equivariant K-theory (which is probably the exemplar for the type of theory you might be interested in) serves as an example of this structure. You might try asking him. – Tyler Lawson Nov 11 '11 at 1:13
• @Tyler "any group $G$" - discrete group or Lie group? – David Roberts Nov 11 '11 at 1:19
• @Tyler: that's interesting; a connection to global equivariant spectra hadn't occurred to me. My naive guess would that any equivariant spectrum could be considered as an "orbispectrum" (why didn't that word occur to me when asking the question?), in the same way that any space with a group action can be considered a "global quotient" orbispace. But I would expect equivariant K-theory for different groups to give different "global quotient" orbispectra. Are you thinking of something else? – Mike Shulman Nov 11 '11 at 2:19
• @David Roberts: I think the collection of groups you use is a parameter of the theory. So you could pick all discrete groups, or all finite groups, or all Lie groups, or all compact Lie groups, or maybe some small subcategories of those. – Mike Shulman Nov 11 '11 at 2:21
• @David: What Mike said. My impression was that there were two different model structures involved. – Tyler Lawson Nov 11 '11 at 6:10

I wrote and thought I posted an answer a few minutes ago, but it didn't appear. I don't know of any connection between orbifolds and global spectra, so this answer is a digression. Global spectra are one $G$-spectrum for each group in a chosen class, suitably related. Since one wants to start with genuine G-spectra, definitions so far are restricted to subclasses of the class of compact Lie groups. The first such definition was given by Gaunce Lewis and myself (II.8.5 in SLN 1213, 1986). A later definition was given by Greenlees and myself (\S5 in Localization and completion theorems for $MU$-modules, Annals 1997). The cited definitions are quite different (I'm forgetful), and in fact there are quite a few sensible choices for both global Mackey functors and global spectra. Various definitions and examples are sorted out in Anna Marie Bohmann's 2011 Chicago PhD thesis, and more work is in progress.

Since Mike asked this question almost 10 years ago, Schwede's work on global equivariant homotopy theory, alluded to by Tyler in the comments, has become a classic. I'm personally not aware of the connection between orbifolds and equivariant homotopy theory having been deeply explored, but maybe it has (and maybe somebody will chime in on this question to talk about it!). It appears that a seminar on the intersection of the two subjects was held at Columbia in 2019. From the syllabus, I would guess that there hadn't been much work bridging the fields to that point, because the topics appear to be pretty clearly delineable between "equivariant homotopy theory" talks and "orbifold" talks.

However, there is recent work of Juran constructing a global equivariant stable homotopy type from any orbifold. This seems like an area ripe for cross-fertilization.

Something about the way both subjects are especially fond of finite groups really screams to me that they must be related.

One would think there'd be an analogy

orbifolds : global equivariant homotopy theory :: manifolds : ordinary homotopy theory