ScSmoothness, Retractions and New Models for Smooth Spaces
Dedicated to Louis Nirenberg on the Occasion of his 85th Birthday
Contents
1 ScSmoothness and MPolyfolds
In paper [9], the authors have described a generalization of differential geometry based on the notion of splicings. The associated Fredholm theory in polyfolds, presented in [9, 10, 11], is a crucial ingredient in the functional analytic foundation of the Symplectic Field Theory (SFT). The theory also applies to the Floer theory as well as to the GromovWitten theory and quite generally should have applications in nonlinear analysis, in particular in studies of families of elliptic pde’s on varying domains, which can even change their topology.
A basic ingredient for the generalization of differential geometry is a new notion of differentiability in infinite dimensions, called scsmoothness. The goal of this paper is to describe these ideas and, in particular, to provide some of the “hard” analysis results which enter the polyfold constructions in symplectic field theory (SFT). The advantage of the polyfold Fredholm theory can be summarized as follows.

Many spaces, though they do not carry a classical smooth structure, can be equipped with a weak version of a smooth structure. The local models for the spaces, be they finite or infinitedimensional, can even have locally varying dimensions.

Since the notion of the smooth structure is so weak, there are many charts so that many spaces carry a manifold structure in the new smoothness category.

Finitedimensional subsets in good position in these generalized manifold inherit an induced differentiable structure in the familiar sense.

There is a notion of a bundle. Smooth sections of such bundles, which, under a suitable coordinate change, can be brought into a sufficiently nice form, are Fredholm sections. A Fredholm section looks nice (near a point) only in a very particular coordinate system and not necessarily in the smoothly compatible other ones. Since we have plenty of coordinate systems, many sections turn out to be Fredholm.

The zero sets of Fredholm sections lie in the smooth parts of the big ambient space, so that they look smooth in all coordinate descriptions (systems). The invariance of the properties of solution sets under arbitrary coordinate changes is, of course, a crucial input for having a viable theory.

There is an intrinsic perturbation theory, and moreover, a version of SardSmale’s theorem holds true. In applications, for example to a geometric problem, one might try to make the problem generic by perturbing auxiliary geometric data. As the GromovWitten and SFTexamples show, this is, in general, not possible and one needs to find a sufficiently large abstract universe, which offers enough freedom to construct generic perturbations. The abstract polyfold Fredholm theory provides such a framework.

Important for the applications is a version of this new Fredholm theory for an even more general class of spaces, called polyfolds. In this case the generic solution spaces can be thought of locally as a finite union of (classical) manifolds divided out by a finite group action. Moreover, the points in these spaces carry rational weights. Still the integration of differential forms can be defined for such spaces and Stokes’ theorem is valid. This is used in order to define invariants. The GromovWitten invariants provide an example.
The current paper develops the analytical foundations for some of the applications of the theory described above. It also provides examples illustrating the ideas.
The organization of the paper is as follows.
The introductory chapter describes the new notions of smoothness for spaces and mappings leading, in particular, to novel local models of spaces, which generalize manifolds and which are called Mpolyfolds. The general Fredholm theory in this analytical setting is outlined and an outlook to some applications is given, the proofs of which are postponed to chapter 3.
The second chapter is of technical nature and is devoted to detailed proofs of the new smoothness results which are crucial for many applications.
The third chapter illustrates the concepts by constructing Mpolyfold structures on a set of mappings between conformal cylinders which break apart as the modulus tends to infinity. A strong bundle over this Mpolyfold is constructed which admits the CauchyRiemann operator as an scsmooth Fredholm section. Its zeroset consists of the holomorphic isomorphisms between cylinders of various sizes. Since the solution set carries a smooth structure, this has interesting functional analytic consequences for the behavior of families of holomorphic mappings.
Acknowledgement: We would like to thank J. Fish for useful comments and suggestions.
1.1 ScStructures on Banach Spaces
Scstructures on Banach spaces generalize the smooth structure from finite dimensions to infinite dimensions. We first recall the definition of an scstructure on a Banach space from [9]. In the following stands for .
Definition 1.1.
An scstructure on a Banach space is a nested sequence of Banach spaces ,
so that the following two conditions are satisfied:

The inclusion maps are compact operators.

The vector space is dense in every .
Points in are called smooth points. What just has been defined is a compact discrete scale of Banach spaces which is a standard object in interpolation theory for which we refer to [17]. Our interpretation as a generalization of a smooth structure on seems to be new. The only scstructure on a finitedimensional vector space is given by the constant structure for all . If is an infinitedimensional Banach space, the constant structure is not an scstructure because it fails property (1).
Definition 1.2.
A linear map between the two scBanach spaces and is called an scoperator if and if is continuous for every .
We shall need the notion of a partial quadrant in an scBanach space .
Definition 1.3.
A closed subset of an scBanach space is called a partial quadrant if there are an scBanach space , a nonnegative integer , and a linear scisomorphism so that .
Given a partial quadrant in an scBanach space , we define the degeneration index
as follows. We choose a linear scisomorphism satisfying . Hence for , we have
where and . Then we define the integer by
It is not difficult to see that this definition is independent of the choice of an sclinear isomorphism .
Let be a relatively open subset of a partial quadrant in an scBanach space . Then the scstructure on induces the scstructure on defined by the sequence equipped with the topology of and called the induced scstructure on . The points of are called smooth points of . We adopt the convention that denotes the set equipped with the scstructure for all . If and are open subsets equipped with the induced scstructures, we write for the product equipped with the scstructure .
Definition 1.4.
If is a relatively open subset of a partial quadrant in an scBanach space , then its tangent is defined by
Example 1.5.
A good example which illustrates the concepts, and is also relevant for SFT, is as follows. We choose a strictly increasing sequence of real numbers starting with . We consider the Banach spaces and where the space consists of those elements in having weak partial derivatives up to order which if weighted by belong to . Using Sobolev’s compact embedding theorem for bounded domains and the assumption that the sequence is strictly increasing, one sees that the sequence defines an scstructure on . We take as the partial quadrant the whole space and let be the open unit ball centered at in . Then the tangent of is given by
The scstructure on is defined by
The notion of a continuous map between two relatively open subsets of partial quadrants in scBanach spaces is as follows.
Definition 1.6.
A map is said to be if for all and if the induced maps are continuous.
Example 1.7.
An important example used later on is the shiftmap. We consider the Hilbert space equipped with the scstructure introduced in Example 1.5. Then we define the map
where
The shiftmap is as proved in Proposition 4.1. It is clearly not differentiable in the classical sense. However, in Proposition 4.2 we shall prove that the map is scsmooth for the new notion of smoothness which we shall introduce next. The shift map will be an important ingredient in later constructions and its scsmoothness will be crucial.
1.2 ScSmooth Maps and MPolyfolds
Having defined an appropriate notion of continuity we define what it means that the map is of class . This is the notion corresponding to a map being in our sc–framework.
Definition 1.8.
Let and be relatively open subsets of partial quadrants and in scBanach spaces and , respectively. An map is said to be if for every there exists a bounded linear operator so that the following holds:

If and , then

The map , called the tangent map of , and defined by
is of class .
In general, the map , will not(!) be continuous if the space of bounded linear operators is equipped with the operator norm. However, if we equip it with the compact open topology it will be continuous. The maps between finite dimensional Banach spaces are the familiar maps.
Proceeding inductively, we define what it means for the map to be or . Namely, an –map is said to be an –map if it is and if its tangent map is . By Definition 1.8, the tangent map of ,
is of class . If the tangent map is , then is said to be , and so on. The map is , if it is for all .
Useful in our applications are the next two propositions which relate the scsmoothness with the familiar notion of smoothness.
Proposition 1.9 (Upper Bound).
Let and be scBanach spaces and let be a relatively open subset of a partial quadrant in . Assume that is an map so that for every and every the induced map
is of class . Then is .
Proposition 1.10 (Lower Bound).
Let and be scBanach spaces and let be a relatively open subset of a partial quadrant in . If the map is , then the induced map
is of class for every and every .
The proofs of the two propositions will be carried out in section 2.1. In view of the following chain rule, the scsmoothness is a viable concept.
Theorem 1.11 (Chain Rule).
Assume that , , and are relatively open subsets of partial quadrants in scBanach spaces and let and be . Then the composition is and
The proof can be found in [9]. We would like to point out that the proof relies on the assumption that the inclusion operators between spaces in the nested sequence of Banach spaces are compact.
The next definition introduces the notions of an scsmooth retraction and an scsmooth retract. This will be the starting point for a differential geometry based on new local models.
Definition 1.12.
Let be a relatively open subset of a partial quadrant in an scBanach space . An scsmooth map is called an retraction provided it satisfies
A subset of a partial quadrant is called an scsmooth or retract (relative to ) if there exists a relatively open subset and an scsmooth retraction so that
If is an retraction, then its tangent map is also an retraction. This follows from the chain rule. Next comes the crucial definition of the new local models of smooth spaces.
Definition 1.13.
A local Mpolyfold model is a triple in which is an scBanach space, is a partial quadrant of , and is a subset of having the following properties:

There is an scsmooth retraction defined on a relative open subset of so that

For every smooth point , the kernel of the map possesses an sccomplement which is contained in .

For every , there exists a sequence of smooth points converging to in and satisfying .
The choice of in the above definition is irrelevant as long as it is an scsmooth retraction onto defined on a relatively open subset of .
A special Mpolyfold model has the form . Such triples can be viewed as the local models for scsmooth space without boundary whereas the more general triples are models for spaces with boundaries with corners. In the case without boundary the conditions (2) and (3) of Definition 1.13 are automatically satisfied.
In our applications the local scmodels quite often arise in the following way. We assume that we are given a partial quadrant in an scBanach space and a relatively open subset of . Moreover, we assume that for every we have a bounded linear projection
into another scBanach space . In general, the projection is not an scoperator. We require that the map
is scsmooth. Then we look at the scBanach space , the partial quadrant , and the relatively open subset of . Finally, we define the map by
Then the map is an sc–smooth retraction and the set
is an sc–smooth retract. We call this particular retraction, due to its partially linear character, a splicing. For more details on splicings we refer to [9].
Lemma 1.14.
Let be an scsmooth local model and assume that and are two scsmooth retractions defined on relatively open subsets and of and satisfying . Then
Proof.
If , then there exists so that . Consequently, , and hence . Similarly, one sees that . If , then for a pair . Moreover, so that . From it follows using the chain rule that
implying . Similarly one shows that and the proof of the proposition is complete. ∎
The lemma allows us to define the tangent of a local Mpolyfold model as follows.
Definition 1.15.
The tangent of a local Mpolyfold model , denoted by , is defined as a triple
in which is the tangent of the partial quadrant and , where is any scsmooth retraction onto .
As we already pointed out, the tangent map of the retraction is an scsmooth retraction. It is defined on the relatively open subset of and is an retract. Thus, the tangent of a local Mpolyfold model is also a a local Mpolyfold model.
It is clear what it means that the map between two local Mpolyfold models is . In order to define –maps between local models we need the following lemma.
Lemma 1.16.
Let be a map between two local Mpolyfold models and let and be scsmooth retractions onto . Then the map is if and only if the same holds true for the map . Moreover, the map
does not depend on the choice of an scsmooth retraction as long as is an scsmooth retraction onto .
Proof.
Assume that is . Since is , the chain rule implies that the composition is . Using the identity , we conclude that is . Interchanging the role of and , the first part of the lemma is proved. If , then and using the identity and the chain rule, we conclude
Now take any sc–smooth retraction defined on a relatively open subset of the the partial quadrant in satisfying . Then so that . Application of the chain rule yields the identity
for all Consequently, and this map is independent of the choice of an scsmooth retraction onto . ∎
In view of the lemma, we define the map between local models to be of class if the composition is of class . If this is the case, we define the tangent map as
where is any scsmooth retraction onto . Similarly, is of class provided that the composition is of class where is any scsmooth retraction defined on relatively open subset of satisfying .
In the following we simply write instead of for the local Mpolyfold model, however, we always keep in mind that there are more data in the background.
With the above definition of –maps between local Mpolyfold models, the next theorem is an immediate consequence of the chain rule stated in Theorem 1.11.
Theorem 1.17 (General Chain Rule).
Assume that and are maps between local Mpolyfold models. Then the composition is an map and
The degeneracy index introduced in Section 1.1 generalizes to local models as follows.
Definition 1.18.
The degeneracy index of the local Mpolyfold model is defined by
The next result shows that scdiffeomorphisms recognize the difference between a straight boundary and a corner. Of course, this is true also for the usual notion of smoothness but not for homeomorphisms.
Theorem 1.19 (Boundary Recognition).
Consider local Mpolyfold models and and let be an scdiffeomorphism. Then
at each point .
Proof.
We slightly modify the argument in [9]. First, assuming that the theorem holds at smooth points , we show that it also holds at points on level . Indeed, take a point . By the condition (3) of Definition 1.13, we find a sequence of smooth points converging to in and satisfying . That is, By assumption, . Since , it follows immediately that . The same argument applied to shows that we must have equality at every point on the level in .
Now we prove the equality for smooth points. Without loss of generality we may assume that and and, similarly, and . Take a smooth point and let be an scsmooth retract defined on the relatively open subset of satisfying . Abbreviate by the kernel of at the point . The kernel is a closed subspace of possessing, by the condition (2) of Definition 1.13, an sccomplement which is contained in . Then is the scdirect sum of two closed scsubspaces, namely,
Indeed, if , then implying that , i.e., . If , then there is a unique decomposition with and implying that and and hence . So,
with and .
For our smooth point , we denote by be the set of indices at which . In particular, . We denote by the subspace of consisting of vectors with for . From the decomposition , we see that has codimension in . The same arguments apply at the point . Abbreviate by the set of indices at which . With the kernel of the map at , we let be the codimension subspace of consisting of all vectors satisfying for . We observe that the subspaces and are precisely the tangent spaces and .
Take any smooth vector . Then for and sufficiently small , so that for . Hence
By assumption, the map is scsmooth. This implies that the map
is smooth for every . Now for every we introduce the bounded linear functional defined by . Then we have
Fix . Then, since where as and , we conclude that
which after letting gives
If , then and the above inequality with replacing implies that . Consequently,
for all and all vectors . At this point we have proved that . Recalling that and are the tangent spaces and and that is an sc–linear isomorphism, we see that the subspace has codimension in . It follows that . Repeating the same argument for the scdiffeomorphism , we obtain the opposite inequality. Hence we conclude that preserves indeed the degeneracy index for smooth points and the proof of Theorem 1.19 is complete. ∎
At this point it is clear that one can take potentially any recipe from differential geometry and construct new objects. We begin with the recipe for a manifold. Let be a metrizable space. A chart for is a triple in which is a local Mpolyfold model, is an open subset of , and is a homeomorphism. Two such charts
are called scsmoothly compatible if the composition
is an scsmooth map between local Mpolyfold models.
Definition 1.20.
An scsmooth atlas for consists of a collection of charts such that the associated open sets cover and the transition maps are scsmooth. Two atlases are equivalent if the union is also an sc–smooth atlas. The space equipped with an equivalence class of sc–smooth atlases is called an Mpolyfold.
Observe that an Mpolyfold inherits a filtration
The same is true for subsets of . The tangent is defined by the usual recipe used in the infinitedimensional situation. Consider tuples
in which is a chart, , and . Two such tuples,
are called equivalent if and
An equivalence class of a tuple is called a tangent vector at . The collection of all tangent vectors at is denoted by and the tangent is defined by
One easily verifies that is an Mpolyfold in a natural way with specific charts given by
Here is the union of all with ,
Remark 1.21.
For SFT we need a more general class of spaces called polyfolds. They are essentially a Morita equivalence class of epgroupoids, where the latter is a generalization of the notion of an étale proper Lie groupoid to the scworld. In a nutshell, this is a category in which the class of objects as well as the collection of morphisms are sets, which in addition carry Mpolyfold structures. Further all category operations are scsmooth and between any two objects there are only finitely many morphisms. A polyfold is a generalization of the modern notion of orbifold as presented in [16]. We note that the notion of proper has to be reformulated for our generalization as is explained in [11].
Next, we illustrate the previous concepts by an example. We shall construct a connected subset of a Hilbert space which is an sc–smooth retract and which has one and twodimensional parts. (By using the fact that the direct sum of two separable Hilbert spaces is isomorphic to itself one can use the following ideas to construct connected sc–smooth subsets which have pieces of many different finite dimensions.)
Example 1.22.
We take a strictly increasing sequence of weights starting at and equip the Hilbert space with the scstructure given by for all . Next we choose a smooth compactly supported function having norm equal to ,
Then we define a family of scoperators as follows. For , we put , and for , we define
In other words, for we take the orthogonal projection onto the dimensional subspace span by the function with argument shifted by . Define
Clearly, . We shall prove below that the map is scsmooth. Consequently, is an scsmooth retraction and the image of , which is the subset
of , is an scsmooth retract. Observe that the above retract is connected and consists of  and dimensional parts.
Out of this example one can define more retractions which have parts of any finite dimension. The above example is enough to show that the subspace of the plane obtained by taking the open unit disk and attaching a closed interval to it by mapping the end points to the unit circle in fact admits an scsmooth atlas, i.e. is a generalization of a manifold in the scsmooth world, see Figure 2.
Observe that with this Mpolyfold structure satisfies , i.e. the induced filtration is constant. This implies that has a tangent space at every point. As a consequence of later constructions one could even construct a family of sc–projections
where and where is the scBanach space from above, such that for and has an infinitedimensional image for , and such that the map is scsmooth. Hence the local jumps of dimension can be quite stunning. Of course, we can even combine the two previous examples in various ways.
We shall prove that the retraction is scsmooth. We recall that taking a strictly increasing sequence of real numbers starting with , the space is equipped with the scstructure where . We choose a smooth function having support contained in the compact interval and satisfying . Define the map by
Lemma 1.23.
The map is of class .
Proof.
It is clear that the restriction is smooth. Abbreviate for and . Observe that for the th derivative of with respect to we have the estimate,
(1) 
where is a polynomial of degree in the variable and of degree in the variable It depends on and has nonnegative coefficients. In addition, the function has its support contained in the interval . Note that if is sufficiently small.
Then if and if is small we can estimate
(2) 
where we have used that
Similarly one finds
(3) 
Using the estimate (2) we shall first show that the map is continuous at a point . Take satisfying and set . By definition, and so,
Since as , one concludes that as in . So far we have proved that the map is . In order to prove that the map is we proceed by induction. Our inductive statements are as follows.
.
The map is and if . Moreover, if is a projection onto the factor of , then the composition at the point