Polyfold Regularization of Constrained Moduli Spaces
aa r X i v : . [ m a t h . S G ] J u l POLYFOLD REGULARIZATION OF CONSTRAINEDMODULI SPACES
BENJAMIN FILIPPENKO
Abstract.
We introduce tame sc-Fredholm sections and slices ofsc-Fredholm sections. A slice is a notion of subpolyfold that is com-patible with the sc-Fredholm section and has finite locally constantcodimension. We prove that a sc-Fredholm section restricted to aslice is a tame sc-Fredholm section with a drop in Fredholm indexgiven by the codimension of the slice. Moreover, we prove that thesubspace of a tame polyfold that satisfies a transverse sc-smoothconstraint in a finite dimensional smooth manifold is a slice of anytame sc-Fredholm section that is compatible with the constraint.As a corollary, we obtain fiber products of tame sc-Fredholm sec-tions. We describe applications to Gromov-Witten invariants, theconstruction [1] of the Piunikhin-Salamon-Schwarz maps for gen-eral closed symplectic manifolds, and avoiding sphere bubbles inmoduli spaces of pseudoholomorphic curves of expected dimension0 and 1.
Contents
1. Introduction 21.1. Results 21.2. Applications 72. Sc-calculus: the normal form of a local sc-smooth submersionto R n R n References 891.
Introduction
Polyfold theory, developed by Hofer, Wysocki, and Zehnder in thegrowing literature [5][6][7][8][9][10][11][12][13], is an analog of classicalnonlinear Fredholm theory designed to realize compact moduli spaces,e.g. Gromov-Witten moduli spaces and Floer trajectory spaces, as zerosets of sc-Fredholm sections of polyfold bundles. See the polyfold sur-vey [3] for an overview and a discussion of applications. The abstractpolyfold machinery provides perturbations such that the perturbed sc-Fredholm section is transverse to zero, and hence the perturbed solu-tion space has smooth structure by a polyfold implicit function theo-rem. Crucially, the perturbations can be chosen so that the perturbedsolution space remains compact. This process, beginning with the de-scription of the compact moduli space as the zero set of a sc-Fredholmsection and ending with the smooth compact perturbed solution space,is colloquially referred to as “polyfold regularization” of a moduli space.Often in symplectic topology we wish to constrain moduli spaces ofpseudoholomorphic curves to consist of those curves satisfying inter-section conditions with submanifolds. For example, Gromov-Witteninvariants can be defined as counts of curves whose marked points eval-uate to submanifolds. Any fiber product of moduli spaces over eval-uation maps is another example of such a constraint. The evaluationmaps are usually not transverse on the moduli space, however they ex-tend to the ambient polyfold and here they are submersive. Using thistransversality, we construct in this paper the constrained polyfold andthe constrained sc-Fredholm section. This provides an abstract toolto regularize constrained moduli spaces, whenever the original modulispace is given as the zero set of a sc-Fredholm section.We state our theorems in Section 1.1. Then in Section 1.2 we explainapplications to Gromov-Witten invariants (Section 1.2.1), constructingthe Piunikhin-Salamon-Schwarz maps (Section 1.2.2) to prove the weakArnold conjecture (see [1] for details), and avoiding sphere bubbles inperturbed moduli spaces of expected dimension 0 and 1 (Section 1.2.3).1.1.
Results.
The main goal of this paper is to prove the M -polyfoldand ep-groupoid (with boundary and corners) versions of the followingclassical Facts 1.1, 1.2 from non-linear Fredholm theory over Banachmanifolds. OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 3
Fact 1.1. (Restrictions of Fredholm sections to sub-Banachmanifolds)
Consider a Banach manifold B , a smooth Banach bundle p : E → B , and a Fredholm section s : B → E with Fredholm index ind x ( s ) for x ∈ B . If ˜ B ⊂ B is a codimension- n sub-Banach manifold,then the restriction ˜ p : ˜ E := p − ( ˜ B ) → ˜ B is a smooth Banach bundleand the restricted section ˜ s := s | ˜ B : ˜ B → ˜ E is a Fredholm section withFredholm index satisfying ind x (˜ s ) = ind x ( s ) − n for x ∈ ˜ B . (cid:3) The proof of Fact 1.1 is an exercise in differential topology and func-tional analysis. Then Fact 1.2 follows from Fact 1.1 together with thecodimension- n Banach manifold charts provided by the normal formof a C local submersion to R n , for which we provide a proof (in thecontext of boundary and corners) in Lemma 2.1 for later use. Fact 1.2. (Transverse preimages are sub-Banach manifolds)
Consider a Banach manifold B , a finite dimensional smooth manifold Y together with a codimension- n submanifold N ⊂ Y , and a smoothmap f : B → Y . Assume that f is transverse to N .Then, ˜ B := f − ( N ) is a codimension- n sub-Banach manifold of B .In particular, if s : B → E is a Fredholm section of a smooth Banachbundle p : E → B , then the restriction ˜ p : ˜ E := p − ( ˜ B ) → ˜ B is asmooth Banach bundle and the restricted section ˜ s := s | ˜ B : ˜ B → ˜ E is aFredholm section with Fredholm index satisfying ind x (˜ s ) = ind x ( s ) − n for x ∈ ˜ B . (cid:3) The M -polyfold versions (with boundary and corners) of the abovefacts are our main theorems, Theorem 1.3 and Theorem 1.5, which westate in this section and prove in Sections 5.2, 5.3. The generalizationsof these theorems to the ep-groupoid case, which are required in ap-plications to handle nontrivial isotropy groups, are Corollary 6.7 andCorollary 6.8. We obtain fiber products of tame sc-Fredholm sectionsas Corollary 7.3.Throughout, we denote M -polyfolds B , strong M -polyfold bundles ρ : E → B , and sc-Fredholm sections σ : B → E .The central objects developed in this paper are tame sc-Fredholmsections σ : B → E (Definition 5.4) and slices ˜
B ⊂ B (Definition 5.7) ofsc-Fredholm sections. A tame sc-Fredholm section is a new notion thatis stronger than a sc-Fredholm section (Definition 5.3). To be tame,we require the change of coordinates that brings the local sc-Fredholmfillers into basic germ form to be a linear sc-isomorphism on the base. Aslice is a new notion of a finite codimension M -polyfold ˜ B embedded in B . These notions are related by our main Theorems 1.3, 1.5. Roughly,given a tame sc-Fredholm section σ : B → E and a sc-smooth map
BENJAMIN FILIPPENKO f : B → Y to a finite dimensional manifold Y that is σ -compatiblytransverse (Definition 5.8) to a submanifold N ⊂ Y , then f − ( N ) isa slice of σ . Moreover, given a slice ˜ B ⊂ B of a sc-Fredholm section σ : B → E , the restriction σ | ˜ B is tame sc-Fredholm. In Section 5.1,we explain why the Cauchy-Riemann section ∂ J : B → E is tame sc-Fredholm and why evaluation maps ev : B → Y at marked points are ∂ J -compatibly transverse to every submanifold N ⊂ Y .Before we state the theorems, we briefly recall some polyfold no-tation. See Section 5 for more detail. Given a M -polyfold B , thereis a filtration B = B ⊃ B ⊃ · · · induced by the sc-structures in localcharts. Each filtration level B m has its own topology which is not thesubspace topology, but the inclusions are continuous and dense. Thesmooth points of B are the subset B ∞ := ∩ m ≥ B m , which is also densein B .For x ∈ B , the degeneracy index d B ( x ) is the number of distinct localboundary faces of B intersecting at x .The following result is the M -polyfold analog of Fact 1.1; see Sec-tion 5.2 for the proof. See Corollary 6.7 for the generalization to ep-groupoids. Theorem 1.3. (Restrictions of sc-Fredholm sections to slices) (I)
Consider a tame M -polyfold B and a slice ˜ B ⊂ B (Definition 5.7).Then, ˜ B is a tame M -polyfold with atlas induced by the sliced chartswith respect to ˜ B ⊂ B . For x ∈ ˜ B , the codimension codim x ( ˜ B ⊂ B ) iswell-defined and locally constant in ˜ B , i.e. it is equal to codim x ( ˜ B ⊂ B ) in an open neighborhood of x in ˜ B . For x ∈ ˜ B ∞ , the degeneracy indexsatisfies d ˜ B ( x ) = d B ( x ) . (II) Consider, in addition, a tame strong bundle ρ : E → B . If ˜ B ⊂ B is a slice of ρ , then the restriction ˜ ρ := ρ | ˜ E : ˜ E := ρ − ( ˜ B ) → ˜ B is atame strong bundle with atlas induced by the sliced bundle charts for ρ with respect to ˜ B ⊂ B . (III) Consider, in addition, a sc-Fredholm section σ : B → E . If ˜ B ⊂ B is a slice of σ , then the restriction ˜ σ = σ | ˜ B : ˜ B → ˜ E is a tamesc-Fredholm section (Definition 5.4) of ˜ ρ with tame sc-Fredholm chartsinduced by the sliced sc-Fredholm charts for σ with respect to ˜ B ⊂ B .For x ∈ ˜ B ∞ , the index satisfies ind x (˜ σ ) = ind x ( σ ) − codim x ( ˜ B ⊂ B ) .If σ − (0) is compact and ˜ B ∞ ⊂ B ∞ is closed, then ˜ σ − (0) is compact. OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 5
Remark 1.4.
There are three notions of a slice ˜ B ⊂ B (Definition 5.7)appearing in Theorem 1.3: (I) a slice ˜ B ⊂ B of a tame M -polyfold,(II) a slice ˜ B ⊂ B of a tame strong bundle ρ : E → B , and (III) aslice ˜ B ⊂ B of a sc-Fredholm section σ : B → E . Each successivenotion requires further compatibility of the subset ˜ B ⊂ B with the addi-tional structure. This is in contrast to the Banach manifold situationin Fact 1.1 where bundles and Fredholm sections automatically restrictto any finite codimension sub-Banach manifold ˜ B ⊂ B .The required compatibilities are roughly as follows. (I) There arecharts on B to R n -sliced sc-retracts O (Definition 3.2) that locally iden-tify ˜ B ⊂ B with the induced tame sc-retract ˜ O ⊂ O from Lemma 3.3.(II) There are bundle charts on ρ to R n -sliced bundle retracts K (Def-inition 3.4) covering R n -sliced sc-retracts O . In this case, ρ − ( ˜ B ) islocally identified with the induced tame bundle retract ˜ K ⊂ K fromLemma 3.5, and the restriction ρ − ( ˜ B ) → ˜ B is locally identified withthe induced tame local bundle model ˜ K → ˜ O . (III) There are sc-Fredholm charts for σ at every x ∈ ˜ B ∞ to R n -sliced sc-Fredholm germs O → K (Definition 3.8). In this case, the restriction σ | ˜ B : ˜ B → ˜ E islocally identified with the induced tame sc-Fredholm germ ˜ O → ˜ K fromLemma 3.10.The reason for the further requirements in the M -polyfold setting isthe non-trivial sc-retractions and sc-Fredholm fillings: compatibility of ˜ B with the sc-retractions on B does not imply compatibility of ˜ B withthe bundle retractions on E or with the local fillings of σ . We now state our main theorem, which is the M -polyfold analog ofthe classical Fact 1.2; see Section 5.3 for the proof, and Corollary 6.8for the generalization to ep-groupoids. See the following Remark 1.6for a discussion of the technicalities in the statement. Given a M -polyfold B , there is a m -shifted M -polyfold B m for each m ≥ B , . . . , B m − of B discussed above. Theorem 1.5. (Transverse preimages are slices of sc-Fredholmsections) (I)
Consider a tame M -polyfold B , a smooth manifold Y together witha codimension- n submanifold N ⊂ Y , and a sc-smooth map f : B → Y . Assume that f is transverse to N (Definition 5.8).Then, there exists an open neighborhood ˜ B ⊂ f − ( N ) ∩ B BENJAMIN FILIPPENKO of f − ( N ) ∩ B ∞ such that ˜ B is a slice of B with codim x ( ˜ B ⊂ B ) = n for every x ∈ ˜ B = ˜ B ∩ B . In particular, ˜ B is a tame M -polyfold withdegeneracy index satisfying d ˜ B ( x ) = d B ( x ) for all x ∈ ˜ B ∞ . (II) Consider, in addition, a tame strong bundle ρ : E → B . Then,there exists a possibly smaller neighborhood ˜ B in (I) that is a slice ofthe bundle ρ | E : E → B . In particular, the restriction ˜ ρ := ρ | ˜ E : ˜ E := ( ρ | E ) − ( ˜ B ) → ˜ B is a tame strong bundle. (III) Consider, in addition, a tame sc-Fredholm section σ : B → E (Definition 5.4) of ρ . Assume that f is σ -compatibly transverse to N (Defintion 5.8). Then, there exists a possibly smaller neighborhood ˜ B in (II) that is a slice of the tame sc-Fredholm section σ | B : B → E .In particular, the restriction ˜ σ := σ | ˜ B : ˜ B → ˜ E is a tame sc-Fredholm section of ˜ ρ with index satisfying ind x (˜ σ ) = ind x ( σ ) − n for all x ∈ ˜ B ∞ . If N is closed as a subset of Y and σ − (0) is compact, then ˜ σ − (0) is compact. Remark 1.6. (i)
The notion of σ -compatibly transverse (Definition 5.8) requirescompatibility between the tangent map D x f at x ∈ f − ( N ) ∩B ∞ with the change of coordinates on the base of the local sc-Fredholm filling of σ at x that brings the filling into basicgerm form. See Section 5.1 for an explanation why evalua-tion maps f = ev at marked points are compatible with theCauchy-Riemann section σ = ∂ J in applications. (ii) The reason Theorem 1.5 holds only in some neighborhood ˜ B ofthe smooth points of the preimage f − ( N ) ∩ B ∞ is as follows.The tame sc-retracts modeling ˜ B are built from the subspaceof the tangent space T x B at x ∈ f − ( N ) that is mapped bythe tangent map D x f : T x B → T f ( x ) Y onto T f ( x ) N , and thistangent space T x B has the structure of a sc-Banach space onlyat smooth points x ∈ B ∞ . So we can only hope to constructa sc-retract modeling a neighborhood of x in f − ( N ) aroundsmooth points x . (iii) The neighborhood ˜ B is open only in the -level of the preim-age f − ( N ) ∩ B because, in the proof of the local submersionnormal form in sc-calculus (Lemma 2.3), we must -shift the OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 7 sc-Banach space to obtain C regularity in order to use theclassical C local submersion normal form (Lemma 2.1). Remark 1.7.
All manifolds Y and submanifolds N ⊂ Y in this paperare smooth, finite dimensional, and without boundary: ∂N = ∅ and ∂Y = ∅ . This suffices for our initial intended applications to evaluation mapswith target a closed symplectic manifold Y .It is possible to generalize our theorems to the case where both N and Y are smooth finite dimensional orbifolds with boundary and cor-ners. This generalization will be useful in applications. For example,polyfolds B constructed for regularization of moduli spaces of pseudo-holomorphic curves in symplectic topology come with an everywheresubmersive sc-smooth forgetful map B → Y to the Deligne-Mumfordspace Y consisting of all domains of curves in B . The Deligne-Mumfordspace Y usually has an orbifold structure with non-trivial isotropy, andwhen the domains have boundary, Y has boundary and corners. Applications.
We discuss applications of our polyfold results topseudoholomorphic curves in symplectic manifolds.First, we provide an alternative construction of the Gromov-Witteninvariants, defined using polyfold theory in [13] as integrals of differ-ential forms over a perturbed moduli space, as counts of points ina 0-dimensional constrained moduli space, where the constraints areevaluation maps at marked points that are required to evaluate tosubmanifolds. Then we describe the construction of the HamiltonianPiunikhin-Salamon-Schwarz maps for general closed symplectic mani-folds, which is carried out in detail in [1] using our theorems to con-struct the fiber product of Morse moduli spaces and Symplectic FieldTheory polyfolds, providing a proof of the weak Arnold conjecture.Last, we describe a method for perturbing expected dimension 0 and1 moduli spaces so that the perturbed moduli space does not containany curves with a sphere bubble.The applicable polyfold results in this paper are the ep-groupoid gen-eralizations of the theorems presented above, because in applicationsthere will be nontrivial isotropy groups. The ep-groupoid results arein Sections 6,7.Throughout, let (
Y, ω ) be a closed symplectic manifold of dimension2 n .1.2.1. Gromov-Witten invariants.
The Gromov-Witten invariants aredefined in [13] for general closed symplectic manifolds as integrals of
BENJAMIN FILIPPENKO differential forms over a perturbed moduli space. The results in thispaper provide an alternative construction as counts of 0-dimensionalperturbed moduli spaces of curves satisfying intersection conditionswith submanifolds.For a given homology class A ∈ H ( Y ) and integers g, m ≥ g + m ≥
3, the Gromov-Witten invariant with respect to thefundamental class [ M g,m ] of the Deligne-Mumford space M g,m of closedgenus g curves with m marked points is a multilinear map Ψ A,g,m : H ∗ ( Y ; R ) ⊗ m → R . This map is defined for α , . . . , α m ∈ H ∗ ( Y ; R ) with degrees satisfying | α | + · · · + | α m | = 2 c ( A ) + (2 n − − g ) + 2 m as follows, and forother choices of α i it is defined to be 0. For an ω -compatible almostcomplex structure J on Y , a sc-Fredholm section ∂ J : X A,g,m → E of a polyfold bundle E → X A,g,m with Fredholm index ind ( ∂ J ) =2 c ( A ) + (2 n − − g ) + 2 m is constructed in [13] such that thesolution set ∂ − J (0) = M g,m ( Y, A, J )is the Gromov compactified moduli space of J -holomorphic curves in Y of genus g with m marked points that represent the class A . Then theabstract perturbation theory in [7] provides a sc + -multisection Λ of thebundle so that the perturbed solution space S ( ∂ J , Λ ) is a smooth com-pact oriented weighted branched orbifold of dimension dim S ( ∂ J , Λ ) = ind ( ∂ J ) over which we can integrate differential forms using the inte-gration theory from [11]. For | α | + · · · + | α m | = ind ( ∂ J ), the Gromov-Witten invariant is defined by Ψ A,g,m ( α ⊗ · · · ⊗ α m ) := Z S ( ∂ J ,Λ ) ev ∗ ( α ) ∧ · · · ∧ ev ∗ m ( α m ) , where ev k : X A,g,m → Y is evaluation at the k -th marked point.We now explain how to use the results in this paper to constructthe Gromov-Witten invariant as a count of points in a 0-dimensionalmoduli space of curves evaluating to submanifolds of Y . For k =1 , . . . , m let L k ⊂ Y be an oriented submanifold such that [ L k ] = P D ( α k ) ∈ H ∗ ( Y ; R ), where P D denotes the Poincar´e dual. Then thecodimension of L k in Y is equal to the degree | α k | , and so L × · · · × L m OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 9 is a codimension- ind ( ∂ J ) submanifold of Y m . The total evaluation map ev × · · · × ev m : X A,g,m → Y m u ( ev ( u ) , . . . , ev m ( u ))records the positions of all the marked points. Consider the subspace X L ,...,L m A,g,m := ( ev × · · · × ev m ) − ( L × · · · × L m )of X A,g,m , which consists of those curves whose k -th marked point eval-uates to L k for every k = 1 , . . . , m . Then M L ,...,L m g,m ( Y, A, J ) = ∂ − J (0) ∩ X L ,...,L m A,g,m is the compactified moduli space of J -holomorphic curves u ∈ M g,m ( Y, A, J )satisfying the point constraints ev k ( u ) ∈ L k for all k = 1 , . . . , m .To perturb this constrained moduli space M L ,...,L m g,m ( Y, A, J ) so thatit is cut out transversely, we first apply Corollary 6.8 to obtain a de-scription of M L ,...,L m g,m ( Y, A, J ) as the zero set of a sc-Fredholm sectionof a polyfold bundle, as follows. The hypotheses of Corollary 6.8 aresatisfied because the total evaluation map ev × · · · × ev m : X A,g,m → Y m is submersive, and moreover it is ∂ J -compatibly transverse to L × · · · × L m as explained in Section 5.1. Hence, applying the corol-lary, there exists an open neighborhood ˜ X ⊂ X L ,...,L m A,g,m of the con-strained moduli space M L ,...,L m g,m ( Y, A, J ) such that the restricted sec-tion ∂ J | ˜ X : ˜ X → E | ˜ X is sc-Fredholm and has Fredholm index 0. More-over, the solution space ∂ J | − X (0) = M L ,...,L m g,m ( Y, A, J )is the constrained moduli space. The perturbation theory in [7] thenprovides a sc + -multisection ˜ Λ of the restricted bundle E | ˜ X → ˜ X sothat the perturbed solution space S ( ∂ J | − X (0) , ˜ Λ ) is a smooth compactoriented weighted branched orbifold of dimension 0.Notice that, even after perturbation by ˜ Λ , all curves u ∈ S ( ∂ J | − X (0) , ˜ Λ )are guaranteed to satisfy the constraints ev k ( u ) ∈ L k for k = 1 , . . . , m since S ( ∂ J | − X (0) , ˜ Λ ) is contained in X L ,...,L m A,g,m . Morally, the weightedcount S ( ∂ J | − X (0) , ˜ Λ ) is the Gromov-Witten invariant, however toprove the equality S ( ∂ J | − X (0) , ˜ Λ ) = Ψ A,g,m ( α ⊗ · · · ⊗ α m ) one wouldneed to construct the perturbations so that ˜ Λ = Λ | ˜ X ; see [17] for aprecise treatment of this comparison. The PSS morphism.
Let H : S × Y → R be a nondegenerateHamiltonian and ( f, g ) a Morse-Smale pair of a Morse function f : Y → R and a Riemannian metric g . The original application [1] thatmotivates this project is the construction of the Piunikhin-Salamon-Schwarz (PSS) morphism P SS : H Morse ∗ ( Y ; f, g ) → H F loer ∗ ( Y ; H ) , where H Morse ∗ ( Y ; f, g ) is the Morse homology of ( f, g ) and H F loer ∗ ( Y ; H )is the Floer homology of H . This map is an isomorphism, proving theweak Arnold conjecture in full generality. This has been done underthe assumption that ( Y, ω ) is semi-positive in [16] and [15].The moduli spaces from which the PSS morphism is constructed areas follows. Consider a critical point p of f , a 1-periodic orbit γ ofthe Hamiltonian vector field associated to H , and a singular homologyclass A ∈ H ( Y ). Let M ( p, Y ) denote the compactified moduli space ofhalf-infinite gradient flow lines τ : ( −∞ , → Y that limit to p on theirinfinite end and evaluate to ev ( τ ) := τ (0) in the unstable manifold of p .The compactification includes broken flow lines that start at p , breakat finitely many other critical points, and end in a half-infinite flow lineoriginating from the final critical point at which breaking occurs andevaluating to its unstable manifold. Then M ( p, Y ) can be given thestructure of a smooth compact manifold with boundary and corners(see for example [18]) equipped with a smooth evaluation map ev p : M ( p, Y ) → Y. Fix smooth capping discs on each periodic orbit of H and an ω -compatiblealmost complex structure J on Y . Then let M ( γ, A ) denote the modulispace appearing in Symplectic Field Theory [2] consisting of smoothmaps C → X satisfying the Cauchy-Riemann equation ∂ J near 0 andthe Floer equation near ∞ (with a fixed interpolation in between givenby a cutoff function that turns off the Hamiltonian term in Floer’sequation near 0), and such that the map glued to the capping discon γ represents the homology class A . The compactified moduli space M ( γ, A ) also includes configurations with broken Floer trajectories andsphere bubble trees. There is an evaluation map ev γ : M ( γ, A ) → Y given by evaluating at 0 ∈ C . The PSS moduli spaces are then thefiber products M ( p, γ, A ) := M ( p, Y ) × ev p ev γ M ( γ, A ) . See Figure 1 for a diagram of an element of M ( p, γ, A ). OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 11 p p ′ γ ′ γ ev p ( τ ) = ev γ ( u ) Figure 1.
An element ( τ, u ) in the moduli space M ( p, γ, A ). The red lines represent an element τ ∈M ( p, Y ) consisting of a Morse trajectory from p to p ′ and a half-infinite Morse trajectory starting at p ′ andevaluating to ev p ( τ ) in the unstable manifold of p ′ . Thegreen near ev γ ( u ) represents the neighborhood of 0 ∈ C on which the map C → Y satisfies the J -holomorphiccurve equation. As the map limits to the Hamiltonianorbit γ ′ , the J -curve equation interpolates in the blue re-gion to Floer’s equation represented in magenta. A Floertrajectory from γ ′ to γ has broken off. The green circlesrepresent bubbled off J -holomorphic spheres. The eval-uations ev p ( τ ) = ev γ ( u ) agree since M ( p, γ, A ) is a fiberproduct.If all choices can be made so that M ( p, γ, A ) is smooth (and of theexpected dimension), then the coefficient of P SS ( p ) on the generator( γ, A ) is defined to be the count M ( p, γ, A ) if the expected dimensionis 0, and the coefficient is 0 otherwise. Now, for general Y , the compactmoduli space M ( γ, A ) will not be cut out transversely for any J , andhence has no reason to be smooth. Moreover, even if M ( γ, A ) is cutout transversely, there is no reason to expect that the fiber productwith M ( p, Y ) is transverse.In [1], these transversality issues are overcome using the results inthis paper as follows. The Symplectic Field Theory polyfolds in [5]include polyfold bundles E ( γ, A ) → X ( γ, A ) and sc-Fredholm sections σ ( γ, A ) : X ( γ, A ) → E ( γ, A )with solution set the SFT moduli space σ ( γ, A ) − (0) = M ( γ, A ) . Here X ( γ, A ) is the polyfold of broken and nodal maps of the same formas those in M ( γ, A ) but not necessarily satisfying any equation, andthe section σ ( γ, A ) is the equation that maps in M ( γ, A ) are required to satisfy. Moreover there is a sc-smooth evaluation map ev γ : X ( γ, A ) → Y which evaluates at 0 ∈ C . This evaluation map is a submersion on theambient space X ( γ, A ), and it restricts to the evaluation map on themoduli space ev γ : M ( γ, A ) → Y .Applying the fiber product result Corollary 7.3 to the zero sectionof the rank-0 bundle over the Morse moduli space M ( p, Y ) and the sc-Fredholm section σ ( γ, A ), we obtain an open neighborhood X ( p, γ, A )of the zero set of the fiber product section M ( p, Y ) × ev p ev γ X ( γ, A ) → E ( γ, A )( τ, u ) σ ( γ, A )( u )such that the restricted section σ ( p, γ, A ) : X ( p, γ, A ) → E ( p, γ, A ) := E ( γ, A ) | X ( p,γ,A ) is sc-Fredholm with index ind ( σ ( p, γ, A )) = dim M ( p, Y )+ ind ( σ ( γ, A )) − n . Its zero set is compact and equal to the PSS moduli space σ ( p, γ, A ) − (0) = M ( p, γ, A ) . The perturbation theory in [7] then provides a sc + -multisection Λ ofthe polyfold bundle E ( p, γ, A ) → X ( p, γ, A ) so that the perturbedsolution space S ( σ ( p, γ, A ) , Λ ) is a smooth compact weighted branchedorbifold; see Figure 2 for a diagram of an element in S ( σ ( p, γ, A ) , Λ ).The weighted count of points in this perturbed moduli space providesthe definition of the PSS map. That is, the coefficient of P SS ( p ) on( γ, A ) is given by h P SS ( p ) , ( γ, A ) i := S ( σ ( p, γ, A ) , Λ ) . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 13 p p ′ γ ′ γ ev p ( τ ) = ev γ ( u ) Figure 2.
An element ( τ, u ) in the perturbed mod-uli space S ( σ ( p, γ, A ) , Λ ) ⊂ M ( p, Y ) × ev p ev γ X ( γ, A ).The red lines represent the broken Morse trajectory τ ∈ M ( p, Y ) with finite end evaluating to ev p ( τ ). Theregion from ev γ ( u ) to γ ′ is a map C → Y limiting tothe Hamiltonian orbit γ ′ at ∞ . The region in between γ ′ and γ represents a cylinder limiting to these orbits on itstwo ends, and the circles represent attached sphere bub-bles. Together the white regions represent the element u ∈ X ( γ, A ). They are colored white, in contrast toFigure 1, to indicate that they do not necessarily satisfyany equation due to the perturbation Λ . The evalua-tions ev p ( τ ) = ev γ ( u ) still agree since S ( σ ( p, γ, A ) , Λ ) iscontained in the fiber product.1.2.3. Avoiding sphere bubbles in expected dimension and . A com-mon mantra in symplectic topology is that “sphere bubbling is a codimension-2 phenomenon” and hence sphere bubbles do not appear in regularizedmoduli spaces of dimension 0 and 1. The notions of a sliced sc-retract(Definition 3.2) and a sliced sc-Fredholm germ (Definition 3.8) intro-duced in this paper provide a method for making this precise in thecontext of polyfold theory.For simplicity, we consider the case of a curve with 1 interior node,for example a curve with a single sphere bubble. This curve naturallysits inside a R -sliced sc-retract ( O , R × C, R × E ) which locally modelsa neighborhood of the curve in an ambient polyfold. The R -sliced sc-retraction r : U → U with image the sc-retract r ( U ) = O is (roughly)the splicing [7, Def. 2.18] obtained from pregluing at the node. Inparticular, O is homeomorphic to the image of the pregluing map, soconceptually we identify them. Since pregluing with gluing parameter0 ∈ R preserves the node, the induced tame sc-retract ˜ O = O ∩ ( { } × C ) (Lemma 3.3) consists of the curves in O that have 1 interior node.This formalizes the notion that a curve with 1 interior node sits insidea codimension-2 stratum consisting of nearby nodal curves. Moreover, the Cauchy-Riemann section ∂ J : O → K of the localbundle model K → O is a R -sliced sc-Fredholm germ, and hence byLemma 3.10 its restriction to ˜ O is sc-Fredholm with index satisfying ind ( ∂ J | ˜ O ) = ind ( ∂ J ) −
2. If the original section satisfies ind ( ∂ J ) ≤
1, i.e. the expected dimension of the moduli space is ≤
1, then itsrestriction to the nodal curves in ˜ O satisfies ind ( ∂ J | ˜ O ) <
0. So, afterperturbing the restricted section, the transversely cut out zero set mustbe empty as a smooth object with negative dimension. Extending thisperturbation over all of O , the perturbed zero set will not intersect ˜ O ,meaning that the perturbed zero set will not contain any nodal curves.This perturbation extension result will be part of a future work,which to be applied must include an inductive procedure that perturbsand extends starting with the highest codimension strata correspond-ing to solutions with the most nodes. Indeed, a curve with k ≥ k distinct codimension-2 strata, and thisintersection is a codimension-2 k stratum. One must perform the localperturbations and extensions coherently with respect to the intersec-tions of these nodal strata. Acknowledgments:
I am deeply grateful to my PhD advisor, Ka-trin Wehrheim, for warmly inviting me into this area of research, lead-ing me to fruitful areas to explore, keeping my vision clear as I navigate,and teaching me how to write. Katrin’s support throughout my grad-uate studies was fundamental to my completion of this research.2.
Sc-calculus: the normal form of a local sc-smoothsubmersion to R n The purpose of this section is to establish the normal form (Lemma 2.3)of a sc-smooth local submersion f : [0 , ∞ ) s × E → R n , where E is asc-Banach space. This is a sc-calculus analog of the classical localsubmersion normal form (Lemma 2.1) in the case where E is an ordi-nary Banach space E . In the Banach case, the normal form followsfrom the inverse function theorem for C maps between open subsetsof quadrants [0 , ∞ ) s × E . The inverse function theorem does not holdin sc-calculus [4], however, using relationships between classical dif-ferentiability and sc-differentiability, we leverage the classical normalform to prove the normal form in sc-calculus. The key ingredient isthat the target of these submersions is finite dimensional R n , on whichall sc-structures are trivial (i.e. all levels are isomorphic to the infinitylevel). OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 15
For completeness, we now provide a proof of the normal form in theclassical Banach case. We view a quadrant [0 , ∞ ) s × E as a Banachspace with boundary and corners. It suffices to consider neighborhoodsin [0 , ∞ ) s × E of a point x that sits in the maximally degenerate corner x ∈ { } × E . Lemma 2.1. (Normal form of a C local submersion to R n ) Consider a Banach space E , an open subset U ⊂ [0 , ∞ ) s × E for some s ≥ , and a C map f : U → R n . Suppose that, for some x ∈ U ∩ ( { } s × E ) , the tangent map ( d x f ) | { } s × E : E → R n of the restrictionto the corner f | U ∩ ( { } s × E ) is surjective.Then, for any complement L of K := ker( d x f ) | { } s × E in E , thereexist open neighborhoods x ∈ ˆ U ⊂ U and U ′ ⊂ R n × [0 , ∞ ) s × K suchthat, writing v ∈ [0 , ∞ ) s and e ∈ E , the map g : ˆ U → U ′ ( v, e ) ( f ( v, e ) , v, pr ( e )) is a C -diffeomorphism, where pr : E = K ⊕ L → K is the projectionalong L .Proof. Denote the restriction of f to the corner by f := f | U ∩ ( { }× E ) : U ∩ ( { } × E ) → R n . Then d x f : E → R n is surjective by hypothesis. Let L be any com-plement of K = ker d x f in E . In particular, note that this means therestriction(1) d x f | L : L → R n is an isomorphism. Writing v ∈ [0 , ∞ ) s and e ∈ E , define the map g : U → R n × [0 , ∞ ) s × K ( v, e ) ( f ( v, e ) , v, pr ( e )) , where pr : E = K ⊕ L → K is the projection along L . Note that g is C since f is C and pr is C ∞ .We claim that the tangent map d x g : R s × E → R n × R s × K ( v, e ) ( d x f ( v, e ) , v, pr ( e ))is an isomorphism. To verify injectivity, suppose (0 , ,
0) = ( d x f ( v, e ) , v, pr ( e )).Then e ∈ ker( pr ) = L , and moreover 0 = d x f (0 , e ) = d x f ( e ) means e ∈ K , hence e = 0. To verify surjectivity, let ( p, v, k ) ∈ R n × R s × K .Since the map (1) is an isomorphism, there exists l ∈ L such that d x f ( l ) = p − d x f ( v, k ). It follows that d x f ( v, k + l ) = d x f ( v, k ) + d x f ( l ) = p and hence d x g ( v, k + l ) = ( d x f ( v, k + l ) , v, pr ( k + l )) =( p, v, k ). So d x g is an isomorphism, as claimed.Since g is a C map whose tangent map d x g at x is an isomorphism,the inverse function theorem for C maps between quadrants of Ba-nach spaces applies: There exists an open neighborhood ˆ U ⊂ U of x and an open set U ′ ⊂ R n × [0 , ∞ ) s × K such that the restriction g | ˆ U : ˆ U → U ′ is a C -diffeomorphism, as claimed. (cid:3) We briefly review basics about sc-calculus on sc-Banach spaces from[7, Sec. 1.1] to prepare for Lemma 2.3. A sc-Banach space [7, Def. 1.1]is a sequence of Banach spaces and continuous linear injections E := (cid:0) E ← ֓ E ← ֓ · · · (cid:1) such that the map E m +1 ֒ → E m is a compact operator for every m ≥ E ∞ := ∩ m ≥ E m is dense in every E m . We call E ∞ the smooth points of E . By a subset S of a sc-Banach space E we mean a subset S of E , which then inducessubsets S m := S ∩ E m ⊂ E m for all m ≥ ,S ∞ := S ∩ E ∞ ⊂ E ∞ . In particular, if U is open in E then U m ⊂ E m is open for all m ≥ U is open in the sc-Banach space E .A sc-subspace F ⊂ E [7, Def. 1.4] is a closed subspace F ⊂ E suchthat the induced subsets F m := F ∩ E m define a sc-Banach space F =( F ← ֓ F ← ֓ · · · ) . Given sc-subspaces F , F ′ ⊂ E such that, for every m ≥
0, the Banach space E m splits as a direct sum E m = F m ⊕ F ′ m ,we say that there is a sc-splitting E = F ⊕ F ′ and that F , F ′ are sc-complements in E . Given a sc-subspace F ⊂ E , the quotient space E / F has the structure of a sc-Banach space with m -level E m /F m ; see[7, Prop. 1.2]. The following fact is established in the proof of [7,Prop. 1.4]. We reproduce the proof here for completeness. Lemma 2.2.
Consider a sc-Banach space E and a sc-subspace F ⊂ E such that the quotient E / F is finite dimensional. Then, there exists asc-complement F ⊕ L = E and moreover L ⊂ E ∞ .Proof. Consider the sc-continuous quotient map p : E → E / F . Since E ∞ ⊂ E is a dense linear subspace and p : E → E /F is surjective, itfollows that p ( E ∞ ) is a dense linear subspace of the finite dimensionalspace E /F , and so we have p ( E ∞ ) = E /F . Hence, choosing any See, for example, [14, Thm. 2.2.4].
OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 17 basis of E /F , there are preimages of the basis elements in E ∞ , andthese preimages span a subspace L ⊂ E ∞ . We claim that L is sc-complementary to F in E . Indeed, for every m ≥ E m ⊂ E is dense and so p m : E m → E /F is surjective. Moreover,we have F m = ker( p m ) and the restriction p m : L → E /F is anisomorphism, so F m ⊕ L = E m holds. (cid:3) For every l ≥ l -shifted sc-Banach space defined by E l := ( E l ← ֓ E l +1 ← ֓ · · · ) , that is, ( E l ) m = E m + l for all m ≥ . Conceptually, we are forgetting about finitely many levels. Note that l -shifting does not change the ∞ -level. Analogously, for any subset S ⊂ E , we define S l ⊂ E l by ( S l ) m := S m + l . In particular, if U ⊂ E isopen, then U l ⊂ E l is open for all l ≥ E × F of sc-Banach spaces has a natural sc-structure with m -level given by ( E × F ) m := E m × F m equipped withany standard Banach norm on a Cartesian product. In this paper,we use the convention that all norms on Cartesian products are thesum norm || ( · , · ) || E m × F m = || · || E m + || · || F m , which is equivalent to allstandard choices.The finite dimensional space E = R n has a canonical sc-structuregiven by E m = R n equipped with the standard Euclidean norm forevery m ≥
0, and where every inclusion E m +1 → E m is the identitymap.The tangent space [7, Def. 1.8] of a sc-Banach space E is the sc-Banach space T E := E × E , with sc-structure given by ( T E ) m = E m +1 × E m for m ≥
0. Given anopen subset U ⊂ [0 , ∞ ) s × E for some s ≥
0, its tangent space is
T U := U × ( R s × E ) . Consider sc-Banach spaces E , F , and open subsets U ⊂ [0 , ∞ ) s × E and V ⊂ [0 , ∞ ) s ′ × F . Then a map f : U → V is called sc or sc-continuous [7, Def. 1.7] if, for all m ≥
0, we have f ( U m ) ⊂ V m and the map f : U m → V m is continuous. A sc map f : U → V is called sc with tangent map [7, Def. 1.9](2) T f : T U → T V defined by
T f : U × ( R s × E ) → V × ( R s ′ × F )( x, ξ ) ( f ( x ) , D x f ( ξ ))if, for every x ∈ U , there exists a bounded linear operator D x f : R s × E → R s ′ × F such that, for ξ ∈ E satisfying x + ξ ∈ U ,lim | ξ | → | f ( x + ξ ) − f ( x ) − D x f ( ξ ) | | ξ | = 0 , holds, and moreover such that T f is sc . Iterating the definition of sc yields the notions of sc k for k ≥ ∞ ); seethe discussion after [7, Def. 1.9].An important note is that, for a sc map f : U → V and x ∈ U , thebounded linear operator D x f : R s × E → R s ′ × F is not necessarilysc-continuous when considered as a map between sc-Banach spaces D x f : R s × E → R s ′ × F ; that is, continuity on levels higher than0 can fail. However, if x ∈ U ∞ is a smooth point, then by [7, Prop. 1.5]the map D x f is indeed a sc-operator [7, Def. 1.2], i.e. a sc-continuouslinear map. For this reason, we consider only smooth points x in thefollowing lemma so that the kernel of D x f is a sc-Banach space as thekernel of a sc-operator. Lemma 2.3. (Normal form of a sc-smooth local submersion to R n ) Consider a sc-Banach space E , an open subset U ⊂ [0 , ∞ ) s × E forsome s ≥ , and a sc-smooth map f : U → R n . Suppose that, for somesmooth point x ∈ U ∞ ∩ ( { }× E ) , the tangent map ( D x f ) | { }× E : E → R n of the restriction to the corner f | U ∩ ( { }× E ) is surjective.Then, for any sc-complement L of K := ker( D x f ) | { }× E in E , thereexist open neighborhoods x ∈ ˆ U ⊂ U and U ′ ⊂ R n × [0 , ∞ ) s × K suchthat, writing v ∈ [0 , ∞ ) s and e ∈ E , the map g : ˆ U → U ′ ( v, e ) ( f ( v, e ) , v, pr ( e )) is a sc-diffeomorphism, where pr : E = K ⊕ L → K is the projectionalong L . Moreover, for all m ≥ , the map g | ˆ U m : ˆ U m → U ′ m is a C m +1 -diffeomorphism.In particular, the following statements hold: A sc-complement L of K exists by Lemma 2.2, since the surjection ( D x f ) | { }× E : E → R n induces an isomorphism E / K ∼ = R n . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 19 • The composition f ◦ g − : U ′ → R n is projection onto the R n -coordinate. • g preserves the [0 , ∞ ) s -coordinate.Proof. We claim that the Banach space E (the 1-level of the sc-Banachspace E ), the open subset U ⊂ [0 , ∞ ) s × E , and the C -map f := f | U : U → R n satisfy the hypotheses of Lemma 2.1 (the local submersion normal formin the classical Banach case) at the given point x . First of all, themap f is indeed C by [7, Prop. 1.7]. By [7, Prop. 1.5], we have d x f = ( D x f ) | R s × E . We now deduce the surjectivity of the tangent map(3) ( d x f ) | { }× E = ( D x f ) | { }× E : E → R n . By hypothesis, the map ( D x f ) | { }× E is surjective. Then since E ⊂ E is dense, it follows that ( D x f ) | { }× E ( E ) is a dense linear subspace of R n and hence is equal to R n , proving the claimed surjectivity of themap (3).Let L ⊂ E ∞ be any sc-complement of K = ker( D x f ) | { }× E in E ,which exists by Lemma 2.2 since the surjection ( D x f ) | { }× E : E → R n induces an isomorphism E / K ∼ = R n . In particular, on the 1-level, wehave K ⊕ L = E . Notice that K is the kernel of the map (3).We have shown that the map f satisfies the hypotheses of the clas-sical local submersion normal form (Lemma 2.1), yielding an openneighbhorhood ˆ U ⊂ U of x and an open subset U ′ ⊂ R n × [0 , ∞ ) s × K such that, writing v ∈ [0 , ∞ ) s and e ∈ E , the map g : ˆ U → U ′ ( v, e ) ( f ( v, e ) , v, pr ( e ))is a C -diffeomorphism, where pr : E = K ⊕ L → K is the projectionalong L .We may view ˆ U and U ′ as open neighborhoods in the sc-calculussense, i.e. ˆ U ⊂ [0 , ∞ ) s × E and U ′ ⊂ R n × [0 , ∞ ) s × K . We claim that g is a sc-diffeomorphism. First of all, it is sc-smoothsince f is sc-smooth by hypothesis and since the projection pr : E = K ⊕ L → K is sc-smooth.To show that g − is sc-smooth, we show that it satisfies the conditionsof [7, Prop. 1.8]. Let m, l ≥
0. We must show that g − induces a map g − | U ′ m + l : U ′ m + l → ˆ U m that is C l +1 . It suffices to show that, for all m ≥ g − restricts to a C m +1 -map g − | U ′ m : U ′ m → ˆ U m , because then given m, l ≥ U ′ m + l g − −−→ ˆ U m + l ֒ → ˆ U m is C m + l +1 since the inclusion ˆ U m + l ֒ → ˆ U m iscontinuous and linear hence C ∞ . So, to complete the proof of thelemma, it suffices to show that g : ˆ U m → U ′ m is a C m +1 -diffeomorphism for all m ≥ f | ˆ U m : ˆ U m = ˆ U ∩ ( R s × E m +1 ) → R n is C m +1 . It follows that the restriction g | ˆ U m : ˆ U m → U ′ m is C m +1 .To see that g | ˆ U m : ˆ U m → U ′ m is a bijection, note first that injectivityholds since it is a restriction of the bijection g . To see surjectivity, notefirst that, since g is surjective onto all of U ′ , it suffices to show that g ( v, e ) ∈ U ′ m = ⇒ ( v, e ) ∈ ˆ U m . Note that U ′ m ⊂ R n × [0 , ∞ ) s × K m +1 .So, from the definition of g , we have g ( v, e ) ∈ U ′ m = ⇒ pr ( e ) ∈ K m +1 ⊂ E m +1 . Since e − pr ( e ) ∈ L ⊂ E ∞ , we conclude that e ∈ E m +1 . Henceindeed ( v, e ) ∈ ˆ U m = ˆ U ∩ ( R s × E m +1 ) holds, proving surjectivity of g | ˆ U m onto U ′ m . The same reasoning shows that the classical tangentmap d ( g | ˆ U m ) : ˆ U m × ( R s × E m +1 ) → U ′ m × ( R n × R s × K m +1 )is bijective.The inverse ( g − ) | U ′ m = ( g | ˆ U m ) − : U ′ m → ˆ U m is C m +1 because it is the inverse of a C m +1 map with invertible deriva-tive. We have shown that g | ˆ U m : ˆ U m → U ′ m is a C m +1 -diffeomorphism,completing the proof of the lemma. (cid:3) We briefly review general partial quadrants, which up to a linearchange of coordinates are the same as the standard quadrants (5), i.e.of the form [0 , ∞ ) s × E for some sc-Banach space E . This level ofgenerality makes constructions more convenient and is equivalent toworking with standard partial quadrants only.A partial quadrant [7, Def. 1.6] in a sc-Banach space E is a closedconvex subset C ⊂ E such that there exists another sc-Banach space E ′ and a linear sc-isomorphism(4) Ψ : E → R s × E ′ satisfying Ψ ( C ) = [0 , ∞ ) s × E ′ for some s ≥ . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 21
That is, all partial quadrants come from applying a linear change ofcoordinates to a partial quadrant in the standard form :(5) C = [0 , ∞ ) s × E ′ ⊂ R s × E ′ = E . The degeneracy index [7, Def. 1.10] d C : C → N is defined for x ∈ C by d C ( x ) := { i ∈ { , . . . , s } | the i -th coordinate of Ψ ( x ) is 0 } , which is independent of the choice of Ψ by [7, Lem. 1.1]. Conceptually,the degeneracy index of a point in a partial quadrant is the local notionof boundary and corner index in a M -polyfold.Later, we need the following properties of the degeneracy index. Let C be a partial quadrant of a sc-Banach space E and n ≥
0. Then R n × C is a partial quadrant of R n × E and(6) d R n × C ( p, x ) = d C ( x ) for all ( p, x ) ∈ R n × C. Let C i be a partial quadrant of E i for i = 1 , . Then C × C is a partialquadrant of E × E and(7) d C × C ( x , x ) = d C ( x ) + d C ( x ) for all ( x , x ) ∈ C × C . We recall from [7, Def. 2.16] the following linear sc-subspace E x ⊂ E associated to a point x in a partial quadrant C ⊂ E . First assume that C is in the standard form (5) and write x = ( x , . . . , x s , e x ) ∈ C. Then,the sc-subspace(8) E x := { ( v , . . . , v s , e ) ∈ R s × E ′ | v i = 0 if x i = 0 } ⊂ E conceptually is the tangent space of the intersection of all of the facesof C that contain x . For a general partial quadrant C ⊂ E and x ∈ C ,the subspace E x ⊂ E is given by(9) E x := Ψ − (( R s × E ′ ) Ψ ( x ) ) , where Ψ is any linear isomorphism of the form (4).3. Slices: the local picture
Sliced sc-retracts.
In this section, we introduce the new notionof R n -sliced sc-retracts (Definition 3.2), which we use later as the localmodels in our definition of a slice ˜ B ⊂ B (Definition 5.7) of a tame M -polyfold B (Definition 5.1). We prove in Lemma 3.3 that a R n -slicedsc-retract O induces a tame sc-retract ˜ O ⊂ O which has codimension- n tangent spaces T x ˜ O ⊂ T x O at every x ∈ ˜ O . The global definition ofa slice ˜ B ⊂ B is then a subspace such that around every point x ∈ ˜ B there is a M -polyfold chart to a R n x -sliced sc-retract O that locallyidentifies ˜ B with the induced tame sc-retract ˜ O . We first recall the local structure of tame M -polyfolds: tame sc-retracts (Definition 3.1). Consider a relatively open subset U of apartial quadrant C in a sc-Banach space E . A sc-smooth map r : U → U satisfying r ◦ r = r is called a sc-smooth retraction (or sc-retraction) [7, Def. 2.1] on U , and the image O := r ( U ) of such a map is called a sc-smooth retract (or sc-retract) . The triple ( O , C, E ) is also called a sc-retract [7, Def. 2.2].We note that the notion of a smooth retract makes sense in the clas-sical Banach space setting, i.e. given an ordinary Banach space E , wecan define a smooth retract O to be any image O = r ( U ) of a smoothmap r : U → U that satisfies r ◦ r = r , where U ⊂ [0 , ∞ ) s × E is open. However, modeling spaces on these smooth retracts repro-duces the definition of a Banach manifold because, by [7, Prop. 2.1], asmooth retract O is a C ∞ -sub-Banach manifold of E . The sc-retractscan have much more complicated structure, including locally varyingdimension. This is a key difference between classical differentiabilityand sc-differentiability which allows M -polyfolds to have local dimen-sion jumps and other non-manifold-like structure. Polyfolds arising inapplications have these local dimension jumps near broken and nodalcurves.A map ϕ : O → O ′ between sc-retracts ( O , C, E ) and ( O ′ , C ′ , E ′ ) iscalled sc-smooth [7, Def. 2.4] if the composition ϕ ◦ r : U → O ′ ⊂ E ′ issc-smooth as a map U → E ′ , where U ⊂ C is open and r : U → U isany sc-retraction onto r ( U ) = O . This definition is independent of thechoice of open set U and sc-retraction r by [7, Prop. 2.3]. The chainrule holds for sc-smooth maps between sc-retracts; see [7, Thm. 2.1].The tangent space [7, Def. 2.3] of a sc-retract ( O , C, E ) is the image(10) T O := T r ( T U ) , where r : U → U is any sc-retraction on some open subset U ⊂ C withimage r ( U ) = O and T r : T U → T U is the tangent map (see (2)) of r .The tangent space T O is well-defined, i.e. independent of U and r , by[7, Prop. 2.2]. The tangent space at x ∈ O is(11) T x O = D x r ( T x U ) . For a smooth point x ∈ O ∞ , the tangent space T x O is a sc-Banach spacesince D x r is a sc-operator. The reduced tangent space [7, Def. 2.15] isthe subspace of T x O defined by(12) T Rx O := T x O ∩ E x , OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 23 where E x ⊂ E is the subspace from (9). Conceptually, T Rx O consists ofthose tangent vectors that point in directions that preserve the degen-eracy index, i.e. they lie along the intersection of all of the local facesthat contain x . Note that in [7, Def. 2.15] the reduced tangent spaceis only defined at smooth points x ∈ O ∞ . This is because T Rx O canbe proven to be invariant under sc-diffeomorphisms ϕ : O → O ′ , i.e. D x ϕ ( T Rx O ) = T Rϕ ( x ) O ′ holds, only for smooth points x ; see [7, Prop. 2.8].This invariance proves that the reduced tangent space at a smoothpoint in a M -polyfold is well-defined, i.e. independent of the chart.The invariance is proven using the characterization [7, Lem. 2.4] of thereduced tangent space T Rx O at smooth points x ∈ O ∞ as the closure ofthe space of derivatives of sc-smooth paths γ : ( − ǫ, ǫ ) → O satisfying γ (0) = x . This only works for smooth points x ∈ O ∞ since the im-age of any sc-continuous map ( − ǫ, ǫ ) → O is contained in O ∞ because( − ǫ, ǫ ) ⊂ R has the trivial sc-structure where all levels are equal.As discussed in [7], we must require sc-retractions to be well-behavedwith respect to the boundary faces of the partial quadrant C in the fol-lowing way in order for the full polyfold machinery to work as requiredin applications. Definition 3.1. [7, Def. 2.17]
Consider an open subset U of a partialquadrant C of a sc-Banach space E . A sc-retraction r : U → U iscalled a tame sc-retraction if it satisfies the following conditions:(1) d C ( r ( x )) = d C ( x ) for all x ∈ U .(2) At every smooth point x ∈ O ∞ = O ∩ E ∞ , there exists a sc-subspace A ⊂ E such that E = T x O ⊕ A and A ⊂ E x (see (9) for E x ).If so, then the sc-retract O = r ( U ) is called a tame sc-retract (andso is the triple ( O , C, E ) ). We introduce the following new notions of R n -sliced sc-retractionsand R n -sliced sc-retracts. Definition 3.2.
Consider a partial quadrant C of a sc-Banach space E and an open subset U ⊂ R n × C for some n ≥ . A tame sc-retraction r : U → U is called a R n -sliced sc-retraction if it satisfies (13) π R n ◦ r = π R n on U, i.e. r preserves the R n -coordinate.If so, then the tame sc-retract O = r ( U ) (and the triple ( O , R n × C, R n × E ) ) is called a R n -sliced sc-retract . In the following lemma, we show that for any R n -sliced sc-retract O in R n × C , the set ˜ O := O ∩ ( { } × C ) is a tame sc-retract. Later, we use the inclusion ˜ O ⊂ O to define the local models for a slice ˜
B ⊂ B (Definition 5.7), which is our new notion of a M -polyfold ˜ B embeddedwith finite codimension in an ambient M -polyfold B . Lemma 3.3.
Consider a partial quadrant C of a sc-Banach space E and a R n -sliced sc-retract ( O , R n × C, R n × E ) .Then, for any open subset U ⊂ R n × C and R n -sliced sc-retraction r : U → U such that r ( U ) = O , the set ˜ U := U ∩ ( { } × C ) is open in C and the restriction ˜ r := r | ˜ U : ˜ U → ˜ U is a tame sc-retraction onto ˜ O := ˜ r ( ˜ U ) . We call ˜ r the tame sc-retraction induced by r .Moreover, (14) ˜ O = O ∩ ( { } × C ) holds, so in particular ˜ O does not depend on the choices of U and r .We may view ˜ O as a subset of C , and we call ( ˜ O , C, E ) the tame sc-retract induced by the R n -sliced sc-retract ( O , R n × C, R n × E ) .At every x ∈ ˜ O , the inclusion ˜ O ⊂ O induces an inclusion oftangent spaces T x ˜ O ⊂ T x O satisfying T x ˜ O = T x O ∩ ( { } × E ) , (15) T Rx ˜ O = T Rx O ∩ ( { } × E ) , (16) and (17) T x O /T x ˜ O ∼ = R n . We say that ˜ O is codimension- n in O .If x ∈ ˜ O ∞ is a smooth point, then the inclusion T x ˜ O ֒ → T x O inducesa linear isomorphism (18) T x ˜ O /T Rx ˜ O ∼ = T x O /T Rx O . Proof.
The defining property (13) of the R n -sliced sc-retraction r im-plies r ( ˜ U ) ⊂ ˜ U , so indeed the map ˜ r := r | ˜ U : ˜ U → ˜ U takes valuesin ˜ U . Moreover, ˜ r inherits sc-smoothness and the retraction property˜ r ◦ ˜ r = ˜ r from the corresponding properties of r . So ˜ r is a sc-retractiononto the sc-retract ˜ O . We prove the other statements in the lemmabefore showing that ˜ r is tame.We now verify that (14) holds. The forwards inclusion is immediatefrom the definitions of the sets involved. To prove the reverse inclusion,let x ∈ O∩ ( { }× C ) . Then since
O ⊂ U we have x ∈ ˜ U and so ˜ r ( x ) ∈ ˜ O .We claim that x = ˜ r ( x ), proving (14). Indeed, since x ∈ O and r is aretraction with image O , it follows that x = r ( x ) = ˜ r ( x ) . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 25
We now verify (15) and (17). Let x ∈ ˜ O . By definition of tangentspace (11) of a sc-retract, we have T x O = D x r ( R n × E ) ⊂ R n × E . Since r preserves the R n -coordinate by the sliced retraction property(13), the same is true for D x r, from which it follows that D x r ( R n × E ) ∩ ( { } × E ) = D x r ( { } × E ). Hence we have T x O ∩ ( { } × E ) = D x ˜ r ( { } × E ) = T x ˜ O , proving (15). Moreover, the projection π : T x O → R n to the first factorof R n × E is a surjection. Since ker( π ) = T x O ∩ ( { } × E ), we concludethat π induces an isomorphism T x O /T x ˜ O ∼ = R n , proving (17).To verify (16), first note that by (8),(9), we have(19) ( R n × E ) x ∩ ( { } × E ) = ( { } × E ) x , and hence we have T Rx ˜ O = T x ˜ O ∩ ( { }× E ) x = T x ˜ O ∩ ( R n × E ) x ∩ ( { }× E ) = T x O ∩ ( R n × E ) x ∩ ( { } × E ) = T Rx O ∩ ( { } × E ), as required.To verify (18), it suffices to consider the case x = 0 and C = [0 , ∞ ) s × E ′ ⊂ R s × E ′ = E is in standard form (5). By (8), we have ( R n × R s × E ′ ) x = R n × { } s × E ′ . Since r preserves the degeneracy index byDefinition 3.1, we claim it follows that D x r ( R n × { } s × E ′ ) ⊂ R n × { } s × E ′ . Indeed, given a smooth point ξ ∈ R n × { } s × E ′∞ , there exists a sc-smooth path α : ( − ǫ, ǫ ) → U ∩ ( R n × { } s × E ′ ) satisfying α (0) = x and α ′ (0) = ξ . Since r preserves degeneracy index we have r ◦ α (( − ǫ, ǫ )) ⊂ R n × { } s × E ′ , and hence we have D x r ( ξ ) ∈ R n × { } s × E ′ . For anarbitrary point ξ ∈ R n × { } s × E ′ the result follows by considering asequence { ξ k } k ≥ of smooth points that converges to ξ , which exists bydensity of the inclusion E ′∞ ⊂ E ′ . Note that this essentially uses thefact that x is a smooth point.Since by definition (12) we have T Rx O = D x r ( R n × R s × E ′ ) ∩ ( R n ×{ } s × E ′ ) and moreover the retraction property D x r ◦ D x r = D x r holds,it follows that T Rx O = D x r ( R n × { } s × E ′ ) . Hence, since D x r preserves the R n -coordinate, the projection π : T x O → R n restricts to a surjection π : T Rx O → R n . The kernel of this surjec-tion is T Rx ˜ O by (16). So we have a short exact sequence of sc-Banachspaces 0 → T Rx ˜ O → T Rx O π −→ R n → → T x ˜ O → T x O π −→ R n . This implies (18).
To prove the lemma, it remains to show that ˜ r is tame. The R n -sliced sc-retraction r is tame by definition. Hence, for all x ∈ ˜ U , wecompute, using (6) and property Definition 3.1(1) of r , d { }× C (˜ r ( x )) = d R n × C (˜ r ( x )) = d R n × C ( r ( x )) = d R n × C ( x ) = d { }× C ( x ) , verifying property Definition 3.1(1) for ˜ r .To verify that ˜ r satisfies property Definition 3.1(2), let x ∈ ˜ O ∞ . Then x ∈ O ∞ , and so by the corresponding property of r and by [7,Prop. 2.9], the sc-subspace A := ( id R n × E − D x r )( R n × E ) of R n × E satisfies(20) R n × E = T x O ⊕ A and A ⊂ ( R n × E ) x . By the sliced retraction property (13) of r andthe definition of A , we conclude that A ⊂ { } × E holds. Then wehave A ⊂ ( R n × E ) x ∩ ( { } × E ) = ( { } × E ) x by (19). We claim that { } × E = T x ˜ O ⊕ A holds, completing the proof that ˜ r is tame. Indeed,it follows from (15), (20), and A ⊂ { } × E that we have { } × E = ( T x O ∩ ( { } × E )) ⊕ A = T x ˜ O ⊕ A. (cid:3) Sliced bundle retracts.
In this section, we introduce the newnotion of R n -sliced bundle retracts (Definition 3.4). We prove in Lemma 3.5that a R n -sliced bundle retract K covering a R n -sliced sc-retract O induces a tame bundle retract ˜ K ⊂ K covering the induced tame sc-retract ˜ O ⊂ O from Lemma 3.3. The global definition of a slice ˜
B ⊂ B of a bundle ρ : E → B (Definition 5.7) is then a subspace such thataround every point x ∈ ˜ B there is a bundle chart for ρ to a R n x -slicedbundle retract K that locally identifies ρ − ( ˜ B ) with the induced tamebundle retract ˜ K .We first recall the local structure of tame strong bundles: tame bun-dle retracts. Consider a relatively open subset U of a partial quadrant C of a sc-Banach space E , and another sc-Banach space F . Then thetrivial bundle(21) U ✁ F → U has total space U ✁ F = U × F as a set, and the map is projection onto U . The triangle ✁ signifies the extra structure of a double filtration onthe set U × F . That is, for 0 ≤ k ≤ m + 1, we have( U ✁ F ) m,k := U m ⊕ F k . Then, for i = 0 ,
1, we define the sc-structure ( U ✁ F )[ i ] by(22) (( U ✁ F )[ i ]) m := U m ⊕ F m + i , m ≥ . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 27
The purpose of defining these two filtrations is that they correspondto the two notions of smoothness of a section of a bundle that areimportant for polyfold theory. Precisely, a section s : U → U ✁ F iscalled sc-smooth if it is sc-smooth as a map to ( U ✁ F )[0]. If, moreover,we have s ( U ) ⊂ ( U ✁ F )[1] and the map s : U → ( U ✁ F )[1] is sc-smooth, then s is called a sc + -section . See [7, Def. 2.24] for a detaileddiscussion.A strong bundle map Φ : U ✁ F → U ′ ✁ F ′ [7, Def. 2.22] is a mapwhich preserves the double filtration and is of the form Φ ( x, ξ ) =( ϕ ( x ) , Γ ( x, ξ )), where the map Γ : U ✁ F → F ′ is linear in ξ . Inaddition, for i = 0 , , we require that the maps Φ : ( U ✁ F )[ i ] → ( U ′ ✁ F ′ )[ i ]are sc-smooth. A strong bundle isomorphism is an invertible strongbundle map whose inverse is also a strong bundle map.To extend (21) to a notion of a trivial bundle over a sc-retract, weemploy the following notion of a retraction in the fibers. A strongbundle retraction is a strong bundle map R : U ✁ F → U ✁ F satisfying R ◦ R = R . As a consequence, the map R has the form(23) R ( x, ξ ) = ( r ( x ) , Γ ( x, ξ )) , where r : U → U is a sc-smooth retraction and Γ ( x, · ) : F → F is alinear projection for every x ∈ U . If r is tame, then R is called a tamestrong bundle retraction . The image K := R ( U ✁ F ) of R is called a strong bundle retract [7, Def. 2.23], as is the triple ( K, C ✁ F , E ✁ F ) . We say that K covers the sc-retract O = r ( U ). If R is tame, then K iscalled a tame strong bundle retract . The projection U ✁ F → U inducesa mapping K → O , which we call a strong local bundle model .We now introduce the new notion of a R n -sliced bundle retract. Definition 3.4.
Consider a partial quadrant C of a sc-Banach space E , an open subset U ⊂ R n × C for some n ≥ , and another sc-Banachspace F .A tame bundle retraction R : U ✁ F → U ✁ F is called a R n -slicedbundle retraction if the tame sc-retraction r : U → U covered by R (see (23) ) is a R n -sliced sc-retraction (Definition 3.2).If so, then the tame bundle retract K = R ( U ✁ F ) (and the triple ( K, R n × C ✁ F , R n × E ✁ F ) ) is called a R n -sliced bundle retract and the tame local bundle model K → O := r ( U ) is called a R n -slicedlocal bundle model . In the following lemma, we show that for any R n -sliced bundle retract K in R n × C ✁ F , the set ˜ K := K ∩ ( { } × C ✁ F ) is a tame bundle retract. Later, we use the inclusion ˜ K ⊂ K to define the local modelsfor the restriction of a bundle to a slice (Definition 5.7). Lemma 3.5.
Consider a partial quadrant C of a sc-Banach space E ,another sc-Banach space F , and a R n -sliced bundle retract ( K, R n × C ✁ F , R n × E ✁ F ) covering a R n -sliced sc-retract ( O , R n × C, R n × E ) .Let π : K → O denote the local bundle model given by restriction ofthe projection along the fiber R n × C ✁ F → R n × C .Then, for any open subset U ⊂ R n × C and R n -sliced bundle retrac-tion R : U ✁ F → U ✁ F covering a R n -sliced sc-retraction r : U → U such that r ( U ) = O and R ( U ✁ F ) = K , the set ˜ U := U ∩ ( { } × C ) isopen in C and the restriction ˜ R := R | ˜ U ✁ F : ˜ U ✁ F → ˜ U ✁ F is a tame bundle retraction onto ˜ K := ˜ R ( ˜ U ✁ F ) covering the inducedtame sc-retraction ˜ r : ˜ U → ˜ U onto the induced tame sc-retract ˜ O = O∩ ( { } × C ) (see Lemma 3.3). We call ˜ R the tame bundle retractioninduced by R .Moreover, (24) ˜ K = K ∩ ( { } × C ✁ F ) = π − ( ˜ O ) holds, so in particular ˜ K does not depend on the choices of U and R .We may view ˜ K as a subset of C ✁ F , and we call ( ˜ K, C ✁ F , E ✁ F ) the tame bundle retract induced by the R n -sliced bundle retract ( K, R n × C ✁ F , R n × E ✁ F ) .In particular, the R n -sliced local bundle model π : K → O restrictedto ˜ K is a tame local bundle model (25) ˜ π := π | ˜ K : ˜ K → ˜ O , which we call the induced tame local bundle model .Proof. The map ˜ R is a strong bundle map and satisfies ˜ R ◦ ˜ R = ˜ R , bythe corresponding properties of R , so ˜ R is a strong bundle retraction.Moreover, ˜ R is tame because it covers the tame sc-retraction ˜ r .We now verify that the first equality in (24) holds. The forwardsinclusion is immediate from the definitions. To see the reverse inclusion,let ( x ✁ ξ ) ∈ K ∩ ( { }× C ✁ F ) . Then x = π ( x ✁ ξ ) ∈ O ∩ ( { }× C ) = ˜ O ,where π : K → O is the local bundle model given by restriction of theprojection along the fiber π : R n × C ✁ F → R n × C . In particular, x ✁ ξ is in the domain of ˜ R = R | ˜ U ✁ F . Since R is a retraction onto K , it follows that ( x ✁ ξ ) = R ( x ✁ ξ ) = ˜ R ( x ✁ ξ ) ∈ im ( ˜ R ) = ˜ K , asrequired. The second equality in (24) holds from the definitions and˜ O = O ∩ ( { } × C ). (cid:3) OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 29
Sliced sc-Fredholm germs.
In this section, we review the stan-dard notion of a local sc-Fredholm germ (Definition 3.6) and we in-troduce the new notions of a tame sc-Fredholm germ (Definition 3.7)and a R n -sliced sc-Fredholm germ (Definition 3.8). We later use tamesc-Fredholm germs as the local models for our new notion of a tamesc-Fredholm section σ : B → E (Definition 5.4) of a tame strong bundle ρ : E → B , and we use R n -sliced sc-Fredholm germs as the local mod-els in our definition of a slice ˜ B ⊂ B (Definition 5.7) of a sc-Fredholmsection σ .We prove in Lemma 3.10 that a R n -sliced sc-Fredholm germ σ : O → K of a tame local bundle model K → O induces a tame sc-Fredholmgerm ˜ σ := σ | ˜ O : ˜ O → ˜ K of the induced tame local bundle model˜ K → ˜ O from (25). The sc-Fredholm index (27) satisfies ind (˜ σ ) = ind ( σ ) − n . The global definition of a slice ˜ B ⊂ B of a sc-Fredholmsection σ : B → E is then a slice of the bundle such that around everysmooth point x ∈ ˜ B ∞ there exists a chart in which σ is a R n x -slicedsc-Fredholm germ and such that the restriction σ | ˜ B : ˜ B → ρ − ( ˜ B ) islocally identified with the induced tame sc-Fredholm germ.All currently known sc-Fredholm sections arising in applications aretame sc-Fredholm by construction; see Section 5.1 for a discussion ofthe tame sc-Fredholm property of the Cauchy-Riemann section ∂ J .We now briefly review sc-germ language. Let C be a partial quadrantof a sc-Banach space E . Then a sc-germ of neighborhoods around ∈ C [7, Def. 3.1], denoted U ( C, , is a sequence U = U (0) ⊃ U (1) ⊃ U (2) ⊃ · · · where U ( m ) is a relatively open neighborhood of 0 in C ∩ E m . We oftenwrite U = U ( C,
0) for brevity. A sc -germ h : U ( C, → F [7, Def. 3.2]into the sc-Banach space F is a continuous map h : U (0) → F suchthat h ( U ( m )) ⊂ F m and h : U ( m ) → F m is continuous for all m ≥ sc -germ h : U ( C, → F is a sc -germ which of is class sc in thesame sense as for a usual map except that the sc-differential D x h isrequired to exist only for x ∈ U (1) (where U (1) can be smaller than U (0) ∩ E in the germ case); see [7, Def. 3.2] for a precise definition.We can in turn define a tangent map on the tangent of a germ andthen iterate the notion of sc to define a sc-smooth germ .It is convenient to denote any section of a trivial bundle by h ( x ) =( x, h ( x )) : U → U ✁ F . We refer to h : U → F as the principal part of h . Recall the standard notion of a local sc-Fredholm germ.
Definition 3.6. [7, Defs. 3.5, 3.6, 3.7]
Consider a tame strong bundleretract ( K, C ✁ F , E ✁ F ) covering the tame sc-retract ( O , C, E ) anda sc-smooth section σ : O → K of the local bundle model K → O .Assume ∈ O .Then σ is called a local sc-Fredholm germ if the following condi-tions hold: (a) There exists a sc-germ of neighborhoods U ( C, around ∈ C anda tame sc-retraction r : U → U onto r ( U ) = O covered by a tamebundle retraction R : U ✁ F → U ✁ F onto R ( U ✁ F ) = K . (b) The principal part σ : O → F of σ has the property that thecomposition σ ◦ r : U ( C, → F possesses a filling h : U ( C, → U ( C, ✁ F , which is a section of the trivial bundle U ( C, ✁ F → U ( C, whoseprincipal part h : U ( C, → F is a sc-smooth germ and such thatthe following conditions ( i ) − ( iii ) hold. Recall that R is of the form R ( x, ξ ) = ( r ( x ) , Γ ( x, ξ )) where Γ ( x, − ) : F → F is a linear projection. (i) σ ( x ) = h ( x ) for x ∈ O . (ii) If x ∈ U and h ( x ) = Γ ( r ( x ) , h ( x )) , then x ∈ O . (iii) The linearization of the map x ( id F − Γ ( r ( x ) , · )) h ( x ) at thepoint , restricted to ker D r , defines a linear sc-isomorphism ker D r → ker Γ (0 , · ) . (c) There exists a sc + -section s : U → U ✁ F satisfying s (0) = h (0) , asc-Banach space W , a sc-germ of neighborhoods U ′ = U ′ ( C ′ , ⊂ C ′ = [0 , ∞ ) s × R k − s × W centered around ∈ C ′ for some k ≥ s ≥ , and a strong bundleisomorphism Ψ : U ✁ F → U ′ ✁ R k ′ × W (for some k ′ ≥ ) covering a sc-diffeomorphism ψ : U → U ′ satisfying ψ (0) = 0 and such that the principal part of the section b := Ψ ◦ ( h − s ) ◦ ψ − : U ′ → U ′ ✁ R k ′ × W is a basic germ . This means that the principal part b : U ′ → R k ′ × W OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 31 of b is a sc-smooth germ satisfying b (0) = 0 and having the followingproperty: Letting P : R k ′ × W → W denote projection onto W , thegerm P ◦ b : U ′ → W is of the form P ◦ b ( a, w ) = w − B ( a, w ) , for ( a, w ) ∈ ([0 , ∞ ) s × R k − s ) × W , where B is a sc-smooth germ and B (0) = 0 . Moreover, B satisfies the contraction property: for every ǫ > and integer m ≥ there exists δ > such that (26) || ( a, w ) || m , || ( a, w ′ ) || m < δ = ⇒ || B ( a, w ) − B ( a, w ′ ) || m ≤ ǫ || w − w ′ || m . The notation || · || m means the Banach norm in the m -level of thesc-structure. We use the convention that the m -norm || · || m on ( R k × W ) m = R k × W m is the sum of the standard Euclidean normon R k plus the W m -norm.The index of the local sc-Fredholm germ σ is the integer (27) ind ( σ ) := k − k ′ , where k and k ′ are the dimensions of the finite dimensional spaces splitoff in the domain U ′ ⊂ ([0 , ∞ ) s × R k − s ) × W and codomain R k ′ × W ,respectively, of the basic germ b . We now introduce a new class of sc-Fredholm germs, called tamesc-Fredholm.
Definition 3.7.
A local sc-Fredholm germ σ : O → K is called a tamesc-Fredholm germ if the structures that exist by Definition 3.6 of localsc-Fredholm germ can be chosen such that the partial quadrant C is inthe standard form (5) , i.e. C = [0 , ∞ ) s × E ′ ⊂ R s × E ′ = E for somesc-Banach space E ′ and integer s ≥ , and such that the required sc-diffeomorphism ψ : U → U ′ in Definition 3.6(c) is the restriction of alinear sc-isomorphism of the form ψ = id [0 , ∞ ) s × ψ : [0 , ∞ ) s × E ′ → [0 , ∞ ) s × R k − s × W , ( v, e ) ( v, ψ ( e )) for some linear sc-isomorphism ψ : E ′ → R k − s × W . The following new class of sc-Fredholm germs, called R n -sliced sc-Fredholm, is defined for sections of R n -sliced local bundle models K →O (Definition 3.4). This convention is equivalent to using any norm on R k and any standard normon a Cartesian product that is equivalent to the sum norm. Definition 3.8.
Consider a R n -sliced bundle retract (Definition 3.4) ( K, R n × C ✁ F , R n × E ✁ F ) covering the R n -sliced sc-retract ( O , R n × C, R n × E ) . Assume ∈ O and that the partial quadrant is in the standard form (5) , i.e. C = [0 , ∞ ) s × E ′ ⊂ R s × E ′ = E for some s ≥ and sc-Banachspace E ′ .Then a local sc-Fredholm germ σ : O → K of the R n -sliced local bundle model K → O is called a R n -sliced sc-Fredholm germ if the structures that exist by Definition 3.6 of localsc-Fredholm germ can be chosen such that the sc-Banach space W andsc-diffeomorphism ψ : U → U ′ from Definition 3.6(c) have the follow-ing form: First, we have W = ˜ W × R n for some other sc-Banach space ˜ W . Moreover, the sc-diffeomorphism ψ : U → U ′ is of the form ψ : R n × [0 , ∞ ) s × E ′ ⊃ U → U ′ ⊂ [0 , ∞ ) s × R k − s × ˜ W × R n (28) ( p, v, e ) ( v, ψ ( e ) , λ ( p, v, e )) for some linear sc-isomorphism ψ : E ′ → R k − s × ˜ W and such that the map ( p, v, e ) λ ( p, v, e ) ∈ R n is C on all levels R n × [0 , ∞ ) s × E ′ m for m ≥ . Remark 3.9.
Given a local sc-Fredholm germ σ : O → K of a localbundle model K → O , all essential properties of the setup are pre-served after restricting to the m -level of the sc-structure for any m ≥ .Precisely, the m -shifted map K m → O m is a local bundle model and σ | O m : O m → K m is a local sc-Fredholm germ [7, Cor. 5.1] with thesame index ind ( σ | O m ) = ind ( σ ) .In particular, if a local sc-Fredholm germ σ : O → K satisfies theproperties required of a R n -sliced sc-Fredholm germ except for the C regularity of the map ( p, v, e ) λ ( p, v, e ) , then the -shift σ | O : O → K is R n -sliced sc-Fredholm since the map λ has the required C regu-larity on all levels m ≥ by [7, Prop. 1.7] . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 33
In the following lemma, we prove that a R n -sliced sc-Fredholm germof a R n -sliced local bundle model K → O (Definition 3.4) restrictsto a tame sc-Fredholm germ of the induced tame local bundle model˜ K → ˜ O from (25). Lemma 3.10.
Consider a R n -sliced bundle retract ( K, R n × C ✁ F , R n × E ✁ F ) covering a R n -sliced sc-retract ( O , R n × C, R n × E ) . Recall theinduced tame local bundle model ˜ K → ˜ O from (25) .Let σ : O → K be a R n -sliced sc-Fredholm germ. Then, the re-striction ˜ σ := σ | ˜ O : ˜ O → ˜ K is a tame sc-Fredholm germ with indexsatisfying ind (˜ σ ) = ind ( σ ) − n .We call ˜ σ the tame sc-Fredholm germ induced by the R n -slicedsc-Fredholm germ σ .Proof. Since σ is a R n -sliced sc-Fredholm germ, the partial quadrant isin the standard form C = [0 , ∞ ) s × E ′ ⊂ R s × E ′ = E and there existsan open subset U ⊂ R n × C and a R n -sliced sc-retraction r : U → U onto r ( U ) = O that satisfies the conditions of Definition 3.2 andDefinition 3.8, which we recall as we use them.Since r satisfies the properties in Definition 3.6 of local sc-Fredholmgerm, we can assume that there exists a sc-germ of neighborhoods U ( R n × C,
0) around 0 satisfying U (0) = U (i.e. the 0-level open set inthe germ is the open set U from above) such that r is covered by a tamebundle retraction R : U ✁ F → U ✁ F such that σ ◦ r : U ( R n × C, → F posesses a filling h : U ( R n × C, → U ( R n × C, ✁ F . Since r is a R n -sliced sc-retraction covered by R , it follows by Def-inition 3.4 that R is a R n -sliced bundle retraction. As in Lemma 3.5,set ˜ U := U ∩ ( { } n × C )and denote the induced tame sc-retraction and tame bundle retractionby ˜ r : ˜ U → ˜ U , ˜ R : ˜ U ✁ F → ˜ U ✁ F , and the induced tame sc-retract and tame bundle retract by˜ O = ˜ r ( ˜ U ) = O ∩ ( { } n × C ) , ˜ K = ˜ R ( ˜ U ✁ F ) = K ∩ ( { } n × C ✁ F ) . Also, let ˜ U ( C,
0) be the sc-germ of neighborhoods given on level- m by˜ U ( m ) := U ( m ) ∩ ( { } n × C ) , or more concisely˜ U ( C,
0) := U ( R n × C, ∩ ( { } n × C ) . We claim that the restriction˜ h := h | ˜ U ( C, : ˜ U ( C, → ˜ U ( C, ✁ F is a filling for the composition ˜ σ ◦ ˜ r : ˜ U ( C, → F . It is a section ofthe trivial bundle ˜ U ( C, ✁ F → ˜ U ( C,
0) and its principal part ˜h is asc-smooth germ, because we have ˜h = h | ˜ U ( C, and the corresponding properties hold for h . It remains to verify thatthe filler properties in Definition 3.6(b).(i)-(iii) hold for ˜ σ and ˜ h . Thesefollow from the corresponding properties of σ and h , as we now describe.Property (i) is immediate since ˜ σ and ˜ h are restrictions of σ and h ,respectively. Write R ( x, ξ ) = ( r ( x ) , Γ ( x, ξ )), as in (23). Consider therestriction ˜ Γ := Γ | ˜ U ✁ F . Then we have ˜ R ( x, ξ ) = (˜ r ( x ) , ˜ Γ ( x, ξ )) . Toverify ( ii ), let x ∈ ˜ U and assume ˜h ( x ) = ˜ Γ (˜ r ( x ) , ˜h ( x )) . It follows that h ( x ) = Γ ( r ( x ) , h ( x )), which implies x ∈ O by property (ii) for h .Hence x ∈ ˜ U ∩ O = ˜ O , as required. It remains to verify (iii) for ˜h . Weclaim that(29) ker D ˜ r = ker D r. The forwards inclusion follows from ˜ r being the restriction of r . To seethe reverse inclusion, let ξ ∈ ker D r. By the defining property (13) ofa R n -sliced sc-retraction, we see that D r preserves the R n -coordinateof ξ and hence ξ ∈ { } n × R s × E ′ . In particular, ξ is in the domainof D ˜ r , and moreover D ˜ r ( ξ ) = D r ( ξ ) = 0. Hence (29) holds. Wenow verify that Definition 3.6(iii) holds for ˜h . From the correspondingproperty of h , the linearization D L at 0 of the map L : U → F x ( id F − Γ ( r ( x ) , · )) h ( x )restricts to a linear sc-isomorphism ker D r → ker Γ (0 , · ). We mustshow that the linearization at 0 of the map˜ L : ˜ U → F x ( id F − ˜ Γ (˜ r ( x ) , · ))˜ h ( x )restricts to a linear sc-isomorphism ker D ˜ r → ker ˜ Γ (0 , · ) = ker Γ (0 , · ).This follows from (29) and since ˜ L and ˜ r are the restrictions of L and OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 35 r , respectively, to ˜ U . This completes the proof that ˜ h is a filling for˜ σ ◦ ˜ r, as claimed.To show that ˜ σ is a local sc-Fredholm germ, it remains to verifythe properties in Definition 3.6(c). Since σ is a R n -sliced sc-Fredholmgerm, the corresponding properties in Definition 3.6(c) hold for σ , andin addition the stronger properties in the definition of R n -sliced sc-Fredholm germ (Definition 3.8) hold: There exists a sc + -section s : U → U ✁ F satisfying s (0) = h (0), a sc-Banach space of the form W =˜ W × R n for some other sc-Banach space ˜ W , a sc-germ of neighborhoods U ′ around 0 ∈ ([0 , ∞ ) s × R k − s ) × ( ˜ W × R n ) , and a strong bundle isomorphism Ψ : U ✁ F → U ′ ✁ R k ′ × ( ˜ W × R n )covering a sc-diffeomorphism ψ : U → U ′ satisfying ψ (0) = 0 and ofthe form ψ : R n × [0 , ∞ ) s × E ′ ⊃ U → U ′ ⊂ [0 , ∞ ) s × R k − s × ˜ W × R n ( p, v, e ) ( v, ψ ( e ) , λ ( p, v, e ))for some linear sc-isomorphism ψ : E ′ → R k − s × ˜ W such that, on all levels R n × [0 , ∞ ) s × E ′ m for m ≥
0, the map(30) ( p, v, e ) λ ( p, v, e ) ∈ R n is C . Moreover, the principal part of the section(31) b := Ψ ◦ ( h − s ) ◦ ψ − : U ′ → U ′ ✁ R k ′ × ( ˜ W × R n )is a basic germ, which means that, for all a ∈ [0 , ∞ ) s × R k − s and z ∈ ˜ W × R n such that ( a, z ) ∈ U ′ , we have(32) P ◦ b ( a, z ) = z − B ( a, z ) , where P : R k ′ × ( ˜ W × R n ) → ( ˜ W × R n ) is projection onto ˜ W × R n and B : U ′ → ˜ W × R n is sc-smooth, satisfies B (0) = 0, and satisfies thecontraction property (26).Now, to verify that ˜ σ = σ | ˜ O inherits the local sc-Fredholm germproperty, consider the restricted sc + section˜ s := s | ˜ U : ˜ U → ˜ U ✁ F , the linear sc-isomorphism given by˜ ψ : [0 , ∞ ) s × E ′ → [0 , ∞ ) s × R k − s × ˜ W ( v, e ) ( v, ψ ( e )) , the open set ˜ U ′ := ˜ ψ ( ˜ U ) , and the strong bundle isomorphism given by˜ Ψ : ˜ U ✁ F → ˜ U ′ ✁ ( R k ′ × R n ) × ˜ W (( v, e ) ✁ ξ ) ( ˜ ψ ( v, e ) ✁ η ( Ψ (( v, e ) ✁ ξ )) , where η : R k ′ × ( ˜ W × R n ) → ( R k ′ × R n ) × ˜ W is the reordering of factorsmap.We claim that the principal part of the section(33) ˜ b := ˜ Ψ ◦ (˜ h − ˜ s ) ◦ ˜ ψ − : ˜ U ′ → ˜ U ′ ✁ ( R k ′ × R n ) × ˜ W is a basic germ, where now ˜ W plays the role of the sc-Banach space W in Definition 3.6(c). Let the maps˜ P : ( R k ′ × R n ) × ˜ W → ˜ W ,π ˜ W : R n × ˜ W → ˜ W , be the projections onto the ˜ W factor in their respective domains. Noticethat we have ˜ P = π ˜ W ◦ P ◦ η − . We write a ∈ [0 , ∞ ) s × R k − s and w ∈ ˜ W . Define the map τ : ˜ U ′ → U ′ ( a, w ) ψ (0 , ˜ ψ − ( a, w )) , and from the definitions observe(34) τ ( a, w ) = ( a, w, λ (0 , ˜ ψ − ( a, w ))) . From the definitions and (32),(34), we compute˜ P ◦ ˜ b ( a, w ) = π ˜ W ◦ ˜ P ◦ π R k ′ × R n × ˜ W ◦ ˜ Ψ ◦ (˜ h − ˜ s ) ◦ ˜ ψ − ( a, w )= π ˜ W ◦ P ◦ π R k ′ × ˜ W × R n ◦ Ψ ◦ ( h − s )(0 , ˜ ψ − ( a, w ))= π ˜ W ◦ P ◦ π R k ′ × ˜ W × R n ◦ Ψ ◦ ( h − s ) ◦ ( ψ − ◦ ψ )(0 , ˜ ψ − ( a, w ))= π ˜ W ◦ P ◦ b ◦ ψ (0 , ˜ ψ − ( a, w ))= π ˜ W ◦ P ◦ b ◦ τ ( a, w )= π ˜ W (( w, λ (0 , ˜ ψ − ( a, w ))) − B ◦ τ ( a, w ))= w − π ˜ W ◦ B ◦ τ ( a, w ) . So, setting ˜ B := π ˜ W ◦ B ◦ τ : ˜ U ′ → ˜ W , OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 37 we have ˜ P ◦ ˜ b ( a, w ) = w − ˜ B ( a, w ). Hence to complete the proof that˜ b is a basic germ it remains to show that ˜ B satisfies the contractionproperty (26).Recall from (34) the map˜ λ : ˜ U ′ → R n ( a, w ) λ (0 , ˜ ψ − ( a, w )) . For all m ≥
0, this map restricted to the m -level ˜ U ′ m → R n is C sincethe map λ from (30) is C on every level by definition of R n -slicedsc-Fredholm germ, and since ˜ ψ is a linear sc-isomorphism and hence C ∞ on every level. So, there exist constants δ ′ m > C m > || ( a, w ) || m , || ( a, w ′ ) || m < δ ′ m , we have the C -estimate(35) || ˜ λ ( a, w ) − ˜ λ ( a, w ′ ) || R n ≤ C m ·|| ( a, w ) − ( a, w ′ ) || m = C m ·|| w − w ′ || m . Then by (34) and our convention that the norm on a Cartesian productis the sum norm (which is equivalent to any standard Banach norm onthe product), we have the estimate || τ ( a, w ) − τ ( a, w ′ ) || m = || ( a, w, ˜ λ ( a, w )) − ( a, w ′ , ˜ λ ( a, w ′ )) || m = || ˜ λ ( a, w ) − ˜ λ ( a, w ′ ) || R n + || w − w ′ || m , ≤ ( C m + 1) · || w − w ′ || m . (36)We now verify the contraction property (26) for ˜ B . Let ǫ > m ≥
0. By the contraction property which is satisfied for B , thereexists δ ′ m > B and ǫ ′ m := ǫ/ ( C m + 1) . Shrink δ ′ m > δ ′ m for which (36) holds. Weclaim that δ m := δ ′ m / ( C m + 1)satisfies (26) for ˜ B and ǫ . Indeed, consider some || ( a, w ) || m , || ( a, w ′ ) || m <δ m . Then by τ (0) = ψ (0 , ˜ ψ − (0)) = 0 and (36), we have || τ ( a, w ) || m , || τ ( a, w ′ ) || m < ( C m + 1) · δ m = δ ′ m . We compute, using the contraction property (26) for B and the esti-mate (36), || ˜ B ( a, w ) − ˜ B ( a, w ′ ) || m = || π ˜ W ◦ B ◦ τ ( a, w ) − π ˜ W ◦ B ◦ τ ( a, w ′ ) || m ≤ || B ◦ τ ( a, w ) − B ◦ τ ( a, w ′ ) || m = || B ( a, w, ˜ λ ( a, w )) − B ( a, w ′ , ˜ λ ( a, w ′ )) || m ≤ ǫ ′ m · || ( w, ˜ λ ( a, w )) − ( w ′ , ˜ λ ( a, w ′ )) || m = ǫ ′ m · || ( a, w, ˜ λ ( a, w )) − ( a, w ′ , ˜ λ ( a, w ′ )) || m = ǫ ′ m · || τ ( a, w ) − τ ( a, w ′ ) || m ≤ ( C m + 1) · ǫ ′ m · || w − w ′ || m = ǫ · || w − w ′ || m . This completes the proof that (26) holds for ˜ B , and hence that ˜ b is abasic germ.We have shown that ˜ σ is a local sc-Fredholm germ. Moreover, ˜ σ is atame sc-Fredholm germ because we have ˜ ψ = id [0 , ∞ ) s × ψ by definitionof ˜ ψ , where ψ : E ′ → R k − s × ˜ W is the linear sc-isomorphism given bythe R n -sliced sc-Fredholm germ property of σ .The claimed index formula holds because, by definition of the sc-Fredholm index (27) and the form of the basic germs b (31) and ˜ b (33), we have ind ( σ ) = k − k ′ and ind (˜ σ ) = k − ( k ′ + n ). (cid:3) Slice coordinates for local submersions to R n The purpose of this section is to prove Lemma 4.2, which generalizesthe local submersion normal form (Lemma 2.3) for sc-smooth maps f : U → R n where the domain U ⊂ [0 , ∞ ) s × E is open to maps f : O → R n where the domain is a tame sc-retract ( O , [0 , ∞ ) s × E , R s × E ).This means that the set O = r ( U ) is the image of a tame sc-retraction r : U → U (see Defintion 3.1), which can have much more compli-cated local structure than the open set U , for example locally varyingdimension. In this case, the local submersion normal form is obtainedby a change of coordinates around any smooth point x ∈ O ∞ at which f is submersive and satisfies f ( x ) = 0 such that the sc-retract in thenew coordinates is a R n -sliced sc-retract (Definition 3.2) with inducedtame sc-retract (see Lemma 3.3) identified with a neighborhood of x in f − (0) ∩ O . For this reason, we call the sc-diffeomorphism with this R n -sliced sc-retract slice coordinates around x .Moreover, given a tame local bundle model K → O and a tamesc-Fredholm germ σ : O → K (Definition 3.7), and assuming that f OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 39 is compatible with σ as explained in Lemma 4.2(III), we prove that σ in the new slice coordinates around x is a R n -sliced sc-Fredholm germ(Definition 3.8) and its induced tame sc-Fredholm germ is identifiedwith the restriction of σ to f − (0) ∩ O . See Section 5.1 for a discussionof why evaluation maps f = ev at marked points are compatible withthe Cauchy-Riemann section σ = ∂ J .For simplicity, in this section we take all partial quadrants C to be instandard form C = [0 , ∞ ) s × E ⊂ R s × E . There is no loss of generalitybecause all partial quadrants are linearly sc-isomorphic to a standardpartial quadrant by definition (see (4)) and hence all sc-retracts aresc-diffeomorphic to sc-retracts in standard partial quadrants.The notion of submersion that we use in Lemma 4.2, as in the lo-cal submersion normal form in sc-calculus (Lemma 2.3), requires sur-jectivity of the tangent map relative to the boundary: For a point x ∈ O ∞ ∩ ( { } × E ), we require that D x f ( T x O ∩ ( { } × E )) = R n .Recall that T Rx O = T x O ∩ ( { } × E ) is the reduced tangent space at x (see (12)). In the following lemma we interpret T Rx O as the tangentspace at x along the corner O ∩ ( { } × E ). Lemma 4.1.
Consider a standard partial quadrant [0 , ∞ ) s × E and atame sc-retract ( O , [0 , ∞ ) s × E , R s × E ) . Then, for any open subset U ⊂ [0 , ∞ ) s × E and tame sc-retraction r : U → U onto r ( U ) = O , therestriction of r to U ∂ := U ∩ ( { } × E ) is a tame sc-retraction r ∂ := r | U ∂ : U ∂ → U ∂ onto O ∂ := O ∩ ( { } × E ) . In particular, the tuple ( O ∂ , E , E ) is a tame sc-retract, which we call the boundary sc-retract associ-ated to the tame sc-retract O .Moreover, for all x ∈ O ∂ , we have (37) T x O ∂ = T x O ∩ ( { } × E ) = T Rx O . Proof.
Since r is tame, for x ∈ U ∂ we have d [0 , ∞ ) s × E ( r ( x )) = d [0 , ∞ ) s × E ( x ) = s, and hence r ( x ) ∈ U ∂ . Thus r ∂ ( U ∂ ) ⊂ U ∂ holds. It is then immediatethat r ∂ is a sc-smooth retraction, as the restriction of the sc-smoothretraction r . Moreover, the domain U ∂ of r ∂ is an open subset of acornerless partial quadrant, i.e. U ∂ ⊂ E , and hence r trivially satisfiesthe tameness hypotheses. We now prove that the image of r ∂ is O ∂ . Indeed, if x ∈ im ( r ∂ ) then x ∈ O and x ∈ U ∂ ⊂ { } × E , hence x ∈ O ∂ . For the reverse inclusion,if x ∈ O ∩ ( { } × E ) then x ∈ U ∂ and hence x = r ( x ) = r ∂ ( x ).It remains to prove that (37) holds. Let ξ ∈ T x O ∂ . Then ξ ∈ { } × E and D x r ( ξ ) = D x r ∂ ( ξ ) = ξ, so ξ ∈ T x O . To prove the reverse inclusion,let ξ ∈ T x O ∩ ( { } × E ) . Then we have D x r ∂ ( ξ ) = D x r ( ξ ) = ξ , whichimplies ξ ∈ T x O ∂ , as required. (cid:3) In the following lemma, we construct the slice coordinates arounda smooth point x in a tame sc-retract O at which a sc-smooth map f : O → R n is submersive on the tangent space T x O ∂ at x to theassociated boundary retract O ∂ from Lemma 4.1, i.e. D x f ( T x O ∂ ) = R n .The statement of the lemma is in three parts: (I) provides slicecoordinates for f − (0), (II) provides slice bundle coordinates for therestriction of a bundle retract to f − (0), and (III) provides slice sc-Fredholm coordinates for the restriction of a sc-Fredholm section. Lemma 4.2. (I)
Consider a tame sc-retract ( O , [0 , ∞ ) s × E , R s × E ) and a sc-smoothmap f : O → R n . Let O ∂ = O ∩ ( { } s × E ) denote the boundary sc-retract associated to O (see Lemma 4.1), and f ∂ := f | O ∂ : O ∂ → R n the restriction of f . Suppose that, at some x ∈ ( O ∂ ) ∞ satisfying f ( x ) = 0 , the tangent map D x f ∂ : T x O ∂ → R n is surjective.Then, there exists an open subset ˆ O ⊂ O , a sc-Banach space K , a R n -sliced sc-retract (38) ( O ′ , R n × [0 , ∞ ) s × K , R n × R s × K ) , and a sc-smooth diffeomorphism g : ˆ O → O ′ satisfying (39) g ( f − (0) ∩ ˆ O ) = O ′ ∩ ( { } n × [0 , ∞ ) s × K ) =: ˜ O ′ . Here, ˜ O ′ is the tame sc-retract induced by O ′ (see Lemma 3.3).We view g as providing slice coordinates ˜ O ′ ⊂ O ′ around x with respect to ( f − (0) ∩ O ) ⊂ O . (II) Consider, in addition, a tame strong bundle retract ( K, [0 , ∞ ) s × E ✁ F , R s × E ✁ F ) covering the tame sc-retract O . Let ˆ K := π − ( ˆ O ) where π : K → O is the -shifted local bundle model. Set K ′ := ( g ✁ id F )( ˆ K ) . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 41
Then, the tuple (40) ( K ′ , R n × [0 , ∞ ) s × K ✁ F , R n × R s × K ✁ F ) is a R n -sliced bundle retract covering the R n -sliced sc-retract (38) . Inparticular, the induced tame strong bundle retract ˜ K ′ (see Lemma 3.5)covers the induced tame sc-retract ˜ O ′ .Moreover, (41) ( g ✁ id F )( π − ( f − (0) ∩ ˆ O )) = ˜ K ′ holds, so we view ( g ✁ id F ) as providing slice bundle coordinates ˜ K ′ ⊂ K ′ for π around x with respect to ( f − (0) ∩ O ) ⊂ O . (III) Consider, in addition, a tame sc-Fredholm germ σ : O → K (Definition 3.7). Assume that x = 0 and that σ has the followingproperty: There exists a choice of sc-Banach space W and linear sc-isomorphism ψ : E → R k − s × W satisfying the conditions in Defini-tion 3.7 of tame sc-Fredholm germ for σ such that, in addition, thereexists a sc-complement L of ker D x f ∂ in T x O ∂ satisfying (42) ψ ( L ) ⊂ ( { } k − s × W ) . Then, the section σ ′ := ( g ✁ id F ) ◦ σ ◦ g − : O ′ → K ′ is a R n -sliced sc-Fredholm germ. In particular, the restriction to theinduced tame local bundle model ˜ σ ′ := σ ′ | ˜ O ′ : ˜ O ′ → ˜ K ′ is a tame sc-Fredholm germ with index satisfying ind (˜ σ ′ ) = ind ( σ ) − n .We view ˜ σ ′ as being in slice sc-Fredholm coordinates around x with respect to ( f − (0) ∩ O ) ⊂ O .Proof. Proof of (I): Consider any open subset U ⊂ [0 , ∞ ) s × E , tamesc-retraction r : U → U onto r ( U ) = O , and the associated retractionon the boundary (see Lemma 4.1) denoted by r ∂ : U ∂ → U ∂ where U ∂ = U ∩ ( { } s × E ) . Set K := ker D x ( f ∂ ◦ r ∂ ) ⊂ T x U ∂ = E . Let L ⊂ E ∞ be a sc-complement of ker D x f ∂ in T x O ∂ , which existsby Lemma 2.2 since the surjection D x f ∂ : T x O ∂ → R n induces anisomorphism T x O ∂ / ker D x f ∂ ∼ = R n . Note that the restriction(43) D x f ∂ | L : L → R n is an isomorphism. We claim that L is a sc-complement of K in E , i.e.(44) E = K ⊕ L. First, note that the sc-splitting E = T x O ∂ ⊕ ker D x r ∂ holds since D x r ∂ : E → E is a linear sc-retraction with image T x O ∂ . Hence we have E = ( L ⊕ ker D x f ∂ ) ⊕ ker D x r ∂ . From this description of E together with the definition of K , we claimthat ker D x f ∂ ⊕ ker D x r ∂ = K follows, which then implies (44). Indeed,write ξ ∈ E as ξ = l + η + ν for some l ∈ L, η ∈ ker D x f ∂ , and ν ∈ ker D x r ∂ . Note that D x r ∂ ( η ) = η and D x r ∂ ( l ) = l since η, l ∈ T x O ∂ .Then if ξ ∈ K we have 0 = D x ( f ∂ ◦ r ∂ )( ξ ) = D x f ∂ ( l ) + D x f ∂ ( η ) = D x f ∂ ( l ), which implies l = 0 by (43), and hence ξ = η + ν ∈ ker D x f ∂ ⊕ ker D x r ∂ . To prove the reverse inclusion, observe that if ξ = η + ν then D x ( f ∂ ◦ r ∂ )( ξ ) = D x f ∂ ( η ) = 0 and so ξ ∈ K . Hence (44) holds.We now prepare to apply the normal form of a sc-smooth local sub-mersion (Lemma 2.3) to the map f ◦ r : U → R n . Notice that the restriction of f ◦ r to U ∂ = U ∩ ( { } s × E ) is equal to f ∂ ◦ r ∂ , and hence by the hypothesized surjectivity of the map D x f ∂ : T x O ∂ → R n and the definition of tangent space T x O ∂ = D x r ∂ ( T x U ∂ ),it follows that the map F := D x ( f ◦ r | U ∩{ } s × E ) : E → R n is surjective with kernel K = ker F . Hence Lemma 2.3 applies to themap f ◦ r and the sc-complement L of K in E . The lemma provides anopen neighborhood ˆ U ⊂ U ⊂ [0 , ∞ ) s × E of x in U , an open set U ′ ⊂ R n × [0 , ∞ ) s × K , and a sc-smooth diffeomorphism of the form g : ˆ U → U ′ ( v, e ) ( f ◦ r ( v, e ) , v, pr ( e )) , where pr : E = L ⊕ K → K is the projection along L , such that onevery level m ≥ g : ˆ U m → U ′ m is a C m +1 -diffeomorphism.In particular,(45) f ◦ r ◦ g − : U ′ → R n is projection onto the R n -coordinateand also g preserves the [0 , ∞ ) s -coordinate. OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 43
We shrink the open sets ˆ U and U ′ in the following way, so thatthe smaller open set ˆ U ⊂ ˆ U has the property r ( ˆ U ) ⊂ ˆ U , making r | ˆ U : ˆ U → ˆ U into a sc-retraction: Setˆ U := r − ( ˆ U ∩ O ) ∩ ˆ U ,U ′ := g ( ˆ U ) . The set ˆ U is open in U because ˆ U ∩ O is open in O and r : U → O is sc-continuous. We claim that r ( ˆ U ) ⊂ ˆ U , so r restricts to a tamesc-retraction r : ˆ U → ˆ U onto the tame sc-retractˆ O := r ( ˆ U ) . Indeed, if y ∈ ˆ U then r ( r ( y )) = r ( y ) ∈ ˆ U ∩ O , and so it follows that r ( y ) ∈ ˆ U . Note that U ′ ⊂ R n × [0 , ∞ ) s × K is open since g is a sc-diffeomorphism, and moreover the restricted map g : ˆ U → U ′ is a sc-diffeomorphism.The map r ′ := g ◦ r ◦ g − : U ′ → U ′ is a sc-smooth retraction since g is a sc-diffeomorphism and r is a sc-smooth retraction. Indeed, r ◦ r = r implies that r ′ ◦ r ′ = r ′ . Set O ′ := r ′ ( U ′ ) . Notice thatˆ O = r ( ˆ U ) = r ◦ g − ( U ′ ) = g − ◦ r ′ ( U ′ ) = g − ( O ′ )holds, so g restricts to a bijection g : ˆ O → O ′ . This restriction is a sc-smooth diffeomorphism, as required, since g ◦ r :ˆ U → U ′ is sc-smooth and g − ◦ r ′ : U ′ → ˆ U is sc-smooth. Indeed,the definition of a map between sc-retracts being sc-smooth is thatthese compositions with the sc-retractions are sc-smooth, where thedomain and codomain are now open subsets of sc-Banach spaces andsc-smoothness is defined as usual.We now prove that r ′ is a R n -sliced sc-retraction, and so O ′ is therequired R n -sliced sc-retract (38). We must show that r is tame andsatisfies the defining property (13) of a R n -sliced retraction, i.e. that r preserves the R n -coordinate. To prove (13), notice from the definitionsand (45) that we have( f ◦ g − ) ◦ r ′ = f ◦ r ◦ g − = π R n . Since r ′ ◦ r ′ = r ′ , we have π R n ◦ r ′ = ( f ◦ g − ) ◦ r ′ ◦ r ′ = ( f ◦ g − ) ◦ r ′ = π R n , completing the proof of (13).We now verify that r ′ is a tame sc-retraction. The property Def-inition 3.1(1) for r ′ = g ◦ r ◦ g − holds because it holds for r andsince g preserves the [0 , ∞ ) s -coordinate. To verify property Defini-tion 3.1(2) for r ′ , let y ′ ∈ O ′∞ and set y := g − ( y ′ ) ∈ ˆ O ∞ . Since r is a tame sc-retraction, there exists a sc-subspace A ⊂ R s × E such that R s × E = T y ˆ O ⊕ A and A ⊂ ( R s × E ) y . We claim that A ′ := D y g ( A ) is the required sc-subspace of R n × R s × K . First,since g is the identity on [0 , ∞ ) s , the tangent map D y g is the iden-tity on R s , and thus A ′ ⊂ ( R n × R s × K ) y ′ by (8). Moreover, bydefinition of r ′ and the tangent space of a retract we have T y ′ O ′ = D y ′ r ′ ( T y ′ U ′ ) = D y g ( D y r ( T y ˆ U )) = D y g ( T y ˆ O ). Hence, since D y g isa linear sc-isomorphism, we have R n × R s × K = D y g ( R s × E ) = D y g ( T y ˆ O ⊕ A ) = T y ′ O ′ ⊕ A ′ , as required. This completes the verifi-cation that r ′ is a R n -sliced sc-retraction and hence (38) is indeed a R n -sliced sc-retract.To complete the proof of (I), it remains to show that (39) holds.First, recall from above that g ( ˆ O ) = O ′ . Let y ∈ g ( f − (0) ∩ ˆ O ) . Since g − ( y ) ∈ ˆ O is in the image of the retraction r , it follows that0 = f ◦ g − ( y ) = f ◦ r ◦ g − ( y ), and hence (45) implies y ∈ { } n × [0 , ∞ ) s × K , proving the forwards inclusion in (39). For the reverseinclusion, let y ∈ O ′ ∩ ( { } n × [0 , ∞ ) s × K ). Then, again by (45), wehave f ◦ g − ( y ) = f ◦ r ◦ g − ( y ) = π R n ( y ) = 0. This proves the reverseinclusion and hence the equality (39).Proof of (II): After shrinking the original open set U ⊂ [0 , ∞ ) s × E ,we can assume that there exists a tame bundle retraction R : U ✁ F → U ✁ F ( y, ξ ) ( r ( y ) , Γ ( y, ξ ))onto R ( U ✁ F ) = K covering r . Since r restricts to a retraction r :ˆ U → ˆ U onto ˆ O as discussed above, it follows that R resticts to a tamebundle retraction R : ˆ U ✁ F → ˆ U ✁ F with image the tame bundle OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 45 retract ˆ K = R ( ˆ U ✁ F ) = π − ( ˆ O ) , where π : K → O is the 1-shifted tame local bundle model.Set Γ ′ : U ′ ✁ F → F ( y ′ , ξ ) Γ ( g − ( y ′ ) , ξ )We claim that the map R ′ := U ′ ✁ F → U ′ ✁ F ( y ′ , ξ ) ( r ′ ( y ′ ) , Γ ′ ( y ′ , ξ ))is a R n -sliced bundle retraction onto K ′ := ( g ✁ id F )( ˆ K ) , proving that (40) is a R n -sliced bundle retract covering the R n -slicedsc-retract (38). It is then immediate from Lemma 3.5 that the inducedbundle retract ˜ K ′ covers the induced sc-retract ˜ O ′ , as claimed. Since r ′ = g ◦ r ◦ g − , we can write R ′ as a composition of strong bundle maps R ′ = ( g ✁ id F ) ◦ R ◦ ( g − ✁ id F ) , and hence R ′ is a strong bundle map. In addition, R ′ ◦ R ′ = R ′ followsfrom R ◦ R = R . Thus R ′ is a bundle retraction covering the R n -slicedsc-retraction r ′ , which implies that R ′ is a R n -sliced bundle retraction,as desired. Moreover, by definition of K ′ we have R ′ ( U ′ ✁ F ) = ( g ✁ id F ) ◦ R ( ˆ U ✁ F ) = ( g ✁ id F )( ˆ K ) = K ′ . Thus K ′ is indeed the desired R n -sliced bundle retract (40).We now prove (41). Let π ′ : K ′ → O ′ denote the R n -sliced localbundle model, which restricts to the induced tame local bundle model˜ π = π ′ | ˜ K ′ : ˜ K ′ → ˜ O ′ (see (25)). Here ˜ K ′ = π ′− ( ˜ O ′ ) is the tamebundle retract induced by the R n -sliced bundle retract K ′ (see (24)).To prove (41), let y ∈ f − (0) ∩ ˆ O and let ξ ∈ π − ( y ) . Then, by (39),we have g ( y ) ∈ ˜ O ′ , and hence ( g ✁ id F )( ξ ) ∈ π ′− ( ˜ O ′ ) = ˜ K ′ , provingthe forwards inclusion. To see the reverse inclusion, let ξ ∈ ˜ K ′ . Set y = π ′ ( ξ ) ∈ ˜ O ′ . Then, again by (39), we have g − ( y ) ∈ f − (0) ∩ ˆ O andhence ( g − ✁ id F )( ξ ) ∈ π − ( f − (0) ∩ ˆ O ) . This proves (41), completingthe proof of the statements in (II).Proof of (III): To prove that the section σ ′ := ( g ✁ id F ) ◦ σ ◦ g − : O ′ → K ′ is a R n -sliced sc-Fredholm germ (Definition 3.8), we must first showthat it is a local sc-Fredholm germ (Definition 3.6). Since the givensection σ is a tame sc-Fredholm germ (Definition 3.7) by hypothesis, byshrinking the original open set U ⊂ [0 , ∞ ) s × E we can assume that thetame sc-retraction r : U → U satisfies the properties guaranteed by thedefinition of a tame sc-Fredholm germ. Moreover, the 1-shifted section σ | O : O → K is also a tame sc-Fredholm germ (see Remark 3.9),and since ˆ U ⊂ U is an open subset of the 1-shift of U , the retraction r : ˆ U → ˆ U onto r ( ˆ U ) = ˆ O and the tame bundle retraction R covering r also satisfy the properties required in the definition of tame sc-Fredholmgerm for σ | ˆ O : ˆ O → ˆ K . We explain these properties as we use them.There exists a sc-germ of neighborhoods ˆ U ([0 , ∞ ) s × E ,
0) around0 such that the 0-level ˆ U (0) in the germ is equal to ˆ U and such thatthe composition σ ◦ r : ˆ U ([0 , ∞ ) s × E , → F possesses a filling (seeDefinition 3.6(b)) h : ˆ U ([0 , ∞ ) s × E , → ˆ U ([0 , ∞ ) s × E , ✁ F , where σ means the principal part in F of σ . By hypothesis and theconstruction of ˆ O , we have x = 0 ∈ f − (0) ∩ ˆ O , and so it follows fromthe form of g above that(46) g (0) = 0 ∈ O ′ . The sc-germ of neighborhoods around 0 given by the image of ˆ U ([0 , ∞ ) s × E ,
0) under g , i.e. U ′ ( R n × [0 , ∞ ) s × K ,
0) := g ( ˆ U ([0 , ∞ ) s × E , , has 0-level U ′ (0) equal to the set open set U ′ = g ( ˆ U ) above. Fromnow on, we abbreviate the sc-germs of neighborhoods simply by their0-level, i.e. by ˆ U and U ′ .We claim that the map h ′ := ( g ✁ id F ) ◦ h ◦ g − : U ′ → U ′ ✁ F . is a filling of σ ′ ◦ r ′ : U ′ → F . The required properties Defini-tion 3.6(b).(i)-(iii) of a filling hold for h ′ because they hold for h , as weexplain now. To verify ( i ) for h ′ , let y ′ ∈ O ′ . Then y := g − ( y ′ ) ∈ ˆ O ,so by ( i ) for h we have σ ( y ) = h ( y ), which implies σ ′ ( y ′ ) = h ′ ( y ′ ) bydefinition of σ ′ and h ′ , as required. To verify (ii), let y ′ ∈ U ′ and as-sume h ′ ( y ′ ) = Γ ′ ( r ′ ( y ′ ) , h ′ ( y ′ )). Then, setting y := g − ( y ′ ) ∈ ˆ U , wehave h ( y ) = h ′ ( y ′ ) = Γ ( r ( y ) , h ( y )). Hence y ∈ ˆ O by property ( ii ) for h , so y ′ ∈ g ( ˆ O ) = O ′ , as required. We now verify property (iii) for h ′ . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 47
Property ( iii ) for h says that the linearization D A at 0 of the map A : ˆ U → F y ( id F − Γ ( r ( y ) , · )) h ( y )restricts to a linear sc-isomorphism ker D r → ker Γ (0 , · ) . We mustshow that the map A ′ : U ′ → F y ′ ( id F − Γ ′ ( r ′ ( y ′ ) , · )) h ′ ( y ′ )restricts to a linear sc-isomorphism ker D r ′ → ker Γ ′ (0 , · ) . Note that,by definition of Γ ′ and by (46), we have ker Γ ′ (0 , · ) = ker Γ ( g − (0) , · ) =ker Γ (0 , · ) . Moreover, from the definition of h ′ we have h ′ ( y ′ ) = h ◦ g − ( y ′ ) , and so by definition of r ′ we have A ′ = A ◦ g − : U ′ → F . Hence D A ′ restricts to the desired isomorphism since D g − restrictsto an isomorphism ker D r ′ → ker D r by definition of r ′ and (46).Hence h ′ is indeed a filling of σ ′ ◦ r ′ , as claimed.To verify that σ ′ is a local sc-Fredholm germ, it remains to checkthe properties in Defintion 3.6(c). Since σ : ˆ O → ˆ K is a tame sc-Fredholm germ, we have the following: There exists a sc + -section s :ˆ U → ˆ U ✁ F satisfying s (0) = h (0), a sc-Banach space W , a sc-germ ofneighborhoods ˆ U ′ around 0 in [0 , ∞ ) s × R k − s × W for some k ≥ s , and a strong bundle isomorphism Ψ : ˆ U ✁ F → ˆ U ′ ✁ R k ′ × W , for some k ′ ≥
0, covering a linear sc-isomorphism ψ : ˆ U → ˆ U ′ of the form ψ = id [0 , ∞ ) s × ψ : [0 , ∞ ) s × E → [0 , ∞ ) s × R k − s × W for some linear sc-isomorphism ψ : E → R k − s × W , such that the principal part of the section(47) b := Ψ ◦ ( h − s ) ◦ ψ − : ˆ U ′ → ˆ U ′ ✁ R k ′ × W is a basic germ. By the hypothesis (42), we can assume that the subspace L ⊂ E ∞ satisfies ψ ( L ) ⊂ { } k − s × W . We consider ψ ( L ) as a sc-subspace of W and let ˜ W be any sc-complement ˜ W ⊕ ψ ( L ) = W which exists by[7, Prop. 1.1] since L is finite dimensional. After fixing any linearisomorphism ψ ( L ) → R n , we obtain a linear sc-isomorphism τ : W = ˜ W ⊕ ψ ( L ) → ˜ W × R n . Define ψ ′ := ( id [0 , ∞ ) s × R k − s × τ ) ◦ ψ ◦ g − : U ′ → [0 , ∞ ) s × R k − s × ˜ W × R n , s ′ := ( g ✁ id F ) ◦ s ◦ g − : U ′ → U ′ ✁ F , ˆ U ′′ := ψ ′ ( U ′ ) = ( id [0 , ∞ ) s × R k − s × τ )( ˆ U ′ ) ,Ψ ′ := (( id [0 , ∞ ) s × R k − s × τ ) ✁ ( id R k ′ × τ )) ◦ Ψ ◦ ( g − ✁ id F ) : U ′ ✁ F → ˆ U ′′ ✁ R k ′ × ˜ W × R n . We claim that the principal part of the section(48) b ′ := Ψ ′ ◦ ( h ′ − s ′ ) ◦ ψ ′− : ˆ U ′′ → ˆ U ′′ ✁ R k ′ × ( ˜ W × R n )is a basic germ, where here ˜ W × R n plays the role of the space W inDefinition 3.6(c). By hypothesis, the principal part b of the map (47)is a basic germ, which means that, letting P : R k ′ × W → W denote projection onto W , we have, for a ∈ [0 , ∞ ) s × R k − s and w ∈ W , P ◦ b ( a, w ) = w − B ( a, w ) , where B : ˆ U ′ → W is a sc-smooth germ satisfying B (0) = 0 and thecontraction property (26). Let P ′ : R k ′ × ( ˜ W × R n ) → ˜ W × R n be projection onto ˜ W × R n . Define B ′ := τ ◦ B ◦ ( id [0 , ∞ ) s × R k − s × τ ) − : ˆ U ′′ → ˜ W × R n . Notice from the definitions that we have P ′ = τ ◦ P × ( id R k ′ × τ ) − . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 49
Then, for a ∈ [0 , ∞ ) s × R k − s and z ∈ ˜ W × R n such that ( a, z ) ∈ ˆ U ′′ ,we compute P ′ ◦ b ′ ( a, z ) = τ ◦ P ◦ ( id R k ′ × τ ) − ◦ π R k ′ × ˜ W × R n ◦ Ψ ′ ◦ ( h ′ − s ′ ) ◦ ψ ′− ( a, z )= τ ◦ P ◦ π R k ′ × W ◦ Ψ ◦ ( h − s ) ◦ ψ − ( a, τ − ( z ))= τ ◦ P ◦ b ( a, τ − ( z ))= τ ( τ − ( z ) − B ( a, τ − ( z )))= z − B ′ ( a, z ) , so to prove that b ′ is a basic germ, it remains to verify that B ′ satisfiesthe contraction property (26). Since τ is a linear sc-isomorphism, forevery integer m ≥ c m , C m > w ∈ W , we have || τ ( w ) || m ≤ C m · || w || m , and for all ( a, z ) ∈ ([0 , ∞ ) s × R k − s ) × ( ˜ W × R n ) we have || ( id R k × τ ) − ( a, z ) || m ≤ c m · || ( a, z ) || m . Fix ǫ > m ≥
0. Then since B satisfies the contraction property(26), there exists δ ′ m > B and ǫ ′ m := ǫ/ ( C m · c m ).We claim that δ m := δ ′ m /c m suffices for B ′ and ǫ . Indeed, given a ∈ [0 , ∞ ) s × R k − s and z, z ′ ∈ ˜ W × R n satisfying || ( a, z ) || m , || ( a, z ′ ) || m < δ m ,we have || ( a, τ − ( z )) || m , || ( a, τ − ( z ′ )) || m < c m · δ m = δ ′ m . So, we compute || B ′ ( a, z ) − B ′ ( a, z ′ ) || m = || τ ( B ( a, τ − ( z )) − B ( a, τ − ( z ′ ))) || m ≤ C m · || B ( a, τ − ( z )) − B ( a, τ − ( z ′ )) || m ≤ C m · ǫ ′ m · || τ − ( z ) − τ − ( z ′ ) || m = ǫ/c m · || τ − ( z − z ′ ) || m ≤ ǫ · || z − z ′ || m , as required. This completes the proof that b ′ is a basic germ, andhence that σ ′ is a local sc-Fredholm germ.We claim that the additional conditions required for σ ′ to be a R n -sliced sc-Fredholm germ are satisfied by construction. We must checkthat ψ ′ : U ′ → ˆ U ′′ is of the required form (28). Consider the projection T : W = ˜ W ⊕ ψ ( L ) → ˜ W along ψ ( L ) ⊂ { } k − s × W . We have L ⊕ K = E by (44) and since L ⊂ E ∞ . Hence the map ψ ′ := ( id R k − s × T ) ◦ ψ | { }⊕ K : K → R k − s × ˜ W is a linear sc-isomorphism. Let ( p, v, κ ) ∈ U ′ ⊂ R n × [0 , ∞ ) s × K . Bythe form of the map g above, we can write g − ( p, v, κ ) =: ( v, e p,v,κ ) ∈ [0 , ∞ ) s × E for some e p,v,κ ∈ E . Moreover, we have pr ( e p,v,κ ) = κ , where pr is theprojection pr : E = L ⊕ K → K along L . In particular, we have e p,v,κ − κ ∈ L , from which we obtain a map into { } × R n ⊂ ˜ W × R n defined by λ : U ′ → R n ( p, v, κ ) τ ( ψ ( e p,v,κ − κ )) . This map is C on every level ( U ′ ) m → R n for m ≥ g is a C -diffeomorphism on every level and the linear sc-isomorphisms τ and ψ are C ∞ on every level. Then we compute ψ ′ ( p, v, κ ) = ( id [0 , ∞ ) s × R k − s × τ ) ◦ ψ ◦ ( v, e p,v,κ )= ( id [0 , ∞ ) s × R k − s × τ ) ◦ ( id [0 , ∞ ) s × ψ ) ◦ ( v, e p,v,κ )= ( v, ( id R k − s × τ ) ◦ ψ ( e p,v,κ ))= ( v, ( id R k − s × τ )( ψ ( κ ) + ψ ( e p,v,κ − κ )))= ( v, ( id R k − s × T )( ψ ( κ )) , τ ( ψ ( e p,v,κ − κ )))= ( v, ψ ′ ( κ ) , λ ( p, v, κ )) . This proves that σ ′ is a R n -sliced sc-Fredholm germ. Hence, by Lemma 3.10,the restriction ˜ σ ′ = σ ′ | ˜ O ′ : ˜ O ′ → ˜ K ′ is a tame sc-Fredholm germ withsc-Fredholm index ind (˜ σ ′ ) = ind ( σ ′ ) − n . Moreover, by definition (27)of index and by the forms of the basic germs (47) and (48), we have ind ( σ ′ ) = ind ( σ ). Hence ind (˜ σ ′ ) = ind ( σ ) − n . This completes theproof of the lemma. (cid:3) Remark 4.3.
We note some additional properties of the objects con-structed in the proof of Lemma 4.2. These properties are required forthe construction of quotients of polyfolds by group actions in [19] .The sc-Banach space K is ker D x ( f ∂ ◦ r ∂ ) ⊂ E , where r ∂ : U ∩ ( { } × E ) → U ∩ ( { } × E ) is the boundary sc-retraction associated to a tamesc-retraction r : U → U onto r ( U ) = O from Lemma 4.1. Thereis a sc-complement L of ker D x f ∂ in T x O ∂ , and moreover L is a sc-complement of K in E . The change of coordinates g : [0 , ∞ ) s × E → R n × [0 , ∞ ) s × K , which is defined in a neighborhood of x , is of theform g ( v, e ) = ( f ◦ r ( v, e ) , v, pr ( e )) where pr : E = L ⊕ K → K isprojection along L . In particular, in the case x = 0 , the sc-differential OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 51 of g satisfies D g ( { } × L ) = R n × { } × { } and D g ( { } × K ) = { } × { } × K . Slicing tame sc-Fredholm sections with transverseconstraints
The theory of sc-Fredholm sections σ : B → E (Definition 5.3) oftame strong bundles ρ : E → B (Definition 5.2) over tame M -polyfolds B (Definition 5.1) is developed in [7]. In this section, we introduce thenew stronger notion of a tame sc-Fredholm section (Definition 5.4),which is the same as a standard sc-Fredholm section with the strongercondition that the local sc-Fredholm germs (Definition 3.6) can be cho-sen to be the tame sc-Fredholm germs introduced in Definition 3.7:roughly, we require that the change of coordinates that brings the localfillers into basic germ form can be chosen to be a linear sc-isomorphism on the base. We also introduce the new notion of slices ˜ B ⊂ B (Defini-tion 5.7) of sc-Fredholm sections σ , which are subspaces of B in whichthe local sc-Fredholm germs are R n -sliced (Definition 3.8): roughly,the dependence of the change of coordinates to basic germ form on thenormal directions to ˜ B in B splits off into a R n -factor in the codomain.Tame sc-Fredholm sections and slices are the main objects of interestin this paper. In this section, we prove our main Theorems 1.3, 1.5.Roughly, given a tame sc-Fredholm section σ : B → E and a sc-smoothmap f : B → Y to a finite dimensional manifold Y that is σ -compatiblytransverse (Definition 5.8) to a submanifold N ⊂ Y , then f − ( N ) isa slice of σ . Moreover, given a slice ˜ B ⊂ B of a sc-Fredholm section σ : B → E , the restriction σ | ˜ B is tame sc-Fredholm. The generalizationsof the concepts and theorems in this section to the ep-groupoid case,which are required to handle isotropy in applications, are developed inSection 6.In Section 5.1, we explain why the Cauchy-Riemann section ∂ J : B → E is tame sc-Fredholm and why evaluation maps ev : B → Y at marked points are ∂ J -compatibly transverse to every submanifold N ⊂ Y .We begin with a review of standard M -polyfold notions. Definition 5.1. [7, Defs. 2.6, 2.19] • A tame M -polyfold chart on a topological space B is a tuple ( V, ϕ, ( O , C, E )) , where V ⊂ B is an open set, ϕ : V → O is a homeomorphism, and ( O , C, E ) is a tame sc-retract (Definition 3.1). • A tame M -polyfold B is a Hausdorff paracompact topologicalspace together with an equivalence class of tame sc-smooth atlases. A tame sc-smooth atlas consists of a cover of B by tame M -polyfoldcharts whose transition maps are sc-smooth. Atlases are equivalentif and only if their union is an atlas. A tame M -polyfold B carries a filtration by a sequence of topologicalspaces B = (cid:0) B ← ֓ B ← ֓ · · · (cid:1) where the maps B m → B m +1 are continuous injections with dense im-age. This filtration is provided locally by the sc-structure on the sc-retracts in charts, and it is independent of the choice of charts due tosc-continuity of the transition maps. The smooth points B ∞ of B arethe subset B ∞ := ∩ m ≥ B m . By forgetting about the first m ≥ B , we obtain the m -shift B m of B , which is the M -polyfold B m := (cid:0) B m ← ֓ B m +1 ← ֓ · · · (cid:1) with charts given by restricting charts on B to the m -shift ϕ : V m = V ∩ B m → O m = O ∩ E m . Notice that a m -shift does not affect thesmooth points, i.e. B ∞ = B m ∞ . The tangent space T x B at a point x ∈ B is defined as the set ofequivalence classes of tangent vectors in all charts around x , with theequivalence given by the tangent map of the transition map betweencharts, which is a linear isomorphism on every tangent space; see [7,Def. 2.11] for a precise treatment. The result is a Banach space struc-ture on T x B and, for any chart ( V, ϕ, ( O , C, E )) containing x ∈ V , atangent map D x ϕ : T x V = T x B → T ϕ ( x ) O which is a bounded lin-ear isomorphism. If x ∈ B ∞ is a smooth point then ϕ ( x ) ∈ O ∞ is asmooth point and T x B inherits the structure of a sc-Banach space fromthe sc-Banach space T ϕ ( x ) O .The reduced tangent space T Rx B [7, Def. 2.20] is defined only forsmooth points x ∈ B ∞ as the subspace of T x B such that(49) D x ϕ ( T Rx B ) = T Rϕ ( x ) O , where T Rϕ ( x ) O is the reduced tangent space to a sc-retract (12). Notethat, since the reduced tangent space to a sc-retract is invariant undersc-diffeomorphisms of sc-retracts only at smooth points, this globalnotion of reduced tangent space T Rx B is well-defined only for smoothpoints x ∈ B ∞ . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 53
The degeneracy index d B : B → N is defined for x ∈ B as theminimum over all charts ( V, ϕ, ( O , C, E )) containing x ∈ V of the de-generacy index in the chart d C ( ϕ ( x )); see [7, Def. 2.13]. For a smoothpoint x ∈ B ∞ , it is explained in [7, Rmk. 2.4] that there is a globalinterpretation of this quantity:(50) d B ( x ) = dim( T x B /T Rx B ) for x ∈ B ∞ . We now recall the definition of a tame strong bundle ρ : E → B . Definition 5.2. [7, Defs. 2.25, 2.26] • Consider a continuous surjective map ρ : E → B from a topologicalspace E onto a topological space B such that, for every x ∈ B , thefiber ρ − ( x ) = E x has the structure of a Banach space. A tamestrong bundle chart for ρ : E → B is a tuple ( ρ − ( V ) , Φ, ( K, C ✁ F , E ✁ F )) in which ( K, C ✁ F , E ✁ F ) is a tame strong bundle retract (Sec-tion 3.2) covering a tame sc-retract ( O , C, E ) , i.e. K → O is a tamestrong local bundle model. In addition, V ⊂ B is the domain of atame M -polyfold chart ( V, ϕ, ( O , C, E )) and Φ : ρ − ( V ) → K is a homeomorphism onto K that covers ϕ , i.e. (51) π O ◦ Φ = ϕ ◦ ρ holds on ρ − ( V ) . Moreover, Φ is linear on the fibers over V , i.e.for all x ∈ V the map Φ : ρ − ( x ) → π − O ( ϕ ( x )) is a bounded linearisomorphism of Banach spaces. • A tame strong bundle ρ : E → B is a continuous surjective mapfrom a paracompact Hausdorff space E onto a tame M -polyfold B such that every fiber has the structure of a Banach space, togetherwith an equivalence class of tame bundle atlases. A tame bundleatlas consists of a cover of B by tame strong bundle charts whosetransition maps are sc-smooth. Atlases are equivalent if and only iftheir union is an atlas. The tame strong bundle charts on a tame strong bundle ρ : E → B induce a double filtration E m,k for m ≥ ≤ k ≤ m + 1 from thedouble filtration (22) on the bundle retracts in the charts. From thiswe distinguish the M -polyfolds E [ i ] for i = 0 , , by the filtrations(52) E [ i ] m = E m,m + i , m ≥ . Both projections ρ [ i ] : E [ i ] → B are sc-smooth maps.There are two notions of smoothness for a section of ρ (see [7,Def. 2.27]), which correspond to the two projections ρ [ i ] for i = 0 , s : B → E is called a sc-smooth section of ρ if s : B → E [0]is a sc-smooth section of the bundle ρ [0] . The section s is called a sc + -section of ρ if s ( B ) ⊂ E [1] and s : B → E [1] is a sc-smooth section ofthe bundle ρ [1] . The sc + -sections become important when perturbat-ing sc-Fredholm sections. They are analogous to compact operatorsin classical Fredholm theory. In particular, adding sc + -sections to sc-Fredholm sections preserves the sc-Fredholm property [7, Thm. 3.2].We now recall the definition of a sc-Fredholm section σ : B → E . Definition 5.3. [7, Def. 3.8] • Consider a section σ : B → E of a tame strong bundle ρ : E → B .Let x ∈ B ∞ be a smooth point. Then a tame strong bundle chart ( ρ − ( V ) , Φ, ( K, C ✁ F , E ✁ F )) for ρ covering a tame M -polyfold chart ( V, ϕ, ( O , C, E )) on B is called a sc-Fredholm chart for σ at x ifit satisfies x ∈ V and ϕ ( x ) = 0 ∈ O , and if the section Φ ◦ σ ◦ ϕ − : O → K of the local bundle model K → O is a local sc-Fredholm germ (Defi-nition 3.6). • A sc-Fredholm section σ : B → E of ρ is a section that has thefollowing properties:(1) σ is sc-smooth.(2) σ is regularizing , which means that if x ∈ B m and σ ( x ) ∈E m,m +1 then x ∈ B m +1 .(3) For every smooth point x ∈ B ∞ , there exists a sc-Fredholmchart for σ at x . The index ind x ( σ ) of σ at x is the index (27) of the local sc-Fredholm germ in any such chart. We now introduce the new stronger notion of a tame sc-Fredholmsection σ : B → E of a tame M -polyfold bundle ρ : E → B . Definition 5.4. • Consider a section σ : B → E of a tame strong bundle ρ : E → B .Let x ∈ B ∞ be a smooth point. Then a tame sc-Fredholm chartfor σ at x is a sc-Fredholm chart (Definition 5.3) such that thelocal sc-Fredholm germ Φ ◦ σ ◦ ϕ − is in addition a tame sc-Fredholmgerm (Definition 3.7). • A tame sc-Fredholm section σ : B → E of ρ is a sc-Fredholmsection (Definition 5.3) such that there exists a tame sc-Fredholmchart for σ at every x ∈ B ∞ . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 55
Remark 5.5.
Given a strong bundle ρ : E → B and a sc-Fredholmsection σ : B → E , we obtain their m -shifts ρ : E m → B m and σ : B m → E m by restricting domains and codomains to the m -shifts E m and B m . Performing this shift preserves all essential features of thepolyfold setup. In particular, σ : B m → E m is a sc-Fredholm section ofthe strong bundle ρ : E m → B m and the sc-Fredholm index is preserved [7, Cor. 5.1] .Crucially, the zero set of a sc-Fredholm section is preserved under m -shifts. Indeed, since σ − (0) ⊂ B ∞ by the regularizing property Def-inition 5.3(2) of σ , it follows that σ − (0) = σ − (0) ∩ B m = σ | − B m (0) .Moreover, compactness of σ − (0) in the topology of B m for any fixed m ≥ is equivalent to compactness in the topology of B ∞ (see [7,Thm. 5.3, Cor. 5.1] ). For this reason, we can unambiguously refer tocompactness of σ − (0) without reference to the levels, and m -shiftingpreserves this compactness. We now introduce the new notions of a slice ˜
B ⊂ B of a tame M -polyfold B , a slice of a tame strong bundle ρ : E → B , and a slice ofa sc-Fredholm section σ : B → E (Definition 5.7). Roughly, a sliceis a subspace ˜ B such that for every point x ∈ ˜ B there is a tame M -polyfold chart that identifies a neighborhood of x in B with a R n -slicedsc-retract (Definition 3.2), and such that the induced tame sc-retract(14) is identified with a neighborhood of x in ˜ B . To be a slice of thesc-Fredholm section σ , we additionally require that, at smooth points x ∈ ˜ B ∞ = ˜ B ∩ B ∞ , we can choose the chart on B such that it iscovered by a bundle chart for ρ in which σ is a R n -sliced sc-Fredholmgerm (Definition 3.8). In such a chart, the induced tame sc-Fredholmgerm (see Lemma 3.10) is identified with the restriction of σ to ˜ B near x . Globally, the result is that a slice of σ is a finite codimension M -polyfold ˜ B embedded in B that is compatible with the local fillers for σ in such a way that σ restricts to a tame sc-Fredholm section over ˜ B (Theorem 1.3(III)).We first introduce the slice notions in charts. Definition 5.6. • Consider a subspace ˜ B ⊂ B of a tame M -polyfold B . A R n -slicedchart with respect to ˜ B ⊂ B is a tame M -polyfold chart on B ofthe form (53) ( V, ϕ, ( O , R n × C, R n × E )) where ( O , R n × C, R n × E ) is a R n -sliced sc-retract (Definition 3.2)with induced tame sc-retract ˜ O = O ∩ ( { } × C ) (see (14) ) satisfying (54) ˜ O = ϕ ( ˜ B ∩ V ) . • Consider a tame strong bundle ρ : E → B . A R n -sliced bundlechart for ρ with respect to ˜ B ⊂ B is a tame strong bundle chartfor ρ of the form (55) ( ρ − ( V ) , Φ, ( K, R n × C ✁ F , R n × E ✁ F )) covering a R n -sliced chart with respect to ˜ B ⊂ B as in (53) . Inparticular, the tame bundle retract ( K, R n × C ✁ F , R n × E ✁ F ) isa R n -sliced bundle retract (Definition 3.4) covering the sc-retract O from (53) . • Consider a section σ : B → E of ρ and a smooth point x ∈ ˜ B ∞ =˜ B ∩ B ∞ . Then a R n -sliced sc-Fredholm chart for σ at x withrespect to ˜ B ⊂ B is a R n -sliced bundle chart with respect to ˜ B ⊂ B as in (55) that satisfies x ∈ V and ϕ ( x ) = 0 , and such that thesection Φ ◦ σ ◦ ϕ − : O → K is a R n -sliced sc-Fredholm germ (Definition 3.8). The following global notions of slices of tame M -polyfolds, tamestrong bundles, and sc-Fredholm sections require covers of ˜ B by R n -sliced charts. Definition 5.7. • Consider a tame M -polyfold B . A subspace ˜ B ⊂ B is called a sliceof B if for every x ∈ ˜ B there exists an integer n x = codim x ( ˜ B ⊂B ) ≥ and a R n x -sliced chart with respect to ˜ B ⊂ B that contains x . For x ∈ ˜ B = ˜ B ∩ B , the integer codim x ( ˜ B ⊂ B ) is called the codimension of the slice at x (and it is well-defined at x andlocally constant in ˜ B , by Theorem 1.3). • Consider a tame strong bundle ρ : E → B . A slice ˜ B ⊂ B of B is called a slice of the bundle ρ if for every x ∈ ˜ B there exists a R n x -sliced bundle chart for ρ with respect to ˜ B ⊂ B that contains x . • Consider a sc-Fredholm section σ : B → E of ρ . A slice ˜ B ⊂ B ofthe bundle ρ is called a slice of the sc-Fredholm section σ if forevery x ∈ ˜ B ∞ = ˜ B ∩ B ∞ there exists a R n x -sliced sc-Fredholm chartfor σ at x with respect to ˜ B ⊂ B . We are now equipped prove one of the main theorems (Theorem 1.3),which says that slices of tame M -polyfolds are tame M -polyfolds with OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 57 locally constant codimension, bundles restrict to slices, and sc-Fredholmsections restrict to slices with a drop in Fredholm index given by thecodimension. See Section 5.2 for the proof. The essential point inthe proof is that sliced charts with respect to ˜
B ⊂ B induce tame M -polyfold charts on ˜ B , sliced bundle charts induce tame bundle charts onthe restricted bundle map, and sliced sc-Fredholm charts induce tamesc-Fredholm charts on the restricted section. This is the M -polyfoldanalog of the classical Fact 1.1 (restrictions of Fredholm sections tosub-Banach manifolds).We now prepare for the proof of our main theorem (Theorem 1.5),which is the M -polyfold analog of the classical Fact 1.2 (transversepreimages are sub-Banach manifolds). The result is roughly as follows.Consider a tame sc-Fredholm section σ : B → E of a tame M -polyfoldbundle ρ : E → B . Given a sc-smooth map f : B → Y to a finitedimensional smooth manifold Y that is σ -compatibly transverse (Def-inition 5.8) to a submanifold N ⊂ Y , we prove that the restriction of σ to the preimage f − ( N ) is a tame sc-Fredholm section. The notionof σ -compatible transversality requires compatibility between the map f , the submanifold N , and the coordinate changes that bring the localfillers of σ into basic germ form. All necessary compatibility is satis-fied in applications. In particular, evaluation maps f = ev at markedpoints are ∂ J -compatibly transverse to every submanifold, where ∂ J isthe Cauchy-Riemann section; see Section 5.1. We now introduce theprecise definition of σ -compatibly transverse. Definition 5.8. • Consider a tame M -polyfold B , a smooth manifold Y together witha codimension- n submanifold N ⊂ Y , and a sc-smooth map f : B → Y . Then f is called transverse to N if, for all x ∈ f − ( N ) ∩ B ∞ , (56) D x f ( T Rx B ) + T f ( x ) N = T f ( x ) Y holds. • Consider a tame sc-Fredholm section σ : B → E (Definition 5.4) ofa tame strong bundle ρ : E → B . Then f is called σ -compatiblytransverse to N if, for all x ∈ f − ( N ) ∩ B ∞ , condition (56) holdsand moreover there exists a tame sc-Fredholm chart (Definition 5.4)for σ at x , denoted ( ρ − ( V ) , Φ, ( K, [0 , ∞ ) s × E ✁ F , R s × E ✁ F )) ,covering a tame M -polyfold chart ( V, ϕ, ( O , [0 , ∞ ) s × E , R s × E )) on B such that the tame sc-Fredholm germ Φ ◦ σ ◦ ϕ − : O → K (Defini-tion 3.7) satisfies the following property: There exists a choice of sc-Banach space W and linear sc-isomorphism ψ : E → R k − s × W satis-fying the conditions in Definition 3.7 of tame sc-Fredholm germ such that, in addition, there exists a sc-complement L of ( D x f ) − ( T f ( x ) N ) ∩ T Rx B in T Rx B satisfying (57) ψ ◦ D x ϕ ( L ) ⊂ { } k − s × W . We are now equipped to prove the main theorem (Theorem 1.5). SeeSection 5.3 for the proof.5.1.
Example: The Cauchy-Riemann section and evaluationmaps at marked points.
Consider a symplectic manifold (
Y, ω ), acodimension- n submanifold N ⊂ Y , and an ω -compatible almost com-plex structure J on Y . In applications, the M -polyfold B in Theo-rem 1.5 (or, in the presence of isotropy, the ep-groupoid X in Corol-lary 6.8) is a space of maps Σ → Y modulo reparameterization of thedomain Σ , where Σ varies in some Deligne-Mumford space of Riemannsurfaces, and σ = ∂ J : B → E is the Cauchy-Riemann section associ-ated to J ; see for example the Gromov-Witten polyfolds constructedin [13]. We consider the evaluation map f = ev : B → Y at a markedpoint that varies in the domains Σ . The purpose of this section is toexplain the following properties of this setup: • σ is a tame sc-Fredholm section (Defintion 5.4), • f is σ -compatibly transverse to N (Definition 5.8).To see that σ is a tame sc-Fredholm section, we consider a smoothpoint x ∈ B ∞ and explain why σ is a tame sc-Fredholm germ (Defini-tion 3.7) in the chart constructed around x when building the polyfold B . The tame sc-retract O ⊂ [0 , ∞ ) s × R t × D × E is homeomorphic to the image of the pregluing map near x . Here the[0 , ∞ ) s -factor is gluing parameters near the broken points of x , the R t -factor is gluing parameters near the nodal points of x , the space D isvariations of the complex structure on the domain Σ of x (i.e. tangentdirections to the Deligne-Mumford space), and the sc-Banach space E corresponds to varying the map x while preserving the matchingconditions at nodes and breaking points. The tame bundle retract K ⊂ [0 , ∞ ) s × R t × D × E ✁ F is homeomorphic to the image of the pregluing map in the fibers near x . From this local bundle model K → O we obtain a tame M -polyfoldchart on B with domain V ⊂ B and chart map ϕ : V → O where x ∈ V The evaluation map at k marked points B → Y k can be treated similarly; weconsider a single marked point here for simplicity. OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 59 is identified with ϕ ( x ) = 0 ∈ O . It is covered by a tame strong bundlechart Φ : E | V → K .The section Φ ◦ σ ◦ ϕ − : O → K is a local sc-Fredholm germ (Def-inition 3.6), and moreover the natural change of coordinates of thefilling to basic germ form satisfies the required conditions of a tamesc-Fredholm germ (Definition 3.7). Indeed, a filling h : [0 , ∞ ) s × R t × D × E → [0 , ∞ ) s × R t × D × E ✁ F of Φ ◦ σ ◦ ϕ − is constructed as in [13, Sec. 4.5]. The change of coor-dinates ψ on the domain of the filling h that brings h into basic germform (Definition 3.6(c)) is obtained as follows; see [13, Prop. 4.8] fordetails. Let P := ker( D h ) ∩ ( { } s + t × { } × E )denote the kernel of the linearization of h at 0 in the directions E . Notethat P is finite dimensional since D h is a sc-Fredholm operator. Then,the sc-Banach space W and linear sc-isomorphism ψ : E ′ := R t × D × E → R k − s × W required in Definition 3.7 are obtained by choosing any sc-splitting E = P ⊕ W and linear isomorphism D × P ∼ = R k − t − s to obtain thelinear sc-isomorphism ψ : E ′ = R t × D × P ⊕ W → R t × R k − t − s × W . That is, the linear sc-isomorphism ψ = id [0 , ∞ ) s × ψ is a suitable choiceof the sc-diffeomorphism in Definition 3.6(c) of local sc-Fredholm germ,so in particular Φ ◦ σ ◦ ϕ − is a tame sc-Fredholm germ. It follows that σ = ∂ J is a tame sc-Fredholm section.Suppose x ∈ f − ( N ) ∩B ∞ . To show that f is σ -compatibly transverseto N at x , we now construct a sc-subspace L ⊂ T Rx B which satisfiesthe required conditions, in particular condition (57).Let Z be any complement of T f ( x ) N in T f ( x ) Y . First we claim thatit suffices to construct L ⊂ T Rx B satisfying(i) D x f ( L ) = Z and D x f : L → Z is an isomorphism,(ii) D x ϕ ( L ) ⊂ { } s + t × { } × E ,(iii) D x ϕ ( L ) ∩ P = { } . Observe that f is transverse to N at x by (i); indeed, the required span-ning property (56) holds since D x f ( L ) + T f ( x ) N = T f ( x ) Y . Moreover,we claim that L is a sc-complement of A := ( D x f ) − ( T f ( x ) N ) ∩ T Rx B in T Rx B . Note that the linear isomorphism D x f | L : L → Z is automat-ically sc-continuous because all norms on finite dimensional spaces areequivalent. Consider the projection π : T f ( x ) Y = Z ⊕ T f ( x ) N → Z andthe composition π ◦ D x f : T Rx B → Z . Then we have A = ker( π ◦ D x f ), and moreover π ◦ D x f maps L isomorphically onto Z . Hence the sc-splitting L ⊕ A = T Rx B holds because the coordinate projections aregiven by the sc-operators Π L := ( D x f ) | − L ◦ ( π ◦ D x f ) : T Rx B → L and( id T Rx B − Π L ) : T Rx B → A . Assuming properties (ii) and (iii) of L , we now explain how to choose W and ψ so that the required property (57) for σ -compatible transver-sality holds. By (ii) and the definition of P , we view both D x ϕ ( L ) and P as subspaces of E . By (iii) and since both D x ϕ ( L ) and P are finitedimensional sc-subspaces of E , their span has a sc-splitting D x ϕ ( L ) ⊕ P .Let W ′ be any sc-complement of D x ϕ ( L ) ⊕ P in E . Then choose anylinear isomorphism D × P ∼ = R k − t − s and set W = D x ϕ ( L ) ⊕ W ′ . Then,the linear sc-isomorphism ψ : E ′ = R t × D × P ⊕ W → R t × R k − t − s × W indeed satisfies ψ ◦ D x ϕ ( L ) ⊂ { } k − s × W , as required.It remains to construct the sc-subspace L ⊂ T Rx B that satisfies theconditions (i)-(iii). The space E consists of sections of the pullbacktangent bundle of Y along x that have matching asymptotic conditionsat nodes and breaking points. Since P ⊂ E is a finite dimensionalsubspace, by Lemma 5.9 (applied to the smooth component of Σ onwhich the marked point lies) there exists a neighborhood U of themarked point in Σ small enough such that the following holds: if ξ ∈ P is a section supported in U , then ξ = 0. Moreover, the marked point isalways in the complement of the nodes, breaking, or any other “specialpoints,” and so we can choose U disjoint from all special points.Choose any basis { z , . . . , z n } of the complement Z of T f ( x ) N in T f ( x ) Y . For each i = 1 , . . . , n , consider a sc-smooth path γ i : ( − ǫ, ǫ ) →B through x = γ i (0) obtained by deforming x to move the image f ( x )of the marked point in the direction z i , i.e. D x f ( γ ′ i (0)) = z i , while onlychanging x in the neighborhood U . The result is that all special pointsare preserved along the path, i.e. nodes and breakings do not get glued,and moreover the complex structure on the domain of x is not varyingalong the path. Hence we have D x ϕ ( γ ′ i (0)) ∈ { } s + t × { } × E . Define L := Span( { γ ′ (0) , . . . , γ ′ n (0) } ) . Then it is clear from the construction that (i) and (ii) hold. Moreover(iii) holds because every ξ ∈ D x ϕ ( L ) is supported in U by constructionof the γ i and hence if ξ ∈ D x ϕ ( L ) ∩ P then ξ = 0 by our choice of U .Note that L ⊂ T Rx B holds by (ii), since the reduced tangent space T Rx B is defined by D x ϕ ( T Rx B ) = T R O ∩ ( { } s × R t × D × E ).The following lemma was used in the preceding arguments. OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 61
Lemma 5.9.
Consider a smooth manifold Σ and a finite rank vectorbundle V → Σ . Let Γ ( V ) denote smooth sections and let q ∈ Σ .Then, if P ⊂ Γ ( V ) is a finite dimensional subspace, there exists anopen neighborhood U of q in Σ such that, if ξ ∈ P is supported in U ,then ξ = 0 .Proof. We assume that U does not exist and prove that P must thenbe infinite dimensional. We will construct a countable set of linearlyindependent elements of P .Consider any open neighborhood U of q . Then there exists some ξ ∈ P supported in U such that ξ = 0. Since ξ = 0, there exists anopen neighborhood U of q such that ξ is not supported in U (becauseif a smooth section is supported in every neighborhood of a point thenit is identically 0). Note that U ⊃ U necessarily holds.Inductively, assume that for some n ≥ U ⊃ · · · ⊃ U n +1 and sections ξ , . . . , ξ n such that, for all 0 ≤ i ≤ n , the section ξ i is supported in U i but is not supported in U i +1 . Then by our assumption that the claimedopen set U does not exist, there must exist some ξ n +1 ∈ P supportedin U n +1 such that ξ n +1 = 0. Then let U n +2 be a neighborhood of q onwhich ξ n +1 is not supported. Hence the inductive hypothesis holds for n + 1.This inductive process constructs a section ξ n ∈ P for all n ≥ U n but not in U n +1 . We claim that the collec-tion { ξ n | n ≥ } is linearly independent, proving the lemma. Indeed,suppose P ∞ n =0 c n · ξ n = 0 for some c n ∈ R . Then for every n ≥ ξ n is supported in U , but ξ is not supported in U , hence c = 0. Hence P ∞ n =1 c n · ξ n = 0. Inductively, we conclude that c n = 0for all n ≥ (cid:3) Proof of Theorem 1.3.
Proof of Theorem 1.3.
Proof of (I): By Definition 5.7 of a slice, for ev-ery x ∈ ˜ B there is an integer n x ≥ R n x -sliced chart withrespect to ˜ B ⊂ B that contains x . We will show that each of thesesliced charts induces a tame M -polyfold chart on ˜ B . The transitionmaps between the tame charts on ˜ B constructed in this way are sc-smooth because they are restrictions of the sc-smooth transition mapsbetween the sliced charts on B . Hence we have a tame sc-smooth atlason ˜ B , providing the claimed tame M -polyfold structure.Given a R n -sliced chart(58) ( V, ϕ, ( O , R n × C, R n × E )) with respect to ˜ B ⊂ B , we must construct the claimed induced tame M -polyfold chart on ˜ B . Use the notation˜ V := ˜ B ∩ V and ˜ ϕ := ϕ | ˜ V : ˜ V → ϕ ( ˜ V ) , and recall the induced tame sc-retract ( ˜ O , C, E ) from Lemma 3.3. Weclaim that the tuple(59) ( ˜ V , ˜ ϕ, ( ˜ O , C, E ))is a tame M -polyfold chart on ˜ B . Indeed, the set ˜ V is open in ˜ B since V ⊂ B is open and ˜ B ⊂ B has the subspace topology. Since ϕ is ahomeomorphism, it follows that its restriction ˜ ϕ is a homeomorphismonto its image, which is ˜ O by (54) and definition of ˜ V . This completesthe construction of the tame M -polyfold structure on ˜ B .Let x ∈ ˜ B . We now verify that the codimension n x = codim x ( ˜ B ⊂ B )of the slice ˜
B ⊂ B at x is well-defined and is locally constant in˜ B . There exists a R n x -sliced chart as in (58) with x ∈ V . It suf-fices to show that, for all y ∈ ˜ V , we have T y B /T y ˜ B ∼ = R n x . Indeed,this proves that the codimension n x is independent of the sliced chartaround x , hence is well-defined, and moreover that it is constant in theopen neighborhood ˜ V of x in ˜ B . For every y ∈ ˜ V , the tangent map D y ϕ : T y V → T ϕ ( y ) O of the chart map ϕ at y is a linear isomorphism.Moreover, D y ϕ restricts to a linear isomorphism D y ˜ ϕ : T x ˜ V → T ϕ ( y ) ˜ O .Hence D y ϕ induces an isomorphism T y V /T y ˜ V ∼ = T ϕ ( y ) O /T ϕ ( y ) ˜ O . Thus T y B /T y ˜ B = T y V /T y ˜ V ∼ = R n x holds by (17). Hence codim y ( ˜ B ⊂ B ) = n x holds for all y ∈ ˜ V .We now prove the claimed degeneracy index formula. Let x ∈ ˜ B ∞ .Since x is a smooth point, the reduced tangent space T Rx B is well-defined and D x ϕ restricts to a sc-isomorphism T Rx B → T Rϕ ( x ) O . Hence D x ϕ induces a sc-isomorphism T x B /T Rx B ∼ = T ϕ ( x ) O /T Rϕ ( x ) O . Similarly, D x ˜ ϕ induces a sc-isomorphism T x ˜ B /T Rx ˜ B ∼ = T ϕ ( x ) ˜ O /T Rϕ ( x ) ˜ O . Then by(18), the inclusion ˜ O ⊂ O induces an isomorphism T ϕ ( x ) ˜ O /T Rϕ ( x ) ˜ O ∼ = T ϕ ( x ) O /T Rϕ ( x ) O . It follows that the inclusion ˜ B ⊂ B induces an isomor-phism T x ˜ B /T Rx ˜ B ∼ = T x B /T Rx B . So by the global description (50) of de-generacy index at smooth points, we conclude d ˜ B ( x ) = dim( T x ˜ B /T Rx ˜ B ) =dim( T x B /T Rx B ) = d B ( x ). This completes the proof of the statementsin (I).Proof of (II): By definition of a slice of a bundle, there is a cover of˜ B by sliced bundle charts with respect to ˜ B ⊂ B . We will show that
OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 63 each of these sliced bundle charts induces a tame strong bundle chartfor ˜ ρ = ρ | ˜ E : ˜ E := ρ − ( ˜ B ) → ˜ B . The transition maps between thetame charts for ˜ ρ constructed in this way are sc-smooth because theyare restrictions of the transition maps between the sliced bundle chartsfor ρ . Hence we have a bundle atlas for ˜ ρ , providing the claimed tamestrong bundle structure.Given a R n -sliced bundle chart(60) ( ρ − ( V ) , Φ, ( K, R n × C ✁ F , R n × E ✁ F ))for ρ with respect to ˜ B ⊂ B that covers a R n -sliced chart with respectto ˜ B ⊂ B as in (58), we must construct the claimed induced tamestrong bundle chart on ˜ ρ . Let ( ˜ K, C ✁ F , E ✁ F ) denote the inducedtame bundle retract from Lemma 3.5 and use the notation˜ Φ := Φ | ˜ ρ − ( ˜ V ) : ˜ ρ − ( ˜ V ) → Φ ( ˜ ρ − ( ˜ V )) . We claim that the tuple(61) ( ˜ ρ − ( ˜ V ) , ˜ Φ, ( ˜ K, C ✁ F , E ✁ F ))is a tame strong bundle chart for ˜ ρ that covers the induced M -polyfoldchart (59). Since π O ◦ Φ = ϕ ◦ ρ by (51) it follows by restriction that π ˜ O ◦ ˜ Φ = ˜ ϕ ◦ ˜ ρ on ˜ ρ − ( ˜ V ). It follows that im ( ˜ Φ ) = π − O ( ˜ O ) = ˜ K , wherethe second equality holds by (24). Hence ˜ Φ is a homeomorphism ontoits image ˜ K because it is the restriction of the homeomorphism Φ , andsimilarly ˜ Φ is linear on fibers as the restriction of Φ . This completesthe proof of (II).Proof of (III): The final statement about compactness of the zerosets holds because ˜ σ − (0) = σ − (0) ∩ ˜ B ∞ is the intersection of a compactset and a closed subset of the Hausdorff space B ∞ .We proceed to verify that ˜ σ = σ | ˜ B : ˜ B → ˜ E is a tame sc-Fredholmsection of ˜ ρ . The section ˜ σ is sc-smooth and regularizing because it isthe restriction of the section σ . So to prove that ˜ σ is a tame sc-Fredholmsection it remains to show that there exists a tame sc-Fredholm chart(Definition 5.4) for ˜ σ at every smooth point x ∈ ˜ B ∞ . Since ˜ B is a slice ofthe sc-Fredholm section σ , there exists a R n x -sliced sc-Fredholm chartfor σ at x with respect to ˜ B ⊂ B . This is a R n x -sliced bundle chart asin (60) such that the local sc-Fredholm germ Φ ◦ σ ◦ ϕ − is R n x -slicedand such that ϕ ( x ) = 0.We claim that the tame bundle chart (61) is a tame sc-Fredholmchart for ˜ σ at x . This requires that the local section ˜ Φ ◦ ˜ σ ◦ ˜ ϕ − : ˜ O → ˜ K is a tame sc-Fredholm germ, which follows from Lemma 3.10 since Φ ◦ σ ◦ ϕ − : O → K is a R n x -sliced sc-Fredholm germ. Moreover, Lemma 3.10 and Definition 5.3(3) provide the sc-Fredholm index formula ind x (˜ σ ) = ind ( ˜ Φ ◦ ˜ σ ◦ ˜ ϕ − ) = ind ( Φ ◦ σ ◦ ϕ − ) − n x = ind x ( σ ) − n x . This completesthe proof that ˜ σ is a tame sc-Fredholm section satisfying the claimedindex formula. (cid:3) Proof of Theorem 1.5.
Proof of Theorem 1.5.
Proof of (I): For every x ∈ f − ( N ) ∩B ∞ we willconstruct a R n -sliced chart (Definition 5.6) with respect to f − ( N ) ∩B ⊂ B that contains x . Then we define ˜ B to be the union of thedomains of these charts intersected with f − ( N ) ∩ B .Let x ∈ f − ( N ) ∩ B ∞ . Let Z ⊂ Y be the domain of a manifold chart Z ∼ −→ R n +dim N on Y containing f ( x ) ∈ N that identifies N ∩ Z with { } n × R dim N . Let γ : Z → R n be the smooth map given in the chartby the projection R n × R dim N → R n . Then 0 = γ ( f ( x )) is a regularvalue of γ and(62) γ − (0) = N ∩ Z. In particular, we have(63) T f ( x ) N = ker D f ( x ) γ. Then f − ( Z ) is an open neighborhood of x in B . Choose V ⊂ f − ( Z )so that, in addition, there is a tame M -polyfold chart(64) ( V, ϕ, ( O , [0 , ∞ ) s × E , R s × E ))on B satisfying ϕ ( x ) ∈ { } s × E . Set x := ϕ ( x ) . By definition (49) of the reduced tangent space T Rx B , we have D x ϕ ( T Rx B ) = T Rx O . So since f is transverse to N , we conclude from (56) that wehave(65) D x f ◦ D x ϕ − ( T Rx O ) + T f ( x ) N = T f ( x ) Y. We claim that the map f := γ ◦ f ◦ ϕ − : O → R n satisfies the hypotheses of Lemma 4.2(I) at x . Let O ∂ := O ∩ ( { } s × E )denote the boundary sc-retract associated to O (see Lemma 4.1) andconsider the restriction f ∂ := f | O ∂ . Recall from (37) that we have T Rx O = T x O ∂ . Then the tangent map D x f ∂ : T x O ∂ → T R n = R n issurjective by (65), (63), and since 0 is a regular value of γ . Hence,the hypotheses of Lemma 4.2(I) are indeed satisfied, yielding slice co-ordinates around x with respect to ( f − (0) ∩ O ) ⊂ O . Precisely, OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 65 this means that we obtain a sc-Banach space K , a R n -sliced sc-retract(Definition 3.2)(66) ( O ′ , R n × [0 , ∞ ) s × K , R n × R s × K ) , an open set ˆ O ⊂ O , and a sc-smooth diffeomorphism g : ˆ O → O ′ such that the induced tame sc-retract˜ O ′ = O ′ ∩ ( { } n × [0 , ∞ ) s × K )satisfies(67) g ( f − (0) ∩ ˆ O ) = ˜ O ′ . Set ˆ V := ϕ − ( ˆ O ) and ˆ ϕ := g ◦ ϕ | ˆ V : ˆ V → O ′ . Then(68) ( ˆ
V , ˆ ϕ, ( O ′ , R n × [0 , ∞ ) s × K , R n × R s × K ))is a M -polyfold chart on B .At this point, for every x ∈ f − ( N ) ∩ B ∞ , we have constructed a M -polyfold chart on B with domain ˆ V x ⊂ B containing x ∈ ˆ V x . Define(69) ˜ B := [ x ∈ f − ( N ) ∩B ∞ ˆ V x ∩ f − ( N ) . We claim that ˜
B ⊂ B is a slice. We must show that the chart (68)is a R n -sliced chart (Definition 5.6) with respect to ˜ B ⊂ B . The sc-retract (66) is given as R n -sliced by Lemma 4.2(I), so it remains toshow that (54) holds, which in the notation of this proof is the state-ment ˜ O ′ = ˆ ϕ ( ˜ B ∩ ˆ V ) . Let p ∈ ˜ O ′ . By (67) and the definition of f , wehave ˆ ϕ − ( p ) = ϕ − ◦ g − ( p ) ∈ ( γ ◦ f ) − (0) ⊂ f − ( N ) . Hence ˆ ϕ − ( p ) ∈ ˆ V ∩ f − ( N ) ⊂ ˜ B ∩ ˆ V .
To see the reverse inclusion, let p ∈ ˆ ϕ ( ˜ B ∩ ˆ V ) . By (67), we must show that f ( g − ( p )) = 0. From the definitions wecompute f ◦ g − ( p ) = γ ◦ f ◦ ˆ ϕ − ( p ) ∈ γ ( f ( ˜ B ∩ ˆ V )) ⊂ γ ( N ∩ Z ) = { } , as required. Hence (68) is indeed a R n -sliced chart with respect to˜ B ⊂ B . This proves that ˜ B ⊂ B is a slice (Definition 5.7). Hence ˜ B isa tame M -polyfold with the claimed degeneracy index by Theorem 1.3.Moreover, the existence of a R n -sliced chart around every y ∈ ˜ B provesthe claim that codim y ( ˜ B ⊂ B ) = n for every y ∈ ˜ B . This completesthe proof of the statements (I).Proof of (II): To prove that ˜ B is a slice of the bundle ρ | E : E → B ,for every x ∈ ˜ B we must construct a sliced bundle chart for ρ | E with respect to ˜ B ⊂ B . We can choose the tame M -polyfold chart (64) sothat it is covered by a tame strong bundle chart(70) ( ρ − ( V ) , Φ, ( K, C ✁ F , E ✁ F ))for ρ . Note that this may require shrinking the open neighborhood V ⊂ B of x , and so the resulting slice ˜ B of B defined by the formula(69) may be smaller than in part (I).The tame bundle retract ( K, C ✁ F , E ✁ F ) covers the tame sc-retract O . Hence Lemma 4.2(II) yields a R n -sliced bundle retract( K ′ , R n × [0 , ∞ ) s × K ✁ F , R n × R s × K ✁ F )that covers the sliced sc-retract (66) and satisfies( g ✁ id F )( ˆ K ) = K ′ , where ˆ K ⊂ K is the preimage of ˆ O in the local bundle model K → O .Set ˆ Φ := ( g ✁ id F ) ◦ Φ | ρ | − E ( ˆ V ) : ρ | − E ( ˆ V ) → K ′ . Then the desired sliced bundle chart for ρ | E with respect to ˜ B ⊂ B isthe tuple(71) ( ρ | − E ( ˆ V ) , ˆ Φ, ( K ′ , R n × [0 , ∞ ) s × K ✁ F , R n × R s × K ✁ F )) . This proves that ˜
B ⊂ B is a slice of the bundle ρ | E . The restriction˜ ρ : ρ | − E ( ˜ B ) → ˜ B is then a tame strong bundle by Theorem 1.3(II).Proof of (III): To prove that ˜ B is a slice of the tame sc-Fredholmsection σ | B : B → E , for every x ∈ ˜ B ∞ we must construct a slicedsc-Fredholm chart for σ | B at x with respect to ˜ B ⊂ B . Since f is σ -compatibly transverse to N , we can assume that the tame strongbundle chart (70) is also a tame sc-Fredholm chart for σ at x such thatwe can choose ψ, W , and L ⊂ T Rx B satisfying (57). Note that this mayrequire shrinking the open neighborhood V of x , and so the resultingslice ˜ B of ρ | E defined by the formula (69) may be smaller than in parts(I) and (II).We now show that W , ψ , and L ′ := D x ϕ ( L ) satisfy the hypothesesof Lemma 4.2(III). Indeed, by (57), L is a sc-complement of the sc-subspace A := ( D x f ) − ( T f ( x ) N ) ∩ T Rx B in T Rx B , and so we have the sc-splitting(72) D x ϕ ( L ) ⊕ D x ϕ ( A ) = T Rx O . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 67
Recall from (37) that we have T Rx O = T x O ∂ . Then from the definitionsand (63), we compute D x ϕ ( A ) = D x ϕ (cid:0) ( D x f ) − ( T f ( x ) N ) ∩ T Rx B (cid:1) = D x ϕ (cid:0) ( D x f ) − (ker D f ( x ) γ ) (cid:1) ∩ T Rx O = ker D x f ∩ T Rx O = ker D x f ∂ , and so by (72) the space L ′ = D x ϕ ( L ) is indeed a sc-complement ofker D x f ∂ in T x O ∂ , as required. Moreover, the required condition (42)holds for L ′ by (57).Hence, Lemma 4.2(III) asserts that the section σ ′ := ( g ✁ id F ) ◦ σ ◦ g − : O ′ → K ′ is a R n -sliced sc-Fredholm germ, which means that (71)is a R n -sliced sc-Fredholm chart. This proves that ˜ B is a slice of σ | B .The restriction ˜ σ : ˜ B → ˜ E is then a tame sc-Fredholm section satisfyingthe claimed index formula by Theorem 1.3(III).Since N ⊂ Y is closed it follows that f − ( N ) ⊂ B is closed, so˜ B ∞ = f − ( N ) ∩ B ∞ is closed in B ∞ . Hence if σ − (0) is compact thenit follows from the final statement of Theorem 1.3(III) that ˜ σ − (0) iscompact. (cid:3) Handling isotropy: the ep-groupoid case
In this section, we generalize the main theorems (Theorem 1.3 andTheorem 1.5) to the case of ep-groupoids in Corollary 6.7 and Corol-lary 6.8. An ep-groupoid is the “orbifold version” of a M -polyfold. Inapplications, we must keep track of finite isotropy groups of points inthe polyfold. The theory of sc-Fredholm sections over ep-groupoids isthe generalization of sc-Fredholm sections over M -polyfolds that incor-porates this isotropy; see [7] for a detailed treatment.To perform these constructions on polyfolds, it suffices to take repre-sentative ep-groupoids and then perform the constructions to obtain arepresentative ep-groupoid of a new polyfold. A polyfold [7, Def. 16.3]is a topological space Z together with an equivalence class of polyfoldstructures [7, Def. 16.1]. A polyfold structure for Z is an ep-groupoidtogether with a homeomorphism of its orbit space with Z . Polyfoldstructures are equivalent [7, Def. 16.2] if they are related by a general-ized isomorphism [7, Def. 10.8] compatible with the homeomorphismswith Z . In particular, every ep-groupoid defines a canonical polyfoldstructure on its orbit space.All of the results in this section follow from the results in the M -polyfold case in the previous sections. In fact, this is a general polyfold philosophy: the M -polyfold situation is where all of the analytic data(e.g. the sc-structures on base and bundle and the sc-Fredholm proper-ties) is stored. So, if an application of polyfold theory works assumingthat everything in sight is a M -polyfold, we expect that there is a suit-able upgrade to ep-groupoids by keeping track of the isotropy using theep-groupoid machinery.This philosophy goes deeper: a polyfold theorist thinks throughany construction first assuming that everything is a finite dimensionalsmooth manifold with boundary and corners. In particular, sc-Fredholmsections in the finite dimensional setting are the same as ordinarysmooth sections. We expect that any construction that is motivated byregularization of some moduli space of pseudoholomorphic curves , andthat works assuming everything is finite dimensional, will go throughin the M -polyfold and ep-groupoid case.We now review tame ep-groupoids (Definition 6.1), bundles overthem (Definition 6.3), and their sc-Fredholm section functors (Defi-nition 6.5) before we introduce the new notion of tame sc-Fredholmsection functors (Definition 6.6) and prove Corollary 6.7 and Corol-lary 6.8.A groupoid X = ( X, X ) is a small category with object set X andmorphism set X such that all morphisms are invertible. Associated toany groupoid are the following structure maps. For a detailed descrip-tion, see for example [7, Def. 7.1]. The source map s : X → X and the target map t : X → X send a morphism to its source and target, respectively. The multipli-cation map m : X × s t X → X composes any pair of morphisms such that the source of the first isthe target of the second, and hence m is defined on the fiber product X × s t X . The unit map u : X → X sends an object to the identity morphism from that object to itself,which exists and is unique since each self-morphism set is a group.The inverse map ι : X → X inverts morphisms. OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 69
Definition 6.1. [7, Defs. 7.3, 7.6] A tame ep-groupoid X = ( X, X ) is a groupoid equipped with tame M -polyfold structures on the objectspace X and on the morphism space X satisfying the following proper-ties: (i) (´etale) The source s and target t maps are surjective local sc-diffeomorphisms, (ii) The unit map u and the inverse map ι are sc-smooth. (iii) (proper) Every x ∈ X possesses an open neighborhood V ( x ) ⊂ X of x such that the map t : s − ( V ( x )) → X is proper. (iv) The multiplication map m is sc-smooth, where the fiber product X × s t X is equipped with the M -polyfold structure given by [7,Prop. 2.15] . (See Remark 6.2 to compare with the fiber productresults in this paper.) Remark 6.2.
There is an essential difference between the fiber productresults in this paper and the fiber product in [7, Prop. 2.15] that is usedto give X × s t X a M -polyfold structure in Definition 6.1(iv).The result in [7, Prop. 2.15] requires one of the maps in the fiberproduct to be a local sc-diffeomorphism. In the case of the fiber product X × s t X from Definition 6.1, both the source map s and the target map t are local sc-diffeomorphisms.In this paper, we construct fiber products (Corollary 7.3) over mapsto a finite dimensional smooth manifold. In applications, these mapswill never be local sc-diffeomorphisms because the M -polyfolds have in-finite dimensional tangent spaces everywhere. For an ep-groupoid X = ( X, X ), the orbit space (73) | X | = X/ ∼ is the quotient of the object space X by the equivalence relation definedby x ∼ y if and only if there exists φ ∈ X such that s ( φ ) = x and t ( φ ) = y . That is, to obtain the orbit space, we identify any twoobjects that have a morphism between them.The degeneracy indices d X : X → N and d X : X → N are definedas usual on the M -polyfolds X and X . The induced degeneracy index [7, Def 7.5] d | X | : | X | → N on the orbit space | X | is defined by d | X | ( | x | ) = d X ( x ), and is well-defined by [7, Prop. 2.7] as discussed above the definition [7, Def 7.5].Now we review the notion of a strong bundle over an ep-groupoid X = ( X, X ); see [7, Sec. 8.3] for more detail. Consider a strong M -polyfold bundle P : E → X over the object space X of X . Since the source map s is by definitiona local sc-diffeomorphism, a M -polyfold structure on the fiber product X × s P E is provided by [7, Prop. 2.15]. Moreover, the projection ontothe first factor π : E := X × s P E → X is a strong M -polyfold bundle. Definition 6.3. [7, Def. 8.4] A tame strong bundle over an ep-groupoid X = ( X, X ) is a pair ( P, µ ) of a tame strong bundle P : E → X over the object M -polyfold X and a strong bundle map µ : X × s P E → E covering the target map t : X → X , i.e. P ◦ µ = t ◦ π , and which satisfies (i) µ (1 x , e ) = e for all x ∈ X and e ∈ E x , (ii) µ ( g ◦ h, e ) = µ ( g, µ ( h, e )) for all g, h ∈ X and e ∈ E satisfying s ( h ) = P ( e ) and t ( h ) = s ( g ) = P ( µ ( h, e )) . We call µ the strong bundle structure map . Remark 6.4.
The standard definition [7, Def. 8.4] of a tame strongbundle requires, in addition to the conditions in Definition 6.3, thatthe structure map µ is a surjective local sc-diffeomorphism. However,this condition is automatically satisfied, as noted in [13] on page 37. Let ( P : E → X, µ ) be a strong bundle over the ep-groupoid X =( X, X ). A sc-smooth section functor σ [7, Def. 8.7] of ( P, µ ) is a sc-smooth section σ : X → E of the strong bundle P : E → X over theobject M -polyfold X satisfying the following compatibility with µ : Forall morphisms φ ∈ X ,(74) σ ( t ( φ )) = µ ( φ, σ ( s ( φ ))) OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 71 holds.
Definition 6.5. [7, Def. 8.7] A sc-Fredholm section functor of astrong bundle ( P : E → X, µ ) over an ep-groupoid X = ( X, X ) is asc-smooth section functor σ : X → E such that, when viewed as asection of the strong bundle M -polyfold bundle P , it is sc-Fredholm inthe M -polyfold sense (Definition 5.3). We introduce the following new class of sc-Fredholm section functors.
Definition 6.6. A tame sc-Fredholm section functor of a strongbundle ( P : E → X, µ ) over an ep-groupoid X = ( X, X ) is a sc-Fredholm section functor such that, when viewed as a section of thestrong bundle M -polyfold bundle P , it is tame sc-Fredholm in the M -polyfold sense (Definition 5.4). We now generalize Theorem 1.3 to the ep-groupoid case.
Corollary 6.7. (I)
Consider a tame ep-groupoid X = ( X, X ) and a slice ˜ X ⊂ X ofthe object M -polyfold X , in the sense of Definition 5.7. Assume that ˜ X is closed under morphisms, i.e. for all φ ∈ X we have (75) s ( φ ) ∈ ˜ X ⇐⇒ t ( φ ) ∈ ˜ X, or equivalently, s − ( ˜ X ) = t − ( ˜ X ) . Then, denoting the subset of morphisms X that have source andtarget in ˜ X by (76) ˜ X := s − ( ˜ X ) = t − ( ˜ X ) , the tuple ˜ X = ( ˜ X, ˜ X ) is a tame ep-groupoid with the tame M -polyfoldatlas on ˜ X induced by the sliced charts (Definition 5.6) with respectto ˜ X ⊂ X . Note that ˜ X is the full subcategory of X with object space ˜ X .For x ∈ ˜ X , the codimension codim x ( ˜ X ⊂ X ) (Definition 5.7) iswell-defined and locally constant in ˜ X , i.e. it is equal to codim x ( ˜ X ⊂ X ) in an open neighborhood of x in ˜ X . For x ∈ ˜ X ∞ , the degeneracyindex satisfies d ˜ X ( x ) = d X ( x ) . (II) Consider, in addition, a tame strong bundle ( P : E → X, µ ) over X and suppose that ˜ X is a slice of the bundle P in the sense ofDefinition 5.7. Then, the tuple ( ˜
P , ˜ µ ) consisting of the restrictions ˜ E := P − ( ˜ X )˜ P := P | ˜ E : ˜ E → ˜ X, ˜ µ := µ | ˜ X × s P ˜ E : ˜ X × s P ˜ E → ˜ E, is a tame strong bundle over ˜ X , where the bundle atlas for ˜ P is in-duced by the sliced bundle charts for P with respect to ˜ X ⊂ X . (III) Consider, in addition, a sc-Fredholm section functor σ : X → E of ( P, µ ) and suppose that ˜ X is a slice of σ in the sense of Defini-tion 5.7.Then, the restriction ˜ σ := σ | ˜ X : ˜ X → ˜ E is a tame sc-Fredholmsection functor of the bundle ( ˜ P , ˜ µ ) with tame sc-Fredholm charts in-duced by the sliced sc-Fredholm charts for σ with respect to ˜ X ⊂ X .For x ∈ ˜ X ∞ , the index satisfies ind x (˜ σ ) = ind x ( σ ) − codim x ( ˜ X ⊂ X ) .If | σ − (0) | is compact and | ˜ X ∞ | ⊂ | X ∞ | is closed, then | ˜ σ − (0) | iscompact.Proof. Proof of (I): First we claim that ˜ X = ( ˜ X, ˜ X ) is a groupoid.Given any subset A of the object set of a groupoid, say A ⊂ X , weobtain a sub-groupoid ( A, A ) of ( X, X ) by defining the morphism setto be A := s − ( A ) ∩ t − ( A ). Hence, with the set of morphisms ˜ X asdefined in (76), the tuple ( ˜ X, ˜ X ) is a groupoid.Since ˜ X ⊂ X is a slice of the tame M -polyfold X , Theorem 1.3(I)provides a tame M -polyfold structure on ˜ X with the claimed degen-eracy index for x ∈ ˜ X ∞ and the claimed locally constant codimension codim x ( ˜ X ⊂ X ) for x ∈ ˜ X .We now equip ˜ X with a tame M -polyfold structure by pulling backthe tame M -polyfold charts on ˜ X through the source and target maps s, t : X → X on X . That is, since s and t are local sc-diffeomorphismsby the ´etale property (Definition 6.1(i)) of X , for every z ∈ ˜ X thereexists a neighborhood V ⊂ X of z such that s | V : V → V is a sc-diffeomorphism, where V := s ( V ). Then V is a neighborhood of s ( z )in X . Since ˜ X ⊂ X is a slice and s ( z ) ∈ ˜ X , after shrinking V we canassume that it is the domain of a sliced chart with respect to ˜ X ⊂ X (Definition 5.6). This induces a tame M -polyfold chart on ˜ X withdomain V ∩ ˜ X , as in the proof of Theorem 1.3. Pulling this back through s , we obtain a sliced chart with respect to ˜ X ⊂ X with domain V , andsince s ( V ∩ ˜ X ) = V ∩ ˜ X , its induced tame M -polyfold chart has domain V ∩ ˜ X . So, we have constructed a tame M -polyfold chart with domain OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 73 an open neighborhood of z in ˜ X . The analogous construction usingthe target map t instead of the source map s produces another suchchart. On overlaps between any sliced charts with respect to ˜ X ⊂ X produced in this way, the transition maps are sc-smooth because theyare compositions of the sc-smooth chart maps on X with the sc-smoothsource and target maps (and their local inverses). Hence the transitionsbetween the induced tame M -polyfold charts on ˜ X are also sc-smooth,as restricitons of the transitions on X . Hence covering ˜ X with these M -polyfold charts provides the claimed M -polyfold structure. Moreover,observe from the construction of the charts that the source ˜ s = s | ˜ X andtarget ˜ t = t | ˜ X maps on ˜ X = ( ˜ X, ˜ X ) are local sc-diffeomorphisms, i.e. ˜ X satisfies the ´etale property (Definition 6.1(i)). Similarly, the unit map,the inverse map, and the multiplication map on ˜ X are sc-smooth sincethey are restrictions of the corresponding maps on X . This verifiesproperties Definition 6.1(ii) and Definition 6.1(iv) for ˜ X .To verify that ˜ X is a tame ep-groupoid, it remains to verify proper-ness (Definition 6.1(iii)). Let x ∈ ˜ X . By properness of X , thereexists an open neighborhood V ( x ) of x in X such that the map t : s − ( cl X ( V ( x ))) → X is a proper map. Shrink V ( x ) so that it is thedomain of a sliced chart (Definition 5.6) with respect to ˜ X ⊂ X . Then V ( x ) ∩ ˜ X is closed in V ( x ) because the chart map that homeomor-phically identifies V ( x ) with the sliced sc-retract ( O , R n × C, R n × E )sends V ( x ) ∩ ˜ X to the induced tame sc-retract O ∩ ( { } × C ), whichis closed in O . Let V ′ ( x ) ⊂ V ( x ) be a smaller open neighborhoodof x such that cl X ( V ′ ( x )) ⊂ V ( x ). Then we have cl X ( V ′ ( x ) ∩ ˜ X ) = cl V ( x ) ( V ′ ( x ) ∩ ˜ X ) ⊂ cl V ( x ) ( V ( x ) ∩ ˜ X ) = V ( x ) ∩ ˜ X. In particular, theclosure of V ′ ( x ) ∩ ˜ X in ˜ X and in X agree, i.e.(77) cl X ( V ′ ( x ) ∩ ˜ X ) = cl ˜ X ( V ′ ( x ) ∩ ˜ X ) . We claim that the open neighborhood V ′ ( x ) ∩ ˜ X of x in ˜ X satisfiesthe required condition, i.e. the map t | ˜ X : s | − X ( cl ˜ X ( V ′ ( x ) ∩ ˜ X )) → ˜ X is proper. Let K ⊂ ˜ X be compact. We must show that the set A := t | − X ( K ) ∩ s | − X ( cl ˜ X ( V ′ ( x ) ∩ ˜ X )) is compact. We have A = t − ( K ) ∩ s − ( cl X ( V ′ ( x ) ∩ ˜ X ))by (77) and since by definition ˜ X = t − ( ˜ X ) = s − ( ˜ X ). Observe that A is closed in X since K ⊂ X is closed as a compact subset of the Haus-dorff space X . By our choice of V ( x ), the set t − ( K ) ∩ s − ( cl X ( V ( x )))is compact. Hence A is compact as a closed subset of the compact set t − ( K ) ∩ s − ( cl X ( V ( x ))). This completes the proof that ˜ X is a tame ep-groupoid.Proof of (II): Since ˜ X is a slice of the tame strong bundle P : E → X in the M -polyfold sense, Theorem 1.3(II) provides a tame strong bundlestructure on the restriction ˜ P . To prove that the tuple ( ˜ P , ˜ µ ) is a tamestrong bundle over the ep-groupoid ˜ X , we must show that the map ˜ µ has the required properties. Recall from the lemma statement that ˜ µ is the restriction of µ , i.e.˜ µ := µ | ˜ X × s P ˜ E : ˜ X × s P ˜ E → ˜ E. First of all, ˜ µ indeed takes values in ˜ E because, since µ covers t , forany ( φ, e ) ∈ ˜ X × s P ˜ E we have P ◦ µ ( φ, e ) = t ◦ π ( φ, e ) = t ( φ ) ∈ ˜ X andhence µ ( φ, e ) ∈ P − ( ˜ X ) = ˜ E . Moreover, ˜ µ covers the target map on˜ X since ˜ µ is the restriction of µ which covers the target map t on X .Finally, the required properties Definition 6.3(i)(ii) of ˜ µ follow immedi-ately from the corresponding properties of µ , since ˜ µ is the restriction of µ . This completes the proof that ( ˜ P , ˜ µ ) is a tame strong bundle over ˜ X .Proof of (III): Since ˜ X is a slice of the sc-Fredholm section σ : X → E in the M -polyfold sense, Theorem 1.3(III) provides the structureof a tame sc-Fredholm section on the restricted section ˜ σ := σ | ˜ X :˜ X → ˜ E with the claimed sc-Fredholm index. Moreover, the section˜ σ is a section functor because the required compatibility (74) with ˜ µ is immediate from the compatibility of σ with µ . Hence ˜ σ is a tamesc-Fredholm section functor.The final statement about compactness of the zero sets holds because | ˜ σ − (0) | = | σ − (0) | ∩ | ˜ X ∞ | is the intersection of a compact set and aclosed subset of the Hausdorff space | X ∞ | (recall that the orbit spaceof an ep-groupoid is Hausdorff by [7, Thm. 7.2]). (cid:3) The generalization of Theorem 1.5 to the ep-groupoid case now easilyfollows by combining Theorem 1.5 with Corollary 6.7.
Corollary 6.8. (I)
Consider a tame ep-groupoid X = ( X, X ) , a smooth manifold Y together with a codimension- n submanifold N ⊂ Y , and a sc-smoothmap f : X → Y that satisfies the compatibility with morphisms (78) f ( s ( φ )) = f ( t ( φ )) for all φ ∈ X . Assume that f is transverse to N (Definition 5.8).Then, there exists an open neighborhood ˜ X ⊂ f − ( N ) ∩ X OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 75 of f − ( N ) ∩ X ∞ such that ˜ X is a slice of X in the M -polyfoldsense (Definition 5.7) satisfying codim x ( ˜ X ⊂ X ) = n for every x ∈ ˜ X = ˜ X ∩ X . In particular, the full subcategory ˜ X := ( ˜ X, ˜ X ) of X is a tame ep-groupoid with degeneracy index satisfying d ˜ X ( x ) = d X ( x ) for all x ∈ ˜ X ∞ . (II) Consider, in addition, a tame strong bundle ( P : E → X, µ ) over X . Then, there exists a possibly smaller neighborhood ˜ X in (I) that isa slice of the bundle P | E : E → X in the M -polyfold sense (Defini-tion 5.7). In particular, the tuple ( ˜ P , ˜ µ ) consisting of the restrictions ˜ E := P | − E ( ˜ X )˜ P := P | ˜ E : ˜ E → ˜ X, ˜ µ := µ | ˜ X × s P ˜ E : ˜ X × s P ˜ E → ˜ E, is a tame strong bundle over ˜ X . (III) Consider, in addition, a tame sc-Fredholm section functor σ : X → E of ( P, µ ) . Assume that f is σ -compatibly transverse to N (Definition 5.8).Then, there exists a possibly smaller neighborhood ˜ X in (II) thatis a slice of the tame sc-Fredholm section σ | X : X → E in the M -polyfold sense (Definition 5.7). In particular, the restriction ˜ σ := σ | ˜ X : ˜ X → ˜ E is a tame sc-Fredholm section functor of the bundle ( ˜ P , ˜ µ ) with indexsatisfying ind x (˜ σ ) = ind x ( σ ) − n for all x ∈ ˜ X ∞ . If N is closed as asubset of Y and | σ − (0) | is compact, then | ˜ σ − (0) | is compact.Proof. We first prove the statements in (I). Since f is transverse to N ,Theorem 1.5(I) provides an open neighborhood ˜ X ⊂ f − ( N ) ∩ X of f − ( N ) ∩ X ∞ that is a slice of X with the claimed codimension atevery point in ˜ X . The compatibility (78) of f with the morphisms X implies that s − ( ˜ X ) = t − ( ˜ X ) holds. Hence applying Corollary 6.7(I)to the tame ep-groupoid X = ( X , X ) and the slice ˜ X ⊂ X , weconclude that the full subcategory ˜ X = ( ˜ X, ˜ X ) of X with object space˜ X is a tame ep-groupoid with the claimed degeneracy index.We now prove the statements in (II). Theorem 1.5(II) provides achoice of ˜ X in (I) that is in addition a slice of the bundle P | E : E → X . Hence applying Corollary 6.7(II), we conclude that the tuple ( ˜ P , ˜ µ )is indeed a tame strong bundle over ˜ X , as claimed. We now prove the statements in (III). Since f is σ -compatibly trans-verse to N , Theorem 1.5(III) provides a choice of ˜ X in (II) that is inaddition a slice of the tame sc-Fredholm section σ | X : X → E . Henceapplying Corollary 6.7(III), we conclude that the restriction ˜ σ = σ | ˜ X is indeed a tame sc-Fredholm section functor of the bundle ( ˜ P , ˜ µ ) withthe claimed index.It remains to prove the final statement about compactness of theorbit spaces of the zero sets. Assume that | σ − (0) | is compact. Itsuffices to show that the inclusion of orbit spaces | ˜ X ∞ | ⊂ | ( X ) ∞ | = | X ∞ | is closed, because then Corollary 6.7(III) implies compactness of | ˜ σ − (0) | , as required. Notice that, by (78), the map f descends to amap on the orbit space | f | : | X ∞ | → Y . Then | ˜ X ∞ | = | f | − ( N ) isclosed in | X ∞ | by continuity of | f | , as required. This completes theproof. (cid:3) Fiber products of tame sc-Fredholm sections
The main result of this section is the construction of fiber products oftame sc-Fredholm section functors (Corollary 7.3). This is a corollaryof the construction of restrictions of tame sc-Fredholm section functorsto transverse preimages of sc-smooth maps, which is the main result ofthis paper; see Theorem 1.5 for the M -polyfold case and Corollary 6.8for the ep-groupoid generalization.In this section, we index the M -polyfolds and ep-groupoids withparenthesis around the subscript, i.e. B ( i ) for i = 1 , , to avoid confusionwith the standard notation B m for the m -level of a M -polyfold B .We first describe the Cartesian product of tame sc-Fredholm sectionsover M -polyfolds. Lemma 7.1. (I)
Consider tame M -polyfolds B ( i ) for i = 1 , . Then, the Cartesianproduct B (1) × B (2) is a tame M -polyfold with charts given by prod-ucts of charts on the factors, and with degeneracy index satisfying d B (1) ×B (2) ( x , x ) = d B (1) ( x ) + d B (2) ( x ) for all ( x , x ) ∈ ( B (1) ) ∞ × ( B (2) ) ∞ . (II) Consider tame strong bundles ρ i : E ( i ) → B ( i ) over B ( i ) for i = 1 , . Then, the product map ρ × ρ : E (1) × E (2) → B (1) × B (2) is a tamestrong bundle over B (1) × B (2) with bundle charts given by products ofbundle charts on the factors. OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 77 (III)
Consider tame sc-Fredholm sections σ i : B ( i ) → E ( i ) (Defini-tion 5.4) of ρ i for i = 1 , . Then, the product map σ × σ : B (1) ×B (2) → E (1) × E (2) is a tame sc-Fredholm section of ρ × ρ with tamesc-Fredholm charts given by products of tame sc-Fredholm charts onthe factors (after reordering factors in the charts), and with indexsatisfying ind ( x ,x ) ( σ × σ ) = ind x ( σ ) + ind x ( σ ) for all ( x , x ) ∈ ( B (1) ) ∞ × ( B (2) ) ∞ . If σ − i (0) is compact for i = 1 , , then ( σ × σ ) − (0) is compact.Proof. Proof of (I): We first show that products of tame M -polyfoldcharts (Definition 5.1) on the factors B ( i ) , i = 1 , , are indeed tame M -polyfold charts on the product B (1) × B (2) , as claimed. For i = 1 , M -polyfold chart(79) ( V i , ϕ i , ( O i , C i , E i ))on B ( i ) , an open subset U i ⊂ C i , and a tame sc-retraction r i : U i → U i with image O i = r i ( U i ). We claim that the tuple(80) ( V × V , ϕ × ϕ , ( O × O , C × C , E × E ))is a tame M -polyfold chart on B (1) × B (2) . We equip B (1) × B (2) with theproduct topology on every level. Recall that the product sc-structure isgiven by ( E × E ) m = ( E ) m × ( E ) m for all m ≥
0. Then the productset V × V is open in B (1) ×B (2) and the product map ϕ × ϕ : V × V →O × O is a homeomorphism. It remains to show that O × O is atame sc-retract. The product map r × r : U × U → U × U is asc-smooth retraction with image O × O . We now show that r × r is tame. To check Definition 3.1(1) for r × r , let ( x × x ) ∈ U × U and then compute using the corresponding property of each r i togetherwith (7) that the degeneracy index satisfies d C × C (( r × r )( x , x )) = d C × C ( r ( x ) , r ( x )) = d C ( r ( x ))+ d C ( r ( x )) = d C ( x )+ d C ( x ) = d C × C ( x , x ) , as required. To check Definition 3.1(2) for r × r , let( x , x ) ∈ ( O × O ) ∞ = ( O ) ∞ × ( O ) ∞ be a smooth point. Then bythe corresponding property of each r i , there exist sc-subspaces A i ⊂ E i such that E i = T x i O i ⊕ A i and A i ⊂ ( E i ) x i , where ( E i ) x i ⊂ E i is thesc-subspace (9). From the definition (9) we have(81) ( E ) x × ( E ) x = ( E × E ) ( x ,x ) . So we have A × A ⊂ ( E × E ) ( x ,x ) . Moreover, we have the sc-splitting E × E = ( T x O ⊕ A ) × ( T x O ⊕ A )= T ( x ,x ) ( O × O ) ⊕ ( A × A ) because, by definition (10) of tangent space to a sc-retract, the tangentspace to a product satisfies T ( x ,x ) ( O × O ) = D ( x ,x ) ( r × r )( T ( x ,x ) ( U × U ))(82) = D x r ( T x U ) × D x r ( T x U )= T x O × T x O . This completes the proof that r × r is a tame sc-retraction and hence O × O is a tame sc-retract.We have shown that the product chart (80) is indeed a tame M -polyfold chart on B (1) × B (2) . Since product charts of this form cover B (1) × B (2) and have sc-smooth transition maps due to the transitionson each factor being sc-smooth, the collection of these product chartsforms a tame atlas on B (1) × B (2) . Then B (1) × B (2) equipped with thisatlas is a tame M -polyfold.We now prove the claimed degeneracy index formula. Let ( x , x ) ∈ ( B (1) ) ∞ × ( B (2) ) ∞ . From the splitting (82) of the tangent spaces in theproduct retract, we conclude T ( x ,x ) ( B (1) × B (2) ) ∼ = T x B (1) × T x B (2) . Furthermore, by definition of reduced tangent space in a retract (12)together with (81) and (82), we conclude T R ( ϕ ( x ) ,ϕ ( x )) ( O × O ) = T Rϕ ( x ) O × T Rϕ ( x ) O , which implies by the global definition of re-duced tangent space (49) that T R ( x ,x ) ( B (1) × B (2) ) ∼ = T Rx B (1) × T Rx B (2) by examining any product chart. Hence, by the global description(50) of degeneracy index at smooth points, we have d B (1) ×B (2) ( x , x ) =dim( T ( x ,x ) ( B (1) × B (2) ) /T R ( x ,x ) ( B (1) × B (2) )) = dim( T x B (1) /T Rx B (1) ) +dim( T x B (2) /T Rx B (2) ) = d B (1) ( x ) + d B (2) ( x ) . This completes the proofof (I).Proof of (II): For i = 1 , , consider a tame M -polyfold chart (79) on B ( i ) covered by a tame strong bundle chart (Definition 5.2)(83) ( ρ − i ( V i ) , Φ i , ( K i , C i ✁ F i , E i ✁ F i ))for ρ i . We now construct a tame strong bundle chart for ρ × ρ over theopen set V × V . For i = 1 , , consider a tame strong bundle retraction(23) R i : U i ✁ F i → U i ✁ F i ( x, ξ ) ( r i ( x ) , Γ i ( x, ξ ))with image K i = R i ( U i ✁ F i ), where Γ i : U i ✁ F → F OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 79 is a linear projection Γ i ( x, · ) for every x ∈ U i . Denote the reorderingof factors map by l : ( E × F ) × ( E × F ) → ( E × E ) × ( F × F ) , and note that l is a linear sc-isomorphism. We claim that the tuple(84)(( ρ × ρ ) − ( V × V ) , l ◦ ( Φ × Φ ) , ( l ( K × K ) , C × C ✁ F × F , E × E ✁ F × F ))is a tame strong bundle chart for ρ × ρ covering the product M -polyfold chart (80). Since for i = 1 , , the map Φ i : ρ − i ( V i ) → K i is ahomeomorphism covering ϕ i that is linear on fibers, it follows that themap l ◦ ( Φ × Φ ) is a homeomorphism from ( ρ × ρ ) − ( V × V ) to l ( K × K ) covering ϕ × ϕ and is linear on fibers. Moreover, the set l ( K × K ) is a tame strong bundle retract because it is the image ofthe tame strong bundle retraction l ◦ ( R × R ) ◦ l − : ( U × U ) ✁ ( F × F ) → ( U × U ) ✁ ( F × F )( x , x , ξ , ξ ) ( r ( x ) , r ( x ) , Γ ( x , ξ ) , Γ ( x , ξ )) . We have shown that the chart (84) is indeed a tame strong bundle chartfor ρ × ρ . Sc-smoothness of bundle transitions between charts con-structed in this way follows from sc-smoothness in each factor. Hencethe collection of these charts forms a tame bundle atlas for ρ × ρ .This completes the proof of (II).Proof of (III): To prove that σ × σ is a tame sc-Fredholm sec-tion (Definition 5.4) of ρ × ρ , first observe that it is sc-smooth andregularizing because σ and σ are sc-smooth and regularizing, andthe double filtration satisfies ( B (1) × B (2) ) m = ( B (1) ) m × ( B (2) ) m and( E (1) × E (2) ) m,m +1 = ( E (1) ) m,m +1 × ( E (2) ) m,m +1 for all m ≥ x , x ) ∈ ( B (1) ) ∞ × ( B (2) ) ∞ . Then sinceeach σ i is tame sc-Fredholm there exist tame bundle charts as in (83)that are in addition tame sc-Fredholm charts for σ i at x i . First of all,this means that the partial quadrants C i ⊂ E i are in the standard form(5), i.e. C i = [0 , ∞ ) s i × E ′ i ⊂ R s i × E ′ i = E i . Now, we will prove thata chart similar to the tame bundle chart (84) is a tame sc-Fredholmchart for σ × σ at ( x , x ). The only issue with (84) is that the partialquadrant C × C ⊂ E × E is not in the standard form. This is easilyremedied as follows. Consider the partial quadrant C := [0 , ∞ ) s × s × E ′ × E ′ of the sc-Banach space E := R s × s × E ′ × E ′ Then the reordering of factors map f : R s × E ′ × R s × E ′ → R s + s × E ′ × E ′ is a linear sc-isomorphism E × E → E that restricts to an isomor-phism C × C → C . Applying f to the sc-retract O × O produces atame sc-retract O := f ( O × O ) ⊂ C with tame sc-retraction r := f ◦ ( r × r ) ◦ f − : U → U,U := f ( U × U ) ⊂ C, onto r ( U ) = O . Moreover, the map ϕ := f ◦ ( ϕ × ϕ ) : V × V → O is a homeomorphism, and so from the tame M -polyfold chart (80) weobtain another tame M -polyfold chart( V × V , ϕ, ( O , C, E )) . Similarly, applying f × id F × F to the tame bundle retract l ( K × K )produces another tame bundle retract K := ( f × id F × F ) ◦ l ( K × K )with tame bundle retraction R := ( f × id F × F ) ◦ l ◦ ( R × R ) ◦ l − ◦ ( f × id F × F ) − : U ✁ F × F → U ✁ F × F onto R ( U ✁ F × F ) = K . Consider the map Γ := ( Γ × Γ ) ◦ l − ◦ ( f × id F × F ) − , and observe that, for ( y, ξ , ξ ) ∈ U ✁ F × F , we have R ( y, ξ , ξ ) = ( r ( y ) , Γ ( y, ξ , ξ )) . The map Φ := ( f × id F × F ) ◦ l ◦ ( Φ × Φ ) : ( ρ × ρ ) − ( V × V ) → K is a homeomorphism that is linear on the fibers and covers ϕ . Hencefrom the tame bundle chart (84) we obtain another tame bundle chart(85) (( ρ × ρ ) − ( V × V ) , Φ, ( K, C ✁ F × F , E ✁ F × F )) . We claim that the bundle chart (85) is a tame sc-Fredholm chart for σ × σ at ( x , x ). We must show that the section(86) τ := Φ ◦ ( σ × σ ) ◦ ϕ − : O → K OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 81 is a tame sc-Fredholm germ. Since for each i = 1 , , the section τ i := Φ i ◦ σ i ◦ ϕ − i : O i → K i is a tame sc-Fredholm germ, we can assume that there is a filling h i : U i → U i ✁ F i of τ i ◦ r i : U i → F i , where τ i : O i → F i denotes the principal part of τ i in the fiber F i . We claim that h := ( f × id F × F ) ◦ l ◦ ( h × h ) ◦ f − : U → U ✁ ( F × F )is a filling of τ ◦ r : U → F × F . The required properties in Def-inition 3.6(b).(i)-(iii) of a filling follow from the corresponding prop-erties for each h i , as we now verify. Given y ∈ O , we compute τ ( y ) = ( f × id F × F ) ◦ l ◦ ( Φ × Φ ) ◦ ( σ × σ ) ◦ ( ϕ ◦ ϕ ) − ◦ f − ( y ) = ( f × id F × F ) ◦ l ◦ ( τ × τ ) ◦ f − ( y ) = ( f × id F × F ) ◦ l ◦ ( h × h ) ◦ f − ( y ) = h ( y ) , which verifies property (i) for τ . To see property ( ii ), let y ∈ U and as-sume h ( y ) = Γ ( r ( y ) , h ( y )). Then since the principal parts satisfy h =( h × h ) ◦ f − , we compute that ( h × h ) ◦ f − ( y ) = Γ ( r ( y ) , h ( y )) =( Γ × Γ ) ◦ l − ◦ ( f × id F × F ) − ( f ◦ ( r × r ) ◦ f − ( y ) , ( h × h ) ◦ f − ( y )) =( Γ × Γ ) ◦ l − (( r × r ) ◦ f − ( y ) , ( h × h ) ◦ f − ( y )) , which implies f − ( y ) ∈ O × O by the corresponding property of τ and τ . Hence y ∈ O , proving property (ii) for τ . We now verify ( iii ) for τ . By thecorresponding property for each τ i , the linearization D L i at 0 of L i : U i → F i y i ( id F i − Γ i ( r i ( y i ) , · )) h i ( y i )restricts to a linear sc-isomorphism from ker D r i to ker Γ i (0 , · ). Wemust show that the linearization at 0 of the map L : U → F × F y ( id F × F − Γ ( r ( y ) , · )) h ( y )restricts to a linear sc-isomorphism from ker D r to ker Γ (0 , · ). This fol-lows from the observations that L = ( L × L ) ◦ f − holds, the map f − is a linear sc-isomorphism that restricts to an isomorphism ker D r → ker D ( f ◦ ( r × r )) = ker D ( r × r ) = ker( D r ) × ker( D r ), andker Γ (0 , · ) = ker Γ (0 , · ) × ker Γ (0 , · ) . Thus h is a filling of τ ◦ r , asclaimed.We now verify the properties of τ required in Definition 3.6(c) ofa local sc-Fredholm germ. We in addition show that τ satisfies thestronger properties of a tame sc-Fredholm germ (Definition 3.7). Since each τ i is a tame sc-Fredholm germ, the corresponding prop-erties hold: There exists a sc + -section s i : U i → U i ✁ F i satisfying s i (0) = h i (0), a sc-Banach space W i , a sc-germ of neighbor-hoods U ′ i around 0 in [0 , ∞ ) s i × R k i − s i × W i for some k i ≥ s i ≥
0, and a strong bundle isomorphism Ψ i : U i ✁ F i → U ′ i ✁ R k ′ i × W i covering a linear sc-isomorphism ψ i = id [0 , ∞ ) si × ψ i : U i → U ′ i satisfying ψ i (0) = 0, where ψ i : E ′ i → R k i − s i × W i is a linear sc-isomorphism, and such that the principal part of thesection(87) b i := Ψ i ◦ ( h i − s i ) ◦ ψ − i : U ′ i → U ′ i ✁ R k ′ i × W i is a basic germ. This basic germ property means that the principalpart b i : U ′ i → R k ′ i × W i is a sc-smooth germ satisfying b i (0) = 0 and such that, for a i ∈ [0 , ∞ ) s i , d i ∈ R k i − s i , w i ∈ W i , we have P i ◦ b i ( a i , d i , w i ) = w i − B i ( a i , d i , w i )where P i : R k ′ i × W i → W i is projection onto W i and B i is a sc-smoothgerm satisfying B i (0) = 0 and the contraction property (26).The section s := ( f × id F × F ) ◦ l ◦ ( s × s ) ◦ f − : U → U ✁ ( F × F )is sc + because each s i is sc + and the other maps in the compositionare linear sc-isomorphisms. Denote the linear sc-isomorphisms givenby reordering the factors by q : ([0 , ∞ ) s × R k − s × W ) × ([0 , ∞ ) s × R k − s × W ) → [0 , ∞ ) s + s × R k − s + k − s × W × W ,Q : ( R k ′ × W ) × ( R k ′ × W ) → R k ′ + k ′ × W × W ,l ′ : ( U ′ ✁ R k ′ × W ) × ( U ′ ✁ R k ′ × W ) → ( U ′ × U ′ ) ✁ ( R k ′ × W × R k ′ × W ) . OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 83
Set U ′ := q ( U ′ × U ′ ) ,ψ := q ◦ ( ψ × ψ ) ◦ f − : U → U ′ ,Ψ := ( q ✁ Q ) ◦ l ′ ◦ ( Ψ × Ψ ) ◦ l − ◦ ( f × id F × F ) − : U ✁ ( F × F ) → U ′ ✁ ( R k ′ + k ′ × W × W ) . We claim that the principal part b of the section(88) b := Ψ ◦ ( h − s ) ◦ ψ − : U ′ → U ′ ✁ ( R k ′ + k ′ × W × W )is a basic germ. Observe that the principal parts satisfy b = Q ◦ π R k ′ × W × R k ′ × W ◦ ( Ψ × Ψ ) ◦ (( h × h ) − ( s × s )) ◦ ( ψ × ψ ) − ◦ q − = Q ◦ ( b × b ) ◦ q − . Let P : R k ′ + k ′ × W × W → W × W denote the projection onto W × W . Notice that P = ( P × P ) ◦ Q − . Then, for ( a , a ) ∈ [0 , ∞ ) s × [0 , ∞ ) s , ( d , d ) ∈ R k − s × R k − s , and ( w , w ) ∈ W × W ,we compute P ◦ b ( a , a , d , d , w , w ) = P ◦ Q ◦ ( b × b ) ◦ q − ( a , a , d , d , w , w )= ( P × P ) ◦ ( b × b )( a , d , w , a , d , w )= ( w − B ( a , d , w ) , w − B ( a , d , w ))= ( w , w ) − ( B ( a , d , w ) , B ( a , d , w ))= ( w , w ) − ( B × B ) ◦ q − ( a , a , d , d , w , w ) , so to prove that b is a basic germ, it remains to verify that the map B := ( B × B ) ◦ q − : [0 , ∞ ) s + s × R k − s + k − s × W × W → W × W satisfies the contraction property (26). Recall our convention that Ba-nach norms on Cartesian products are chosen to be the sum norm(which is equivalent to choosing any standard choice of norm on aproduct). The contraction property (26) for B then follows from thecontraction property for each B i , as we now verify. Let ǫ > m ≥
0. Then, for i = 1 , , there exists δ i > B i with the same choice of ǫ. Set δ := min( δ , δ ). Then, given || ( a , a , d , d , w , w ) || m , || ( a , a , d , d , w ′ , w ′ ) || m , < δ we have || ( a i , d i , w i ) || m , || ( a i , d i , w ′ i ) || m < δ i for i = 1 , , from which we compute, using property (26) for the B i , || B ( a , a , d , d , w , w ) − B ( a , a , d , d , w ′ , w ′ ) || m = || ( B ( a , d , w ) , B ( a , d , w )) − ( B ( a , d , w ′ ) , B ( a , d , w ′ )) || m = || ( B ( a , d , w ) − B ( a , d , w ′ ) , B ( a , d , w ) − B ( a , d , w ′ ) || m = || ( B ( a , d , w ) − B ( a , d , w ′ ) || m + || B ( a , d , w ) − B ( a , d , w ′ ) || m ≤ ǫ ( || w − w ′ || m + || w − w ′ || m )= ǫ · || ( w − w ′ , w − w ′ ) || m = ǫ · || ( w , w ) − ( w ′ , w ′ ) || m , as required. This completes the proof that τ is a local sc-Fredholmgerm.We claim that, in addition, τ is a tame sc-Fredholm germ. We mustshow that ψ is in the form required by Definition 3.7. Given ( a i , e i ) ∈ [0 , ∞ ) s i × E ′ i for i = 1 , , write ψ i ( e i ) = ( d i , w i ) ∈ R k i − s i × W i andcompute ψ ( a , a , e , e ) = q ◦ ( ψ × ψ )( a , e , a , e )= q ◦ ( id [0 , ∞ ) s × ψ × id [0 , ∞ ) s × ψ )( a , e , a , e )= ( a , a , d , d , w , w ) . So indeed ψ = id [0 , ∞ ) s s × ψ is of the required form, where ψ : E × E → R k − s + k − s × W × W is the linear sc-isomorphism given by ψ ( e , e ) = ( d , d , w , w ) . We have verified that σ × σ is a tame sc-Fredholm section of ρ × ρ .To verify the claimed index formula, note first that by definition of sc-Fredholm index (27) of a local sc-Fredholm germ and the forms of thebasic germs (87) and (88) we have ind ( τ i ) = k i − k ′ i and ind ( τ ) = ( k + k ) − ( k ′ + k ′ ) = ind ( τ ) + ind ( τ ). Then by definition Definition 5.3(3)of index, for ( x , x ) ∈ ( B (1) ) ∞ × ( B (2) ) ∞ we have ind ( x ,x ) ( σ × σ ) = ind ( τ ) = ind ( τ ) + ind ( τ ) = ind x ( σ ) + ind x ( σ ) . The final statement about compactness holds because the zero set( σ × σ ) − (0) = σ − (0) × σ − (0) ⊂ ( B (1) ) ∞ × ( B (2) ) ∞ is equipped withthe product topology in every level ( B (1) × B (2) ) m = ( B (1) ) m × ( B (2) ) m ,and for i = 1 , , the subspace σ − i (0) ⊂ ( B ( i ) ) m is compact for all m ≥ (cid:3) We now generalize the above Cartesian product construction to theep-groupoid setting.
Lemma 7.2.
OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 85 (I)
Consider tame ep-groupoids X ( i ) = ( X ( i ) , X ( i ) ) for i = 1 , . Then,the Cartesian product X (1) × X (2) = ( X (1) × X (2) , X (1) × X (2) ) is atame ep-groupoid with degeneracy index satisfying d X (1) × X (2) ( x , x ) = d X (1) ( x ) + d X (2) ( x ) for all ( x , x ) ∈ ( X (1) ) ∞ × ( X (2) ) ∞ . (II) Consider tame strong bundles ( P i : E ( i ) → X ( i ) , µ i ) over X ( i ) for i = 1 , . Denote the reordering of factors map by l : ( X (1) × X (2) ) × s × s P × P ( E (1) × E (2) ) → ( X (1) × s P E (1) ) × ( X (2) × s P E (2) ) and set µ := ( µ × µ ) ◦ l : ( X (1) × X (2) ) × s × s P × P ( E (1) × E (2) ) → E (1) × E (2) . Then, the tuple ( P × P : E (1) × E (2) → X (1) × X (2) , µ ) is a tamestrong bundle over X (1) × X (2) . (III) Consider tame sc-Fredholm section functors σ i : X ( i ) → E ( i ) (Definition 6.6) of ( P i , µ i ) for i = 1 , . Then, the product map σ × σ : X (1) × X (2) → E (1) × E (2) is a tame sc-Fredholm section functor of ( P × P , µ ) with index satisfying ind ( x ,x ) ( σ × σ ) = ind x ( σ ) + ind x ( σ ) for all ( x , x ) ∈ ( X (1) ) ∞ × ( X (2) ) ∞ . If | σ − i (0) | is compact for i = 1 , , then | ( σ × σ ) − (0) | = | σ − (0) | × | σ − (0) | is compact.Proof. We prove the statements in (I). The product X (1) × X (2) is agroupoid with structure maps (source, target, multiplication, unit, andinverse) as we describe below. First note that Lemma 7.1(I) providestame M -polyfold structures on the object space X (1) × X (2) and on themorphism space X (1) × X (2) with the claimed degeneracy index. So, toprove (I), it remains to describe the structure maps on X (1) ×X (2) , verifythat they are sc-smooth, and verify the ´etale property Definition 6.1(i)and properness Definition 6.1(iii).For i = 1 ,
2, let ( s i , t i , m i , u i , ι i ) denote the structure maps on X ( i ) .The source s × s : X (1) × X (2) → X (1) × X (2) , target t × t : X (1) × X (2) → X (1) × X (2) , unit u × u : X (1) × X (2) → X (1) × X (2) , and inversion ι × ι : X (1) × X (2) → X (1) × X (2) maps on the product are productsof those in each factor, so sc-smoothness follows from sc-smoothnessin each factor. Moreover, the ´etale property Definition 6.1(i) holdsbecause products of surjective local sc-diffeomorphisms are surjectivelocal sc-diffeomorphisms.To see that the multiplication map on the product is sc-smooth, firstnote that the reordering of factors map ( X (1) × X (2) ) × ( X (1) × X (2) ) → ( X (1) × X (1) ) × ( X (2) × X (2) ) is sc-smooth. It restricts to a bijection q : ( X (1) × X (2) ) × s × s t × t ( X (1) × X (2) ) → ( X (1) × s t X (1) ) × ( X (2) × s t X (2) ) , which is sc-smooth by [7, Prop. 2.15] and [7, Prop. 2.6(1)]. The multi-plication map on X (1) × X (2) is the composition( m × m ) ◦ q : ( X (1) × X (2) ) × s × s t × t ( X (1) × X (2) ) → X (1) × X (2) , hence is sc-smooth.We now check properness Definition 6.1(iii). Let ( x , x ) ∈ X (1) × X (2) and let V ( x i ) ⊂ X ( i ) be open neighborhoods of x i such that t i : s − i ( V ( x i )) → X ( i ) are proper maps. Then V ( x ) × V ( x ) is an openneighborhood of ( x , x ) in X (1) × X (2) and we have( s × s ) − ( V ( x ) × V ( x )) = s − ( V ( x )) × s − ( V ( x )) . Hence t × t : ( s × s ) − ( V ( x ) × V ( x )) → X (1) × X (2) is proper, asthe product of proper maps. This completes the proof that X (1) × X (2) is a tame ep-groupoid, and so the statements in (I) are proved.We now prove the statements in (II). For i = 1 , , the map P i : E ( i ) → X ( i ) is a tame strong bundle over the M -polyfold X ( i ) , so Lemma 7.1(II)provides a tame strong bundle structure on the product map P × P : E (1) × E (2) → X (1) × X (2) . The map µ is a strong bundle map as thecomposition of the strong bundle maps µ × µ and l , and the requiredproperties Definition 6.3(i)(ii) of µ follow immediately from those of µ i for i = 1 ,
2. Hence ( P × P , µ ) is a tame strong bundle over X (1) × X (2) ,as claimed.We now prove the statements in (III). For i = 1 , , the tame sc-Fredholm section functor σ i : X ( i ) → E ( i ) is in particular a tamesc-Fredholm section of the bundle P i , in the M -polyfold sense. SoLemma 7.1(III) provides the product map σ × σ with the structureof a tame sc-Fredholm section of the bundle P × P with the claimedsc-Fredholm index. Moreover, σ × σ satisfies the required property(74) of a section functor of ( P × P , µ ) by the corresponding prop-erty of the sections functors σ i of ( P i , µ i ). Indeed, for all morphisms( φ × φ ) ∈ X (1) × X (2) , we compute( σ × σ ) ◦ ( t × t )( φ , φ ) = ( σ ( t ( φ )) , σ ( t ( φ )))= ( µ ( φ , σ ( s ( φ ))) , µ ( φ , σ ( s ( φ ))))= ( µ × µ ) ◦ l ( φ , φ , σ ( s ( φ )) , σ ( s ( φ )))= µ ( φ , φ , ( σ × σ ) ◦ ( s × s )( φ , φ )) . This completes the proof that σ × σ is a tame sc-Fredholm sectionfunctor of ( P × P , µ ).The final statement about compactness holds because the orbit spaceof X (1) × X (2) is equal to the Cartesian product of the orbit spaces ofthe X ( i ) equipped with the product topology. (cid:3) OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 87
We proceed to construct fiber products of tame sc-Fredholm sec-tion functors over ep-groupoids. As usual, the result specializes to thecase of M -polyfolds by considering a M -polyfold B as an ep-groupoidwith the trivial groupoid structure: the object space is B and the mor-phism space consists of the identity morphisms { x | x ∈ B} whichis a M -polyfold by declaring the bijection x x with B to be asc-diffeomorphism. Corollary 7.3. (I)
Consider tame ep-groupoids X ( i ) = ( X ( i ) , X ( i ) ) for i = 1 , , asmooth manifold Y , and sc-smooth maps f i : X ( i ) → Y for i = 1 , , that satisfy the compatibility with morphisms f i ( s i ( φ )) = f i ( t i ( φ )) for all φ ∈ X ( i ) . Assume that the product map f × f : X (1) × X (2) → Y × Y is transverse(Definition 5.8) to the diagonal ∆ = { ( y, y ) | y ∈ Y } ⊂ Y × Y , anddenote the fiber product of object spaces by X (1) × f f X (2) := { ( x , x ) ∈ X (1) × X (2) | f ( x ) = f ( x ) } ⊂ X (1) × X (2) . Then, there exists an open neighborhood ˜ X ⊂ ( X (1) × f f X (2) ) ∩ (( X (1) ) × ( X (2) ) ) of ( X (1) × f f X (2) ) ∩ (( X (1) ) ∞ × ( X (2) ) ∞ ) such that ˜ X is a slice of X × X in the M -polyfold sense (Definition 5.7) satisfying codim x ( ˜ X ⊂ X × X ) = dim Y for every x ∈ ˜ X = ˜ X ∩ (( X (1) ) × ( X (2) ) ) .In particular, the full subcategory ˜ X := ( ˜ X, ˜ X ) of X is a tame ep-groupoid with degeneracy index satisfying d ˜ X ( x , x ) = d X (1) ( x ) + d X (2) ( x ) for all ( x , x ) ∈ ˜ X ∞ . (II) Consider, in addition, tame strong bundles ( P i : E ( i ) → X ( i ) , µ i ) over X ( i ) for i = 1 , . Then, there exists a possibly smaller neighbor-hood ˜ X in (I) that is a slice of the bundle P | E × P | E : E × E → X × X in the M -polyfold sense (Definition 5.7). In particular,the tuple ( ˜ P , ˜ µ ) consisting of the restrictions ˜ E := ( P | E × P | E ) − ( ˜ X )˜ P := ( P × P ) | ˜ E : ˜ E → ˜ X, ˜ µ := ( µ × µ ) ◦ l | ˜ X × s × s P ˜ E : ˜ X × s × s ˜ P ˜ E → ˜ E, is a tame strong bundle over ˜ X , where l : ( X (1) × X (2) ) × ( E (1) × E (2) ) → ( X (1) × E (1) ) × ( X (2) × E (2) ) is the reordering of factors map. (III) Consider, in addition, tame sc-Fredholm section functors (Defi-nition 6.6) σ i : X ( i ) → E ( i ) of ( P i , µ i ) for i = 1 , . Assume that f × f is ( σ × σ ) -compatibly transverse to ∆ (Definition 5.8).Then, there exists a possibly smaller neighborhood ˜ X in (II) that is aslice of the tame sc-Fredholm section ( σ × σ ) | X × X : X × X → E × E in the M -polyfold sense (Definition 5.7). In particular, therestriction ˜ σ := ( σ × σ ) | ˜ X : ˜ X → ˜ E is a tame sc-Fredholm section functor of the bundle ( ˜ P , ˜ µ ) with indexsatisfying ind ( x ,x ) (˜ σ ) = ind x ( σ ) + ind x ( σ ) − dim Y for all ( x , x ) ∈ ˜ X ∞ . If | σ − i (0) | is compact for i = 1 , , then | ˜ σ − (0) | is compact.Proof. We prove the statements in (I). Lemma 7.2(I) provides a tameep-groupoid structure on X (1) × X (2) with degeneracy index satisfying d X (1) × X (2) ( x , x ) = d X (1) ( x )+ d X (2) ( x ) for ( x , x ) ∈ ( X (1) ) ∞ × ( X (2) ) ∞ .We claim that Corollary 6.8(I) applies to the product map f × f : X (1) × X (2) → Y × Y and the codimension dim Y submanifold ∆ ⊂ Y × Y . Indeed, f × f is transverse to ∆ by hypothesis and the requiredmorphism compatibility ( f × f ) ◦ ( s × s )( φ , φ ) = ( f × f ) ◦ ( t × t )( φ , φ ) holds by the hypothesis f i ( s i ( φ i )) = f i ( t i ( φ i )). Since thefiber product is the preimage of the diagonal X (1) × f f X (2) = ( f × f ) − ( ∆ ) , the result of Corollary 6.8(I) is exactly the assertions in (I).Similarly, to prove (II), we note that Lemma 7.2(II) provides a tamestrong bundle structure on ( P × P , ( µ × µ ) ◦ l ), and then Corol-lary 6.8(II) provides the desired result.We now prove (III). Lemma 7.2(III) shows that σ × σ : X (1) × X (2) → E (1) × E (2) is a tame sc-Fredholm section functor of ( P × P , ( µ × µ ) ◦ l )with index satisfying ind ( x ,x ) ( σ × σ ) = ind x ( σ )+ ind x ( σ ) and with | σ − (0) | × | σ − (0) | compact. Then by Corollary 6.8(III), we concludethat ˜ σ is a tame sc-Fredholm section functor with index satisfying ind ( x ,x ) (˜ σ ) = ind ( x ,x ) ( σ × σ ) − dim Y = ind x ( σ ) + ind x ( σ ) − dim Y and such that | ˜ σ − (0) | is compact. (cid:3) OLYFOLD REGULARIZATION OF CONSTRAINED MODULI SPACES 89
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