Sc-Smoothness, Retractions and New Models for Smooth Spaces

# Sc-Smoothness, Retractions and New Models for Smooth Spaces

H. Hofer⁢*⁢*⁢*Funding provided by NSF grant DMS-0603957.
USA
hofer@ias.edu
K. Wysocki ⁢*⁢*⁢*Funding provided by NSF grants DMS-0606588, DMS-0906280, and the Oswald Veblen Fund.
Penn State University
USA
wysocki@math.psu.com
E. Zehnder⁢*⁢*⁢*Funding provided by the Fund for Mathematics and the Oswald Veblen Fund.
ETH-Zurich
Switzerland
zehnder@math.ethz.ch

Dedicated to Louis Nirenberg on the Occasion of his 85th Birthday

## 1 Sc-Smoothness and M-Polyfolds

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 Gromov-Witten 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 sc-smoothness. 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 infinite-dimensional, 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.

• Finite-dimensional 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 Sard-Smale’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 Gromov-Witten and SFT-examples 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 Gromov-Witten 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 M-polyfolds. 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 M-polyfold structures on a set of mappings between conformal cylinders which break apart as the modulus tends to infinity. A strong bundle over this M-polyfold is constructed which admits the Cauchy-Riemann operator as an sc-smooth Fredholm section. Its zero-set 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 Sc-Structures on Banach Spaces

Sc-structures on Banach spaces generalize the smooth structure from finite dimensions to infinite dimensions. We first recall the definition of an sc-structure on a Banach space from [9]. In the following stands for .

###### Definition 1.1.

An sc-structure on a Banach space is a nested sequence of Banach spaces ,

 E=:E0⊃E1⊃…⊃E∞:=⋂m∈N0Em,

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 sc-structure on a finite-dimensional vector space is given by the constant structure for all . If is an infinite-dimensional Banach space, the constant structure is not an sc-structure because it fails property (1).

###### Definition 1.2.

A linear map between the two sc-Banach spaces and is called an sc-operator if and if is continuous for every .

We shall need the notion of a partial quadrant in an sc-Banach space .

###### Definition 1.3.

A closed subset of an sc-Banach space is called a partial quadrant if there are an sc-Banach space , a nonnegative integer , and a linear sc-isomorphism so that .

Given a partial quadrant in an sc-Banach space , we define the degeneration index

 dC:C→N0

as follows. We choose a linear sc-isomorphism satisfying . Hence for , we have

 T(x)=(r1,…,rk,w),x∈C,

where and . Then we define the integer by

 dC(x)=♯{i∈{1,…,k}|ri=0}.

It is not difficult to see that this definition is independent of the choice of an sc-linear isomorphism .

Let be a relatively open subset of a partial quadrant in an sc-Banach space . Then the sc-structure on induces the sc-structure on defined by the sequence equipped with the topology of and called the induced sc-structure on . The points of are called smooth points of . We adopt the convention that denotes the set equipped with the sc-structure for all . If and are open subsets equipped with the induced sc-structures, we write for the product equipped with the sc-structure .

###### Definition 1.4.

If is a relatively open subset of a partial quadrant in an sc-Banach space , then its tangent is defined by

 TU=U1⊕E.
###### 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 sc-structure 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

 TBE=(BE)1⊕E={(u,h)|u∈H1,δ1,|u|L2<1, h∈L2}.

The sc-structure on is defined by

 (TBE)m={(u,h) | u∈Hm+1,δm+1,|u|L2<1,h∈Hm,δm}.

The notion of a continuous map between two relatively open subsets of partial quadrants in sc-Banach 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 shift-map. We consider the Hilbert space equipped with the sc-structure introduced in Example 1.5. Then we define the map

 Φ:R2⊕L2(R×S1)→L2(R×S1),(R,ϑ,u)↦(R,ϑ)∗u

where

 ((R,ϑ)∗u)(s,t)=u(s+R,t+ϑ).

The shift-map 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 sc-smooth for the new notion of smoothness which we shall introduce next. The shift map will be an important ingredient in later constructions and its sc-smoothness will be crucial.

### 1.2 Sc-Smooth Maps and M-Polyfolds

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 sc-Banach 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

 lim|h|1→01|h|1⋅|f(x+h)−f(x)−Df(x)h|0=0.
• The map , called the tangent map of , and defined by

 (x,h)↦(f(x),Df(x)h),

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 ,

 T2f:=T(Tf):T2(U)=T(TU)→T2(V)=T(TV),

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 sc-smoothness with the familiar notion of smoothness.

###### Proposition 1.9 (Upper Bound).

Let and be sc-Banach 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

 f:Um+l→Fm

is of class . Then is .

###### Proposition 1.10 (Lower Bound).

Let and be sc-Banach spaces and let be a relatively open subset of a partial quadrant in . If the map is , then the induced map

 f:Um+l→Fm

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 sc-smoothness is a viable concept.

###### Theorem 1.11 (Chain Rule).

Assume that , , and are relatively open subsets of partial quadrants in sc-Banach spaces and let and be . Then the composition is and

 T(g∘f)=(Tg)∘(Tf).

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 sc-smooth retraction and an sc-smooth 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 sc-Banach space . An sc-smooth map is called an -retraction provided it satisfies

 r∘r=r.

A subset of a partial quadrant is called an sc-smooth or -retract (relative to ) if there exists a relatively open subset and an sc-smooth retraction so that

 O=r(U).

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 M-polyfold model is a triple in which is an sc-Banach space, is a partial quadrant of , and is a subset of having the following properties:

• There is an sc-smooth retraction defined on a relative open subset of so that

 O=r(U).
• For every smooth point , the kernel of the map possesses an sc-complement 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 sc-smooth retraction onto defined on a relatively open subset of .

A special M-polyfold model has the form . Such triples can be viewed as the local models for sc-smooth 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 sc-models quite often arise in the following way. We assume that we are given a partial quadrant in an sc-Banach space and a relatively open subset of . Moreover, we assume that for every we have a bounded linear projection

 πv:F→F

into another sc-Banach space . In general, the projection is not an sc-operator. We require that the map

 V⊕F→F,(v,f)↦πv(f)

is sc-smooth. Then we look at the sc-Banach space , the partial quadrant , and the relatively open subset of . Finally, we define the map by

 r(v,f)=(v,πv(f)).

Then the map is an sc–smooth retraction and the set

 K={(v,f)|πv(f)=f}

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 sc-smooth local model and assume that and are two sc-smooth retractions defined on relatively open subsets and of and satisfying . Then

 Tr(TU)=Tr′(TU′).
###### 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

 Tr′(x,h)=Tr′∘Tr(y,k)=T(r′∘r)(y,k)=Tr(y,k)=(x,h)

implying . Similarly one shows that and the proof of the proposition is complete. ∎

The lemma allows us to define the tangent of a local M-polyfold model as follows.

###### Definition 1.15.

The tangent of a local M-polyfold model , denoted by , is defined as a triple

 T(O,C,E)=(TO,TC,TE),

in which is the tangent of the partial quadrant and , where is any sc-smooth retraction onto .

As we already pointed out, the tangent map of the retraction is an sc-smooth retraction. It is defined on the relatively open subset of and is an -retract. Thus, the tangent of a local M-polyfold model is also a a local M-polyfold model.

It is clear what it means that the map between two local M-polyfold 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 M-polyfold models and let and be sc-smooth retractions onto . Then the map is if and only if the same holds true for the map . Moreover, the map

 T(f∘r)|Tr(TU):TO→TO′

does not depend on the choice of an sc-smooth retraction as long as is an sc-smooth 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

 T(f∘r)(x,h)=T(f∘r)(Ts)(x,h)=T(f∘r∘s)(x,h)=T(f∘s)(x,h)

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

 T(f∘r)(x,h)=T(q∘f∘r)(x,h)=Tq∘T(f∘r)(x,h)

for all Consequently, and this map is independent of the choice of an sc-smooth 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

 Tf=T(f∘r)|Tr(TU),

where is any sc-smooth retraction onto . Similarly, is of class provided that the composition is of class where is any sc-smooth retraction defined on relatively open subset of satisfying .

In the following we simply write instead of for the local M-polyfold model, however, we always keep in mind that there are more data in the background.

With the above definition of –maps between local M-polyfold 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 M-polyfold models. Then the composition is an -map and

 T(g∘f)=Tg∘Tf.

The degeneracy index introduced in Section 1.1 generalizes to local models as follows.

###### Definition 1.18.

The degeneracy index of the local M-polyfold model is defined by

 d(x):=dC(x),x∈O.

The next result shows that sc-diffeomorphisms 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 M-polyfold models and and let be an sc-diffeomorphism. Then

 dC(x)=dC′(f(x))

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 sc-smooth 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 sc-complement which is contained in . Then is the sc-direct sum of two closed sc-subspaces, namely,

 N=(N∩({0}⊕W))⊕{(q,p)∈N | (0,p)∈M}=:N1⊕N2.

Indeed, if , then implying that , i.e., . If , then there is a unique decomposition with and implying that and and hence . So,

 (p,q)=(0,q−B)+(A,B)

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

 f∘r(x+τ(p,q))∈O′for all 0<τ<ε.

By assumption, the map is sc-smooth. This implies that the map

 [0,ε)→E′m,τ↦f∘r(x+τ(q,p))

is smooth for every . Now for every we introduce the bounded linear functional defined by . Then we have

 0≤λj(f∘r(x+τ(q,p))),0<τ<ε.

Fix . Then, since where as and , we conclude that

 0≤1τλj(f(x)+τDf(x)(p,q)+om(τ))=λj(Df(x)(p,q)+om(τ)τ)

which after letting gives

 0≤λj(Df(x)(p,q)).

If , then and the above inequality with replacing implies that . Consequently,

 λj(Df(x)(p,q))=0

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 sc-diffeomorphism , 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 M-polyfold model, is an open subset of , and is a homeomorphism. Two such charts

 (φ,U,(O,C,E))and(φ′,U′,(O′,C′,E′))

are called sc-smoothly compatible if the composition

 φ′∘φ−1:O→O′

is an sc-smooth map between local M-polyfold models.

###### Definition 1.20.

An sc-smooth atlas for consists of a collection of charts such that the associated open sets cover and the transition maps are sc-smooth. 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 M-polyfold.

Observe that an M-polyfold inherits a filtration

 X=X0⊃X1⊃…⊃X∞=⋂m∈N0Xi

The same is true for subsets of . The tangent is defined by the usual recipe used in the infinite-dimensional situation. Consider tuples

 t=(x,φ,U,(O,C,E),h)

in which is a chart, , and . Two such tuples,

 (x,φ,U,(O,C,E),h)and(x′,φ′,U′,(O′,C′,E′),h′),

are called equivalent if and

 T(φ′∘φ−1)(φ(x),h)=(φ′(x),h′).

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

 TX=⋃x∈X1{x}×TxX.

One easily verifies that is an M-polyfold in a natural way with specific charts given by

 Tφ:TU→TO,Tφ([x,φ,U,(O,C,E),h])=(φ(x),h).

Here is the union of all with ,

 TU=⋃x∈U1{x}×TxX.
###### Remark 1.21.

For SFT we need a more general class of spaces called polyfolds. They are essentially a Morita equivalence class of ep-groupoids, where the latter is a generalization of the notion of an étale proper Lie groupoid to the sc-world. 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 M-polyfold structures. Further all category operations are sc-smooth 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 two-dimensional 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 sc-structure given by for all . Next we choose a smooth compactly supported function having -norm equal to ,

 ∫Rβ(s)2ds=1.

Then we define a family of sc-operators as follows. For , we put , and for , we define

 πt(f)=⟨f,β(⋅+e1t)⟩L2⋅β(⋅+e1t),f∈E.

In other words, for we take the -orthogonal projection onto the -dimensional subspace span by the function with argument shifted by . Define

 r:R⊕E→R⊕E,r(t,f)=(t,πt(f)).

Clearly, . We shall prove below that the map is sc-smooth. Consequently, is an sc-smooth retraction and the image of , which is the subset

 {(t,0) | t≤0}∪{(t,s⋅β(⋅+e1t)) | t>0,  s∈R}

of , is an sc-smooth 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 sc-smooth atlas, i.e. is a generalization of a manifold in the sc-smooth world, see Figure 2.

Observe that with this M-polyfold 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

 ρt:E→E

where and where is the sc-Banach space from above, such that for and has an infinite-dimensional image for , and such that the map is sc-smooth. 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 sc-smooth. We recall that taking a strictly increasing sequence of real numbers starting with , the space is equipped with the sc-structure where . We choose a smooth function having support contained in the compact interval and satisfying . Define the map by

 Φ(t,u)={⟨u,β(⋅+e1t)⟩L2⋅β(⋅+e1t)t>00t≤0.
###### 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,

 |F(k)(t,s)|≤p(t), (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

 ⟨u,F(t,⋅)⟩2L2=∫R|u(s)|2F(t,s)2 ds=∫It|u(s)|2F(t,s)2e2δmse−2δms ds≤Ce−2δme1t∫It|u(s)|2e−2δms ds (2)

where we have used that

 maxs∈It|F(s,t)|2e2δms≤Ce−δme1t.

Similarly one finds

 ⟨u,F(j)(t,⋅)⟩2L2=∫R|u|2F(j)(t,s)2 ds=∫It|u|2F(j)(t,s)2e2δmse−2δms ds≤Cp(t)2e−2δme1t∫It|u|2e−2δms ds. (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,

 |Φ(t,u)−Φ(0,u0)|2m=⟨u,F(t,⋅)⟩L2∑i≤m∫It|∂isF(t,s)|2e−2δms ds≤Ce−2δme1t(∫It|u|2e−2δms ds)⋅(∫Ite−2δms ds)≤C∫It|u|2e−2δms ds≤C(∫It|u0|2e−2δms ds+∫It|h|2e−2δms ds)≤C∫It|u0|2e−2δms ds+C⋅ε.

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.

###### (Sk).

The map is and if . Moreover, if is a projection onto the factor of , then the composition at the point