The notion of the smooth loop space of a (smooth) manifold is a way to make the set of smooth maps from the circle to the target manifold into an object amenable to the tools of differential topology.
Let be a smooth finite dimensional manifold. Then the set of smooth maps is not usefully the underlying set of a finite dimensional manifold. Therefore to make it into an object that can be studied in differential topology, one has to allow for smooth spaces other than finite dimensional smooth manifolds. That is to say, one has to work in a larger category than just that of finite dimensional smooth manifolds.
There are a variety of suitable categories listed at generalized smooth spaces.
The categories of smooth spaces discussed at generalized smooth spaces are all cartesian closed and thus all admit smooth loop spaces of all objects. However, by restricting to a subcategory that might not be cartesian closed (for example, finite dimensional smooth manifolds) one can ask for another subcategory that is large enough to contain all the resulting smooth loop spaces. For finite dimensional smooth manifolds it turns out that one can give their smooth loop spaces the structure of an infinite dimensional Fréchet manifold.
Depending on taste and application, certain smooth quotient objects of this free loop space may be and are often considered.
In this section, we will describe the structure of a smooth loop space of a smooth manifold as a Fréchet manifold in detail. The construction is quite standard, and a partial list of references and further reading can be found at the end of this page.
We start with a smooth manifold, , of dimension . Note that here manifolds definitely do not have a boundary. For simplicity, we assume that it is orientable. The point of this assumption is that it allows us to identify the model space of with the Fréchet space . In the unorientable case, some components would have this as model space whereas others would have a twisted version of this space.
The key piece of structure needed on is that of a local addition, .
Let be a local addition on . Let be the image of the map . Although (as yet) we know nothing about the topologies of or of , we can at least say that the looped map, , is a bijection.
Let . Define the set by:
Then the preimage of under is naturally identified with . In particular, the zero section of maps to .
We claim that there is a diagram:
such that the bijection at the top takes the image of the left-hand vertical map to the image of the right-hand one. Both of the vertical maps are injective - the right-hand one obviously so, we shall investigate the left-hand one in a moment - and thus the bijection induces a bijection from the lower-left to the lower-right.
The left-hand vertical map, , is defined as follows: the total space of is:
It is an embedded submanifold of . Therefore, a map into is smooth if and only if the compositions with the projections to and to are smooth. Now a map is a section if and only if it projects to the identity on . Therefore, there is a bijection (of sets):
In particular, the map is injective.
We apply to the defining condition for and see that is the preimage under this map of everything of the form in . By construction, is such that if and only if . Hence identifies with .
Finally, note that the zero section of maps to the image of under the zero section of . Since composed with the zero section of is the identity on , the image under the zero section of in is as required.
The resulting map, let us write it , has the following concrete description. Let be a section and let be the corresponding loop in (so that when viewing as a submanifold of ). Then so .
As we have assumed to be orientable, can be trivialised. A smooth such trivialisation defines a linear homeomorphism . We use this to impose a smooth structure on , noting that any two such trivialisations induce the same structure.
To investigate the transition functions, we need two loops. In fact, let’s have two of everything.
Let be local additions with corresponding neighbourhoods , of the diagonal in . Let , be smooth loops in . Let and be the corresponding maps defined as above.
The transition function:
is a diffeomorphism.
We start by characterising the space within . Let be the set:
Note that this is open in as it is the preimage of the open set by the continuous map .
Let and let be the image of (so that . Then for all if and only if for all . That is to say, if and only if . This is precisely the condition that . Since , we see that takes values in if and only if . Since , we conclude that .
Let us define similarly. The idea of the proof that is a diffeomorphism is to show that it is induced by a diffeomorphism .
Let be the map:
The definition of ensures that for and this is the image of . Hence is well-defined. Define similarly. These are both smooth maps.
Notice that is the projection on to the first factor and is the projection on to the second. Thus . Hence is such that for all . Then:
so . Hence we have a map given by:
Similarly we have a map . These are both smooth since the composition with the inclusion into is smooth.
Consider the composition . Expanding this out yields:
The penultimate line is because so . Hence is the inverse of and so is a diffeomorphism. Thus the map is a diffeomorphism from to . We just need to show that this is the transition function. To do this, we show that . The right-hand side is, by definition, which satisfies:
On the other side,
Thus and so the transition functions are diffeomorphisms.
This construction easily generalises quite widely. Very little of the structure of was used at all: that mainly came in in the smooth structure of . The key structure of was the local addition and thus one could regard this as a construction of locally additive spaces. For more on the possible extensions, see the references.
There is a functor
from Fréchet manifolds to diffeological spaces defined in the same way as the well-known functor from smooth manifolds to diffeological spaces: a plot is precisely a smooth map , where is an object in the domain category, e.g. an open subset of some .
(Notice that a theorem of M. Losik says that the functor F is full and faithful, just like that including manifolds into diffeological spaces!)
(where denotes the Yoneda embedding).
This means that a map is a plot if and only if the associated map is smooth. The second diffeology is the one obtained from the functor .
These two diffeologies coincide – in the sense that every plot of one is a plot of the other. In particular, they have the same sets of smooth functions.
A detailed proof of this is in (Waldorf, lemma A.1.7)
The usual notions of G-structures for manifolds, such as orientation, spin structure, string structure, etc. do not carry over directly to their smooth loop spaces, but they are closely related by transgression: a spin structure on is supposed to induce a kind of orientation structure on , a string structure on is supposed to induce a kind of spin structure on .
We say this in detail now.
The ordinary procedure is that we regard a topological space as an object in the (∞,1)-category Top and produce its loop space at a chosen base point as the (∞,1)-pullback of the base point along itself:
In words this is the simple statement that is the space of all homotopies in the (∞,1)-category Top from the map to itself. Any one such homotopy is itself a continuous map from the standard interval to , such that restricted to its endpoints it produces the map . Clearly, these are precisely the loops in , based at .
You might think: well, now let be a smooth manifold regarded as a representable object in the (∞,1)-category of (∞,1)-sheaves of ∞-stacks on Diff or maybe better in on CartSp – i.e. of Lie ∞-groupoids – , we play the same trick and compute the homotopy pullback
to obtain the smooth loop space. But it doesn’t: the here is ! That’s because regarded as a representable object in is really a categorically discrete Lie groupoid: it has a smooth space of objects, but no nontrivial morphisms. And the “homotopies” in the homotopy pullback are homotopies as seen by the morphisms in . There is just the identity morphism in going from to itself, so the homotopy pullback is the point.
Under this identification, a topological space is not identified with a representable object! The only representable object in is of course the point itself, the terminal object. Instead, as the notation above already suggests, under this identification a topological space is really identified with its singular simplicial complex . But that’s really to be thought of as the topological fundamental ∞-groupoid of . We should write
If instead we had interpreted the topological space as a representable object, hence as a categorically discrete object in the -category of topological ∞-groupoids, we would have seen the same phenomenon as for the smooth above: its loop space object would have been the point. From this perspective now it is clear how the abstract notion of loop space object corresponds to the geometrically expected one: for a geometric space , its loop space is the loop space object of its fundamental ∞-groupoid.
This statement can be given sense in all contexts where the underlying topos of our ambient (∞,1)-topos of spaces is a lined topos: we need to know which object is the standard line or interval object. This determines the geometric paths in a space. Taking these geometric paths to be the morphisms of a fundamental ∞-groupoid then makes the geometric paths into “categorical paths”, i.e. into morphisms. These then are what the abstract definition of loop space object can see.
And indeed, whenever the underlying topos of spaces that we are looking at is a lined topos the corresponding (∞,1)-topos comes equipped with a generalization of the topological fundamental ∞-groupoid construction: we can associate to every space its path ∞-groupoid : the morphisms of are given by paths in as seen by the given interval object . All entirely analogous to the familiar situation for Top, only that now we are testing our generalized spaces over test objects in an arbitrary site and are using a correspondingly different notion of interval object.
So for a smooth manifold, regarded as a representable object in the (∞,1)-topos of Lie ∞-groupoids we have now that the homotopy pullback of any point along itself in the path ∞-groupoid does indeed produce the expected Lie ∞-groupoid in
smooth space of objects is the smooth space of smooth loops in based at ;
smooth space of morphisms is the smooth space of smooth -homotopies between smooth loops in
Here we want not the full loop ∞-groupoid, but just some sort of truncation to a 0-groupoid just of loops. There are several choices for how exactly to do this, depending on which higher morphisms we just discard, and which we use to identify 1-morphisms. Whatever we do, we end up with some notion of smooth path groupoid of , whose 1-morphisms are certain smooth quotient space of the smooth space of 1-morphisms in .
which is the smooth subspace of the smooth space of morphisms in of those morphisms that start and end at .
When unwrapping what all this means, one sees that the object that we obtain this way is nothing but the image under the embedding of ordinary sheaves into -stacks of some quotient of the internal hom in the closed monoidal structure on sheaves. Being an internal hom of representables, this is a concrete sheaf and as such it is precisely the smooth loop space regarded as a diffeological space.
A general standard reference on generalized smooth spaces is
The structure of loop spaces as Fréchet manifolds is covered in chapter 42 of KM and in various other articles, many of which cover extensions of the basic construction to other mapping spaces. In particular,
Discussion of G-structures on smooth loop spaces is in the following articles.
There are also sketchy notes in
This entry was created in parallel with this MO thread from which parts of it is taken.