Foliation

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so thank you for the introduction thanks

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for sticking around for this last

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let's talk here so yes the basic complex

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of a singular affiliation

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um so this is a result I worked out for

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part of my thesis which

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will soon be completed

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um and uh so it has to do with singular

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foliations so we'll begin with the

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definition of a singular foliation

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especially these days the term could

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maybe be ambiguous so I have to pin down

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what I mean so for me a singular

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affiliation will be a partition of a

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manifold

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into leaves and these or by Leaf will

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mean a connected and weakly embedded sub

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manifold

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so it's a partition of a manifold into

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these sub manifolds and there's a

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smoothness condition so they need to fit

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together smoothly

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in the following sense

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so it's a bit wordy but I don't think

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it's too bad for each point in the

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manifold you can find a collection of

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vector fields

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and

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these Vector Fields will locally span

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the tangent spaces to the leaves in the

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neighborhood

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so in particular they have to spend the

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attention space to the leaf at X but

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also the leaves near X

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and the reason this is a singular

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foliation or potentially singular one is

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that there's no there were Vector Fields

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can depend on the point x so at

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different points you might have lower

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dimensional leaves

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things can change around as you go

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through the manifold

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there's a small comment on this weekly

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embedded assumption uh it's a bit

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stronger than immersed all it means is

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that the uh or consequence for us is

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that the smooth structure on the leaves

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is unambiguous it's completely

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determined by the leaf as a set and the

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ambience manifold structure so I don't

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need to worry about equipping them with

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um I don't need to worry about this

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restructure on the leaves

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specifying what it is

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uh so give a definition here's an

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example I started off with sort of the

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most

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one of the more General ways you can

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approach it so if you have a lead

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algebraid then the integral sub

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manifolds will assemble into a singular

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foliation

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and

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more Elementary examples are sort of can

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be deduced from this so for instance uh

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connected components of the orbits of a

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lead group action will partition your

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manifold into a singular foliation and

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also the synclectic leaves of a poisson

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manifold will give a singular foliation

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Okay so

Motivating question

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what will we do with these uh the

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question I'm sort of interested in is

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taking a singular foliation and looking

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at the leaf space

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and well what I'll do is I want to use

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What's called the diffiological

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structure on this Leaf space to probe

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information about the transverse

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geometry so here we're going to do a bit

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of a comparison

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which will make precise in a second

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so the leaf space will denote by m mod f

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and like I said it has a natural

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so-called diffiological structure

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so I think Eugene mentioned ethiology

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briefly in his talk in the beginning uh

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it's one of the various Notions of

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generalized smooth structure we can put

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on a set

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I don't want to dwell too much on the

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definition so I just put it a bit

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vaguely here

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due to Chen and Syria independently so a

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diffiology will have a set and we'll

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equip it with a set of maps from various

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euclidean spaces or open subsets of

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euclidean spaces into X and these Maps

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which are called plots will essentially

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determine what it means to be smooth

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so if we have a function from a

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difficult Space X we'll use the plots to

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Define what it means for that function

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to be smooth

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Etc

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in the footnote I gave a brief

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description of these axioms that a

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collection of plots must satisfy but

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again I think I don't want to go too

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much into it because that's basically a

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talk on its own or a 15-minute talk on

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its own

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but the idea is that this Leaf space has

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more structure than its apology it has a

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diffiology

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and so I can give a put a vague

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formulation of The Guiding question here

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what aspects of the foliation sitting

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upstairs can we recover from the leaf

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space viewed as a diffiological space

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and just to give a small motivation as

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to why this might make sense to ask if

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we consider an orthogonal group action

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on RN

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for different ends topologically this

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quotient is just a half Ray and you

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can't tell the different half Rays apart

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from the different dimensions that you

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started with but the diffiological

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structure will be distinct for each of

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these no two will be diffiomorphic

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so I can see these are all

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the theologically

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distinct

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so it's a small reason to believe that

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this question is not hopeless

Downstairs: the leaf space

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so we're going to approach this using

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basic complex so first downstairs on the

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leaf space itself we have the following

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facts that we get from equipping it with

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the diffiology so first the quotient map

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is a smooth map as it probably should be

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moreover it's actually what's called a

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subduction which is a diffiological

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version of submersion

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um diffiological spaces always come

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equipped with a complex of differential

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forms

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so this is that side of things works

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very well

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there's nothing special about it being a

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quotient for these

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complex to exist

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it's just the diffiological structure

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and finally the

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smooth quotient map gives a pullback of

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complexes of differential forms so we

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have the forms on the oops

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yeah we have the forums on there Leaf

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space and they pull back to right now

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it's just forms on the manifold

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so this is a complex we'll look at

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downstairs this Omega M mod f

Upstairs: basic forms

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on the other hand if we don't want to

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pass to the leaf space we can stay with

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the foliation upstairs and we can

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consider What's called the complex of

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basic forms the second part of the title

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this was first introduced by Reinhardt

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in 59th uh he did he was working with

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regular foliations at the time but we

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can make the same definition for

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singular ones so we'll call a

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differential form on the manifold basic

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if it is

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horizontal so this interior derivative

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Vanishing is called horizontal and

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invariant

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this lead derivative Vanishing is called

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invariance and the X we take are those

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which are tangent to the foliation

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if you have a foliation induced by a

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connected Lee group action then

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invariance really just means invariance

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under the group action

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but the horizontal condition just stays

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the ways

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and the notation so we'll call this

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Omega sub b m f

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okay

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so now we have two complexes of

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differential forms

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morally whoops

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these basic forms should capture

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transverse geometry but so should these

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forms on the quotient since that is the

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leaf space itself

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and we also have conveniently the

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pullback by the quotient so we can ask

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how does the pullback compare these two

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complexes and we begin with this little

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Lemma again this is a consequence of

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General results in diffiology so the

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pullback will restrict to amorphism of

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these complexes

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so it will always go into basic forms

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and it's also injective

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uh yeah yeah yeah yeah yeah yeah yeah

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yeah yeah

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and uh yeah so here you have injective

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and then sort of the more formal version

Specific question (and partial answer)

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of the question I started with uh we're

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going to specifically ask when is it an

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isomorphism

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and in particular uh because of this

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earlier Lemma we're really asking when

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is it on to

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and that's the result that we have

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so I did this maybe not a little like a

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year ago or so

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I guess that's two years now anyway uh

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so we have the quotient will induce an

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isomorphism

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or will be subjective

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in the following three cases so first

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when f is regular

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and I later found that this was also

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done by Hector marcius Virgos and San

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Martin carbon in 2009

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and also uh Elin and chandong yang also

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have an argument for this to be true in

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one of their recent preprints

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uh so when f is a regular foliation

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we are onto

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more generally though uh we can now go

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to singular foliations and the following

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condition suffices so

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for each

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k for each dimension of the leaves look

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at all the leaves of that dimension

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take the union so it's a subset of the

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manifold and require that this Union is

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a sub manifold

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now I'm a bit General in what I mean by

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sub manifolds but we can just take

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weekly embedded or you could think

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embedded if you want

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uh most of the examples it's an embedded

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manifold so for each K that Union of

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leaves is a sub manifold then the

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pullback will be an isomorphism for that

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singular foliation

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I'd like to call this stratified you

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know the foliation is stratified by

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Dimension or something but I don't know

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that it's a stratification in the sense

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that that word usually used

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that's why I don't write it here

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and there's a finally a third strange

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sort of condition that we can have which

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is more General than the previous two uh

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if you discard all the zero leaves

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from your manifold that's what is M

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bigger than zero is

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and then you equip it with the foliation

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where you remove those zero leaves

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and if the pullback is an isomorphism

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for that

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open subset of your manifold

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then you can throw the zero leaves back

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in and it will remain an isomorphism

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so somehow this question of when the

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pullback is an isomorphism you can

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sprinkle in zero leaves however you like

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in any closed subset and you'll remain

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in isomorphism if you started off as one

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which I think it's a bit strange to

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you can make some strange examples with

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this

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okay and uh yeah irmrn is where this can

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appear

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how about that five minutes

Small example

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okay

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um I just wanted to give a little

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example of me how to to see how the

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diffiology

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sort of works or how you might use it

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um consider the action of sln on

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matrices by left multiplication

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then we want to sort of ask what are the

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basic forms for this foliation of the

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set of matrices and we make the

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following diagram so we split the

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determinants into the quotient map and

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this map I just call it C

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then

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on the difficulty theology side you can

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prove that this C induces an isomorphism

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on the complexes of differential forms

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from R to X X so the diffiology is here

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to make sense of that

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and then

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the theorem comes in because we're in

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case two there's a sort of decomposition

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by dimension for this action and so then

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we know that the quotient will give an

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isomorphism

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and when you put these together we can

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see that pulling back by the

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determinants gives you all the basic

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forms

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for this little example

Bigger examples

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some more sophisticated examples I guess

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I call them bigger uh if we have a

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properly groupoid

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then linearization theorems tell us that

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we're in this decomp decomposed by

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Dimension case

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this item two

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and this uh you can use that fact to

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sort of reproduce work by karshan and

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Watts

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um on the same question in a more a more

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special case of a group acting properly

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or more generally a group whose identity

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component acts properly so you can

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reproduce some earlier results with this

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um

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you can also say for a Romanian singular

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foliation

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results by Molino from the 80s tell you

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that these foliations are decomposed by

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dimension so for each K the union of K

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leaves is a sub manifold in fact in that

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case it's embedded

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and I just want to make a little note

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that this second example actually

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subsumes the first one uh using the hoyo

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and Fernandez's linearization uh you can

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say that every properly groupoid induces

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a singular singular Romanian foliation

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on the base

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the its orbits are singular Romanian

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affiliation so then the second one

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really gives you the first one

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and finally using this sort of stranger

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third part where the zero leaves don't

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matter so to speak any one-dimensional

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singular foliation uh the quotient will

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have this

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property of being an isomorphism when

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you pull back

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because when you remove the zero leaves

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you just have a one a regular

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one-dimensional foliation

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so you can use this to generate a lot of

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examples where the quotient is an

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isomorphism

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but of course that's not all the

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singular relations that could exist so

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there's two sort of big

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well big for me I guess unknown

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questions here so first I don't know of

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an example where Pi star is an

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isomorphism

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and it's not covered by this theorem

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so I don't know of a singular

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affiliation that's not one of those

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three types earlier where Pi star is

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still a an isomorphism

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and

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bigger than that I don't know if there's

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a counter example

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it would be I think interesting if there

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was one uh what type affoliation that

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would look like

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and I'm not sure

Key points in the proof

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so in the last I think of 15 seconds I

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guess the only item I'll highlight here

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is that

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um to do this decomposed by Dimension

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case uh the fact requiring them to be

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sub manifolds allows us to do induction

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and sort of on each level for each K you

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have a manifold with a foliation

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so you use the fact you know it's a

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isomorphism when you have a foliation

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and you can sort of piece it together to

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get the entire quotient as an

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isomorphism

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so that's yeah thank you for your

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attention

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[Applause]