FREDHOLM REALIZATIONS OF ELLIPTIC SYMBOLS ON MANIFOLDS WITH BOUNDARY II: FIBERED BOUNDARY

PIERRE ALBIN AND RICHARD MELROSE

0.8; Revised: January 19, 2009; Run: May 6, 2009

Abstract.

Contents

Introduction 1 1. Φ-b and Φ-c algebras of pseudodifferential operators 4 2. Adiabatic limit of edge to ϕ calculi 8 3. Fredholm perturbations of the identity 12 4. Six term exact sequence 15 Appendix A. Triple adiabatic space 16 References 18

Introduction

If the boundary of a manifold is the total space of a fibration

(1)  [figure: Z; ∂X; Φ; Y]

one can quantize an invertible symbol

σ ∈ C^{∞}(S^{∗}X; hom(π^{∗}E, π^{∗}F))

as an elliptic pseudodifferential operator in the Φ-b or edge calculus, Ψ_{Φ-b}^{0}(X; E, F), introduced in [7] or alternately as an elliptic operator in the Φ-c or ϕ calculus, Ψ_{Φ-c}^{0}(X; E, F), introduced in [8]. As on a closed manifold, either of these operators will induce a bounded operator acting between natural L^{2}-spaces of sections but, in contrast to closed manifolds, these operators need not be Fredholm.

A well-known analysis of Atiyah and Bott established that in order to quantize σ as an operator on X for which Fredholm local boundary conditions exist, it is necessary and sufficient that the Atiyah-Bott obtruction of σ vanish, i.e.,

[σ] ∈ ker (K_{c}(T^{∗}X) → K_{c}^{1}(T^{∗}∂X)) .

For pseudodifferential calculi associated to a fibration ϕ as above, with Z = {pt}, namely for Φ-b the zero calculus of [6] and for Φ-c the scattering calculus of [11], the vanishing of the Atiyah-Bott obstruction is a necessary an sufficient condition for the existence of a Fredholm quantization of σ. At the other extreme, when Y = {pt}, the Φ-b calculus is the b-calculus of [10] and the Φ-c calculus is the cusp calculus of [8] and in either case there is no obstruction to finding a Fredholm realization of σ. The case of general ϕ lies in between these two extremes. Indeed, it was established in [13] that one can use Φ and the families index to induce a map K_{c}(T^{∗}X) → K_{c}^{1}(T^{∗}Y) (see §4) and that σ has a Fredholm quantization in the Φ-c calculus if and only if

[σ] ∈ ker (K_{c}(T^{∗}X) → K_{c}^{1}(T^{∗}Y)) .

One of our main goals in this paper we establish the corresponding result for the edge calculus.

To this end we consider the even ‘smooth K-theory’ group of the edge calculus K_{Φ-b}(X) consisting of equivalence classes of Fredholm edge operators where two operators are equivalent if they can be joined by bundle stabilization, bundle isomorphims, or smooth homotopy (see §1). This has a natural subgroup K_{Φ-b,−∞}(X) of equivalence classes whose prinicipal symbol is the identity, and there are analogous odd smooth K-theory groups defined by suspension. These groups are defined and identified for the Φ-c calculus in [13].

The analysis of these groups for the Φ-c calculus is simpler for two related reasons. The first is that the Φ-c calculus is ‘asymptotically commutative’ whereas the Φ-b calculus is ‘asymptotically non-commutative’. More precisely, the behavior of a Φ-c operator near the boundary is modeled by a family of operators, parametrized by Y, acting on the Lie group ℝ^{h+1} times the closed manfiold Z. On the other hand, the behavior of a Φ-b operator near the boundary is modeled by a family of operators, parametrized by Y, acting on the Lie group ℝ^{+} ⋉ ℝ^{h} times the closed manifold Z. To see the effect of this difference on the analysis of these calculi, one can compare the relative simplicity of identifying K_{sc}(X) in [13, §2] versus the corresponding identification of K_{0}(X) in [1].

The other simplification is that the Φ-c calculus is closed under the smooth functional calculus, while the Φ-b calculus is not. This means that for the Φ-c calculus one can study the smooth K-theory groups using much the same constructions one would use to study the K-theory of its C^{∗}-algebra (see for instance the constructions used in [13, §4] to define KK-classes). One could remove this difficulty by passing to a C^{∗}-closure of the Φ-b calculus, but at the cost of losing much of the structure of the Φ-b calculus.

Instead, we will study the smooth K-theory groups of the Φ-b-calculus by constructing an ‘adiabatic’ calculus of pseudo-differential operators that interpolates between the Φ-b and Φ-c calculi and induces maps, labeled ad, between their smooth K-theory groups. We work more generally in the context of a fibration X −M ⟶^{ϕ} B where the fibers are manifolds with fibered boundaries, thus altogether we have

(2)  [figure: X; M; ϕ; Z; ∂X; ∂M; Φ; B; Y; D]

Using our analysis of the smooth K-theory groups of the zero calculus in [1], we establish the following theorem.

Theorem 1. The maps

K_{Φ-c}(ϕ) ⟶^{ad} K_{Φ-b}(ϕ), K_{Φ-c}^{1}(ϕ) ⟶^{ad} K_{Φ-b}^{1}(ϕ)

are isomorphisms, and restrict to isomorphisms

K_{Φ-c,−∞}(ϕ) ⟶^{ad} K_{Φ-b,−∞}(ϕ), K_{Φ-c,−∞}^{1}(ϕ) ⟶^{ad} K_{Φ-b,−∞}^{1}(ϕ).

Furthermore, these groups fit into a commutative diagram

[figure: K_{Φ-c,−∞}^{0} (ϕ); ad; K_{Φ-c}^{0} (ϕ); ad; K_{c}^{0} (T^{∗}M/B); =; K_{Φ-b,−∞}^{0} (ϕ); K_{Φ-b}^{0} (ϕ); K_{c}^{0} (T^{∗}M/B); K_{c}^{1} (T^{∗}M/B); K_{Φ-b}^{1} (ϕ); K_{Φ-b,−∞}^{1} (ϕ); K_{c}^{1} (T^{∗}M); =; K_{Φ-c}^{1} (ϕ); ad; K_{Φ-c,−∞}^{1} (ϕ); ad]

wherein the inner and outer six term sequences are exact.

In [13], the groups K_{Φ-c}^{q}(ϕ) are identified in terms of the KK-theory of the C(B)- module

C_{Φ}(M) = {f ∈ C(M) : f|_{∂M} ∈ Φ^{∗}C(D)}

and the K_{Φ-c,−∞}^{q}(ϕ) groups are identified with the K-theory of the vertical cotangent bundle T^{∗}D/B. Below, we establish the analogue of the latter directly and deduce the analogue of the former from Theorem 1.

Corollary 2. The groups K_{Φ-b}^{q}(ϕ) are given by

K_{Φ-b}^{q}(ϕ) ≅ KK_{B}^{q}(C_{Φ}(M), C(B))

and the groups K_{Φ-b,−∞}^{q}(ϕ) are given by

K_{Φ-b,−∞}^{q}(ϕ) ≅ K_{c}^{q}(T^{∗}D/B).

Also, we can read off from the six-term exact sequence and this corollary the topological obstruction to realizing an elliptic symbol via a family of Fredholm Φ-b operators.

Corollary 3. An elliptic symbol σ ∈ C^{∞}(S^{∗}M/B; hom(π^{∗}E, π^{∗}F)) can be quantized as a Fredholm family of Φ-b operators if and only if its K-theory class satisfies

[σ] ∈ ker (K_{c}(T^{∗}M/B) → K_{c}^{1}(T^{∗}D/B)) .

Finally, in [13] the second author and Frédéric Rochon defined a topological index as a map K_{Φ-c}(ϕ) → K(B) and showed that it coincided with the analytic index. Thus the group K_{Φ-b}(ϕ) inherits a topological index map from its isomorphism with K_{Φ-c}(ϕ) and, since the adiabatic limit commutes with the analytic index, we also have the following K-theoretic index theorem.

Corollary 4. The analytic and topological indices coincide as maps

K_{Φ-b}(ϕ) → K(B).

In section 1, we review the Φ-b and Φ-c algebras of pseudodifferential operators and the definition of the smooth K-theory groups. In section 2, we set up the adiabatic calculus and prove that it is closed under composition in an appendix. In section 3 we use an excision lemma and the analysis of the smooth K-theory of the zero calculus from [1] to identify the groups K_{Φ-b,−∞}^{q}(ϕ). Finally, in section 4, we establish the six term exact sequence and finish the proof of Theorem 1.

1. Φ-b and Φ-c algebras of pseudodifferential operators

We recall some of the outstanding features of these calculi and refer the reader to, e.g., [10], [7], and [8] for more details. For the moment we restrict attention to the case of a single operator (i.e., B = {pt}).

We start by describing the vector fields that act as differential operators in the two calculi. Let {x, y_{1}, …, y_{h}, z_{1}, …, z_{v}} be local coordinates near the boundary with y_{i} horizontal with respect to the fibration (1) and z_{i} vertical. Here x is a boundary defining function or bdf, i.e. a non-negative function on \overline{X} with \{x = 0} = ∂X and dx ≠ 0 on the boundary. We will always assume that the boundary is embedded which implies the existence of a globally defined bdf. The fibered cusp structure depends mildly on the choice of bdf.

The Lie algebra, V_{Φ-b}, of vector fields tangent to the fibers of the fibration over the boundary is locally spanned by the vector fields

⟨x∂_{x}, x∂_{y_{1}}, …, x∂_{y_{h}}, ∂_{z_{1}}, … ∂_{z_{v}}⟩.

Fibered boundary differential operators are polynomials in these vector fields. That is, any P ∈ Diff_{Φ-b}^{k} (X) can be written locally as

P = ∑_{j+|α|+|β|≤k} a_{j,α,β} (x, y, z) (x∂_{x})^{j} (x∂_{y})^{α} (∂_{z})^{β} .

There is a vector bundle ^{Φ-b}T X, the Φ-b-tangent bundle, whose space of smooth sections is precisely V_{Φ-b}. It plays the role of the usual tangent bundle in the study of the Φ-b calculus. For instance, Φ-b one forms are elements of the dual bundle, Φ-bT^{∗}X, and a Φ-b metric is a metric on ^{Φ-b}T X, e.g. locally

(1.1)  g_{Φ-b} = \frac{dx^{2}}{x^{2}} + \frac{Φ^{∗}g_{Y}}{x^{2}} + g_{Z}.

‘Extreme’ cases of fibered boundary calculi are the b-calculus (where Y is a point) and the 0-calculus (where Z is a point). The b-calculus models non-compact manifolds with a cylindrical end. It was used in [10] to prove the APS-index theorem. The 0-calculus models non-compact manifolds that are asymptotically hyperbolic. It has applications to conformal geometry through the Fefferman-Graham construction and to physics (e.g. holography) through the AdS/CFT correspondence [4].

There is also a Φ-c-tangent bundle, whose space of sections V_{Φ-c} is locally spanned by

⟨x^{2}∂_{x}, x∂_{y_{1}}, …, x∂_{y_{h}}, ∂_{z_{1}}, … ∂_{z_{v}}⟩.

Thus a fibered cusp differential operator P ∈ Diff_{Φ-c}^{k} (X) can be written locally as

P = ∑_{j+|α|+|β|≤m} a_{j,α,β} (x, y, z) (x^{2}∂_{x})^{j} (x∂_{y})^{α} (∂_{z})^{β} .

The ‘extreme’ cases of fibered cusp calculi are known as the cusp-calculus (where Y is a point) and the scattering-calculus (where Z is a point). The cusp-calculus models the same geometric situation (asymptotically cylindrical manifolds) as the b-calculus, and indeed these calculi are very closely related (see [1, §3]). The scattering-calculus models non-compact, asymptotically locally Euclidean manifolds. Indeed, if one compactifies ℝ^{n} radially to a half-sphere, the metric near the boundary takes the form

\frac{dx^{2}}{x^{4}} + \frac{h_{x}}{x^{2}} ,

and so defines a metric on the scattering tangent bundle.

As an illustration of the difference between the Φ-b and Φ-c calculi we point out that the scattering Lie algebra is asymptotically commutative in the sense that

[V_{sc}, V_{sc}] ⊂ xV_{sc},

while for the zero calculus,

[V_{0}, V_{0}] ⊂ V_{0}.

This asymptotic commutativity of horizontal vector fields in the Φ-c calculus lies behind the simplifications over the Φ-b calculus.

As on a closed manifold, certain interesting operations (e.g. powers, parametrices, or inverses) require passing to a larger calculus of pseudodifferential operators. Pseudodifferential operators mapping sections of a bundle E to sections of a bundle F are denoted Ψ_{Φ-b}^{∗} (X; E, F) and Ψ_{Φ-c}^{∗} (X; E, F) respectively. Operators in these calculi act by means of an distributional integral kernel as in the Schwartz kernel theorem, i.e.

P f (ζ) = ∫_{X} K_{P} (ζ, ζ′) f (ζ′) .

These integral kernels have singularities along the diagonal and when ζ, ζ′ ∈ ∂X. The latter can be resolved by lifting the kernel to an appropriate blown-up space, denoted respectively X_{Φ-b}^{2} and X_{Φ-c}^{2}. We will describe these spaces below, in §2, as part of the construction of the adiabatic calculus.

These pseudodifferential calculi posses two symbol maps. On a closed manifold, It is well-known that the highest order terms in the local expression of a differential operator define invariantly a function on the cosphere bundle of the manifold. This same construction yields the principal symbol of a differential (and more generally pseudodifferential) Φ-b operator as a function on the Φ-b cosphere bundle, i.e., the bundle of unit vectors in ^{Φ-b}T^{∗}X. This interior symbol fits into the short exact sequence,

Ψ_{Φ-b}^{k−1} (X; E, F) ↪ Ψ_{Φ-b}^{k} (X; E, F) \begin{matrix}σ \\ ↠ \end{matrix}C^{∞} (^{Φ-b}S^{∗}X; (N^{∗}X)^{k} ⊗ hom (π^{∗}E, π^{∗}F)) ,

where (N^{∗}X)^{k} denotes a line bundle whose sections have homogeneity k and π is the projection map ^{Φ-b}S^{∗}X → X. The interior symbol in the fibered cusp calculus works in much the same way, and the corresponding sequence

Ψ_{Φ-c}^{k−1} (X; E, F) ↪ Ψ_{Φ-c}^{k} (X; E, F) \begin{matrix}σ \\ ↠ \end{matrix}C^{∞} (^{Φ-c}S^{∗}X; (N^{∗}X)^{k} ⊗ hom (π^{∗}E, π^{∗}F)) ,

is also exact. The interior symbol is used to define ellipticity: an operator is elliptic if and only if its interior symbol is invertible.

The second symbol map, known as the normal operator, models the behavior of the operator near the boundary. Following [2], at any point p ∈ Y let I_{Φ^{−1}(p)} ⊂ V_{Φ-b} be the subspace of Φ-b vector fields that vanish along the fiber Φ^{−1} (p) (as Φ-b vector fields). These vector fields form an ideal and the quotient ^{Φ-b}T_{p}X is a Lie algebra. The projection

V_{Φ-b} → ^{Φ-b}T_{p}X

lifts to a map of enveloping algebras

Diff_{Φ-b}^{k} (X) → D (^{Φ-b}T_{p}X) ,

and the image of P ∈ Diff_{Φ-b}^{k} (X) is known as the normal operator of P at p, N_{Φ-b,p} (P). The normal operator extends to pseudodifferential operators and can be realized either globally (its kernel is obtained by restricting the kernel of P to a certain boundary face in X_{Φ-b}^{2}) or locally by means of an appropriate rescaling as follows.

Assume that we have chosen a product neighborhood of the boundary and, in a neighborhood U_{Y} of a point p ∈ Y, we have chosen a local trivialization of Φ, so that U ⊂ X looks like

(1.2)  U ≅ [0, ε) × U_{Y} × Z ⟶^{Φ} U_{Y} ,

with coordinates {x}, {y_{i}}, and {z_{i}} on the respective factors, and a corresponding decomposition of the tangent bundle

Φ-bT X = ^{Φ-b}N∂X ⊕ V ∂X.

Using (1.2) we can define the dilation

(x, y_{i}, z_{j}) ↦^{M_{δ}} (δx, δy_{i}, z_{j}) .

If V ∈ V_{Φ-b} then in local coordinates

V_{δ} := (M_{δ}^{−1})^{∗} V

is a smooth vector field defined in a neighborhood of zero (increasing as δ decreases). The map

\begin{matrix}V_{Φ-b} & → T (^{Φ-b}N∂Z) ⊕ V ∂Z \\ V & ↦ lim_{δ→0} V_{δ} \end{matrix}

has kernel I_{p,Φ-b} and realizes ^{Φ-b}T_{p}Z as a Lie algebra of smooth vector fields on T_{p}Z. More generally, Let L_{U} be a chart at p mapping into N_{p}^{+}Y, and define the normal operator of P ∈ Ψ_{Φ-b}^{0} (X) at p to be

N_{p,Φ-b} (P) u := lim_{δ→0} (M_{δ})^{∗} L_{U}^{∗}P (L_{U}^{−1})^{∗} (M_{\frac{1}{δ}} )^{∗} u.

The Lie group structure on ^{Φ-b}N∂X at the point p ∈ Y is that of ℝ^{+} ⋉ ℝ^{h}, i.e.

(s, u) · (s′, u′) = (ss′, u + su′) .

The kernel of the normal operator at the point p ∈ Y is invariant with respect to this action, so the normal operator acts by convolution in the associated variables. We refer to this as a non-commutative suspension, and denote these operators by Ψ_{Nsus}^{∗} (^{Φ-b}N∂X; E, F ). They fit into a short exact sequence

0 → xΨ_{Φ-c}^{k} (X; E, F) → Ψ_{Φ-c}^{k} (X; E, F) ⟶^{N_{Φ-b}} Ψ_{Nsus}^{k} (^{Φ-b}N∂X; E, F ) → 0.

The same construction yields a model operator at every point p ∈ Y for the fibered cusp calculus. In this case the Lie group is commutative and isomorphic to ℝ × ℝ^{h}; the resulting calculus is referred to as the suspended calculus. The corresponding short exact sequence is

0 → xΨ_{Φ-c}^{k} (X; E, F) → Ψ_{Φ-c}^{k} (X; E, F) ⟶^{N_{Φ-c}} Ψ_{sus}^{k} (^{Φ-c}N∂X; E, F ) → 0.

The volume form of a Φ-b metric (e.g. (1.1)), together with Hermitian metrics on E and F, defines a space of L^{2} sections. An operator P ∈ Ψ_{Φ-b}^{k} (X; E, F) acts linearly on L^{2} sections of E; boundedly if k ≤ 0 and as an unbounded operator otherwise. The domains of elliptic operators define a scale of Φ-b Sobolev spaces, H_{Φ-b}^{s} (X; E), and an element P ∈ Ψ_{Φ-b}^{k} (X; E, F) defines a bounded linear operator

H_{Φ-b}^{s} (X; E) ⟶^{P} H_{Φ-b}^{s−k} (X; F) .

This operator is Fredholm if and only if both symbol maps σ (P) and N_{Φ-b} (P) are invertible operators, in which case P is said to be fully elliptic.

In order to carry out our analysis below, we will need a good understanding of the normal operators of elements of Ψ_{0}^{−∞}. A construction introduced in [5] realizes normal operators of the zero calculus as families of ‘b,c-calculus’ on the interval, referred to as reduced normal operators. Operators in this calculus behave like boperators at one end of the interval and like cusp operators at the other end. The reduced normal operator of an operator in Ψ_{0}^{−∞} (X; E) lies inside the space

Ψ_{b,c}^{−∞,−∞} ([0, 1] , π^{∗}E)

of operators that behave like b-operators of order −∞ near 0 and vanish to infinite order order near 1. In the following result from [5], π denotes the canonical projection S^{∗}∂X → ∂X.

Proposition 1.1 ([5], Prop. 4.4.1). The reduced normal operators of elements of Ψ_{0}^{−∞} (X; E) are precisely those functions

\begin{aligned}N (y′, η; τ, ρ) & ∈ C^{∞} (S^{∗}∂X) ˆ⊗_{π}Ċ^{∞} ([−1, 1] , Ω^{1/2}) ˆ⊗_{π}S (ℝ_{+},^{b}Ω^{1/2}) \\ & ⊗_{C^{∞}\binom{}{I^{2}}} C^{∞} (Ĩ^{2}, β^{∗} (Hom (π^{∗}E ⊗^{b,c}Ω^{−1/2}))) \end{aligned} ; (1.3)

with N (y′, η; ·, ρ) the lift of a density on T^{∗}∂X, and which for each η ∈ S^{∗}∂X extend to an element of Ψ_{b,c}^{−∞,−∞} ([0, 1] , π^{∗}E) in such a way that

(1.4)  N ∈ C^{∞} (S^{∗}∂X, Ψ_{b,c}^{−∞,−∞} ([0, 1] , π^{∗}E)) .

Remark. Notice that the fact that N (y′, η; ·, ρ) is the lift of a density on T^{∗}∂X implies in particular that the b-normal operator is a family defined on S^{∗}∂X but actually only depending on ∂X.

Next, we define K-theory groups of Φ-b operators. Following [13, Definition 2], define

A_{Φ-b} (M; E) = {(σ (A) , N (A)) : A ∈ Ψ_{Φ-b}^{0}(M; E) Fredholm}

so that K_{Φ-b}^{0} (M) consists of equivalence classes of elements in A_{Φ-b} (M; E), where two elements are equivalent if there is a finite chain consisting of the following:

(1.5)  (σ, N) ∈ A_{Φ-b} (E) ∼ (σ′, N′) ∈ A_{Φ-b} (E′)

if there exists a bundle isomorphism E → E′ over M, that intertwines (σ, N) and (σ′, N′),

(1.6)  (σ, N) ∈ A_{Φ-b} (E) ∼ (σ̃, Ñ) ∈ A_{Φ-b} (E)

if there exists a homotopy of Fredholm operators A_{t} in Ψ_{Φ-b}^{0} (M; E), with (σ, N) = (σ (A_{0}) , N (A_{0})) and (σ̃, Ñ) = (σ (A_{1}) , N (A_{1})), and

(1.7)  (σ, N) ∈ A_{Φ-b} (E) ∼ (σ ⊕ Id_{F} , N ⊕ Id_{F}) ∈ A_{Φ-b} (E ⊕ F) .

Similarly, we define K_{Φ-b}^{1} (M) as equivalence classes of elements in the space of based loops

ΩA_{Φ-b} (E) = {s ∈ C^{∞} (𝕊^{1}, A_{Φ-b} (E)) : s (1) = Id},

where the equivalences are finite chains of (1.5), (1.6), (1.7) with bundle transformations and homotopies required to be the identity at 1 ∈ 𝕊^{1}.

Analogously, we can describe K_{Φ-b,−∞}^{0} (M) and K_{Φ-b,−∞}^{1} (M) as equivalence classes of elements in

A_{Φ-b,−∞} (E) = {N (Id +A) : A ∈ Ψ_{Φ-b}^{−∞} (M; E) , Id +A Fredholm},

and

ΩA_{Φ-b,−∞} (E) = {s ∈ C^{∞} (𝕊^{1}, A_{Φ-b,−∞} (E)) : s (1) = Id}

respectively.

2. Adiabatic limit of edge to ϕ calculi

For a compact manifold with boundary, with a specified fibration of the boundary, we show that there is an adiabatic limit construction passing from the fibered boundary (for ε > 0) calculus to the fibered cusp calculus in the limit. Every elliptic Fredholm (i.e. totally elliptic) operator in each of the calculi occurs in a totally elliptic family in the adiabatic calculus, so with constant index. This allows the K-theory of the two algebras to be identified and the (families) index map for one to be reduced to that of the other.

Let X be the compact manifold with boundary, with boundary fibration [figure: Z; ∂X; Y] . We construct a resolution of X^{2} × [0, ε_{0}] which carries the adiabatic calculus. First we blow up the corner and consider

X_{(1)}^{2} := [X^{2} × [0, ε_{0}] ; (∂X)^{2} × {0}] ⟶^{β_{(1)}} X^{2} × [0, ε_{0}] .

Then we blow-up the two sides to get

X_{(2)}^{2} := [X_{(1)}^{2}; X × ∂X × {0}; ∂X × X × {0}] ⟶^{β_{(2)}} X_{(1)}^{2}

the lifts of these manifolds are disjoint in X_{(1)}^{2} and commutativity of blow-ups shows that there is a smooth map which is in fact a b-fibration, to the rescaled single space

X_{ε} := [X × [0, ε_{0}] ; ∂X × {0}]

in either factor:

X_{(2)}^{2} ⇉^{π_{L,ε}}_{π_{R,ε}} X_{ε}.

Understanding this will lead to the action of the adiabatic operators on C^{∞} (X_{ε}).

The choice of the fibered-cusp structure on X corresponds to the singling out of a class of boundary defining functions which all induce the same trivialization of the normal bundle to the boundary along the fibres of Φ. Let x be such a boundary defining function, on the left factor of X in X^{2} and let x′ denote the same function on the right factor. The hypersurface x = x′ is smooth near (∂X)^{2} ⊂ X^{2} and lifts to be smooth near the front face of X_{(2)}^{2}, i.e. the lift of (∂X)^{2} × {0}. In fact it meets the front face in a smooth hypersurface; we denote by 𝔉 its intersection with the fibre diagonal

𝔉 ⊂ ff (X_{(2)}^{2})

(in fact it is already smooth in X_{(1)}^{2} and does not meet the side faces blown up to produce X_{(2)}^{2}).

Note that 𝔉 is not a p-submanifold because of its intersection with the lift of (∂X)^{2} × [0, ε_{0}] where it meets the boundary of the fibre diagonal

𝔅 ⊂ X_{(2)}^{2}

(really of course the lift of the fibre diagonal inside (∂X)^{2} × [0, ε_{0}] to X_{(2)}^{2}). The blow-up of 𝔅 resolves 𝔉 to a p-submanifold, so we can set

X_{εΦ}^{2} := [X_{(2)}^{2}; 𝔅; 𝔉] .

Proposition 2.1. The diagonal, Diag × [0, ε_{0}] ⊂ X^{2} × [0, ε_{0}] lifts to an interior p-submanifold of X_{εΦ}^{2} and the projections lift to b-fibrations

X_{εΦ}^{2} ⇉^{π_{L}}_{π_{R}} X_{ε}

which are transverse to the lifted diagonal.

For any ℤ_{2}-graded vector bundle 𝔼 = (E_{+}, E_{−}) over X the space Ψ_{εΦ}^{k} (X; 𝔼) of adiabatic-Φ pseudodifferential operators on X, between sections of E_{+} and sections of E_{−}, is defined to be

Ψ_{εΦ}^{k} (X; E_{+}, E_{−}) = \{A ∈ I^{k− \frac{1}{4}} (X_{εΦ}^{2}, Diag × [0, ε_{0}] ; Hom (E_{+}, E_{−}) ⊗ Ω_{εΦ}) :

A ≡ 0 at the lifts of ∂X×X×[0, ε_{0}] , X×∂X×[0, ε_{0}] , ∂X×X×{0}, X×∂X×{0}}.

Proposition 2.2. These kernels define continuous linear operators

C^{∞} (X_{ε}; E_{+}) → C^{∞} (X_{ε}; E_{−}) .

Proof. This follows from the push-forward and pull-back theorems of [9] together with proposition 2.1. Indeed, given a section f ∈ C^{∞} (X_{ε},^{εΦ}Ω^{1/2}) and an operator A ∈ Ψ_{εΦ}^{k} (X;^{εΦ}Ω^{1/2}) we have

Af = (π_{L})_{∗} (A · π_{R}^{∗}f) .

Since A vanishes to infinite order at all faces of X_{εΦ}^{2} not meeting the diagonal, and each boundary face meeting the diagonal maps down to a unique boundary face in X_{ε} it follows directly that smooth functions are mapped to smooth functions. In the same way, if E is a smooth index set for X_{ε} and A^{E} is the associated set of polyhomogeneous functions then

A^{E} ⟶^{A} A^{E}.

The same argument holds with bundle coefficients with only notational differences. □ We will give a similar geometric proof of the composition

Ψ_{εΦ}^{k} (X; G, H) ∘ Ψ_{εΦ}^{k′} (X; E, G) ⊂ Ψ_{εΦ}^{k+k′} (X; E, H) .

in the appendix (Proposition A.2), after we describe the corresponding triple space.

Operators in the adiabatic calculus, along with the usual interior symbol, have four ‘normal operators’ from restricting their kernels to the different boundary faces. These restrictions in turn have natural interpretations as operators in simpler calculi derived from the suspended and non-commutative suspended calculi in which the normal operators of the fibered cusp and fibered boundary calculi respectively lie. All together, we have five surjective homomorphisms

(\begin{matrix}σ \\ N_{𝔅} \\ N_{𝔉} \\ R_{0} \\ R_{1} \end{matrix}) : Ψ_{Φ-a}^{m,k}(X; 𝔼) ⟶ (\begin{matrix}S^{m}(^{Φ-a}T^{∗}X, hom(𝔼)) \\ C^{∞} ([0, 1]_{ε} , Ψ_{Φ−Nsus}^{k} (X; 𝔼)) \\ C^{∞} ([−1, 1]_{τ} , Ψ_{Φ−sus}^{k} (X; 𝔼)) \\ Ψ_{Φ-c}^{m}(X; 𝔼) \\ Ψ_{Φ-b}^{m}(X; 𝔼) \end{matrix})

with range respectively the homogeneous sections, the noncommutative Φ-suspended calculus, the Φ-suspended calculus, the fibered cusp and fibered boundary calculi respectively.

The model adiabatic calculus, where N_{𝔉} takes values, is a bundle of suspended pseudodifferential algebras on Y ×(0, 1). Namely the adiabatic front face of X_{εΦ}^{2} as a bundle over Y ×[−1, 1] = ff (X_{ε}) /Z (where Z is the fiber of Φ) with fiber over {y}× \{\frac{x−ε}{x+ε}\} the product of RC(W) × Z_{y} × Z_{y} with RC(W) the radial compactification of the subspace

W = ker (^{εΦ}T X_{ε} ↪ T X_{ε}) .

It is useful to see how the product on the fibers evolves as ε → 0. For simplicity assume we are in the conformally compact case. In projective coordinates away from ε = 0,

(x, y, s, u, ε) := (x, y, \frac{x′}{x} , \frac{y′ − y}{x} , ε) ,

the product on the fiber of the zero front face over the point (0, y_{0}, ε) is given by

(s, u) · (s′, u′) = (ss′, u + su′) .

In the perhaps less natural coordinates

(x, y, S, U, ε) := (x, y, \frac{x′ − x}{x}, \frac{y′ − y}{x} , ε)

this product takes the form

(S, U) · (S′, U′) = (S + S′ + SS′, U + U′ + SU′) .

Near the adiabatic front face we can use coordinates

(x, y, ε, τ, S_{ε}, U_{ε}) := (x, y, ε, \frac{x − ε}{x + ε}, \frac{x′ − x}{x (x + ε)}, \frac{y′ − y}{x} )

and obtain a family of products over the point (0, y_{0}, 0, τ_{0})

(S_{ε}, U) · (S′_{ε}, U′) = (S_{ε} + S′_{ε} + εS_{ε}S′_{ε}, U + U′ + εS_{ε}U′) .

Note that if ε = 1 this product coincides with that on the zero front face, while at the adiabatic front face (ε = 0) it coincides with the product on the scattering front face.

Note that this construction extends to families: If X − M ⟶^{ϕ} B is a fibration with X as above, then carrying out the above blow-ups on M ×_{B} M (the fibered product of M with itself) we obtain a fibration

[figure: X_{εΦ}^{2}; M_{εΦ}^{2}; B]

which carries the kernels of families of adiabatic operators, Ψ_{εΦ}^{∗} (M/B; 𝔼).

Proposition 2.3 (cf. [13], Proposition C.1). If P ∈ Ψ_{εΦ}^{m} (M/B; 𝔼) and the normal operators σ (P), N_{𝔅} (P) and N_{𝔉} (P) are invertible then R_{0} (P) and R_{1} (P) are both fully elliptic and have the same families index.

Proof. The invertibility of σ (P) allows us to construct a parametrix Q_{0} ∈ Ψ_{εΦ}^{−m} (M/B; E_{−}, E_{+}) with residues

Id −Q_{0}P ∈ Ψ_{εΦ}^{−∞} (M/B; E_{+}) , Id −P Q_{0} ∈ Ψ_{εΦ}^{−∞} (M/B; E_{−}) .

Each invertible normal operator allows us to refine to a parametrix whose residues vanish to infinite order at the corresponding boundary face. Thus the hypothesis of the proposition permit us to find Q ∈ Ψ_{εΦ}^{−m} (M/B; 𝔼_{−}) with residues

Id −QP ∈ x^{∞}Ψ_{εΦ}^{−∞} (M/B; E_{+}) , Id −P Q ∈ x^{∞}Ψ_{εΦ}^{−∞} (M/B; E_{−}) .

The induced equations for R_{0} (P) and R_{1} (P) show that these operators are fully elliptic hence Fredholm.

The residues above are families of compact operators over [0, 1] in the intersection of the fibered cusp and fibered boundary calculi hence it is possible to stabilize. That is, we can find an operator

A ∈ x^{∞}Ψ_{εΦ}^{−∞} (M/B; 𝔼) = C^{∞} ([0, 1]_{ε} , Ψ̇^{−∞} (M/B; 𝔼))

such that the null space of P + A is in Ċ^{∞} (M/B; E_{+}) and forms a trivial smooth bundle over [0, 1] × B (see [13, Lemma 1.1]). It follows that the index bundles for R_{0} (P) and R_{1} (P) coincide in K (B). □

Theorem 2.4. The adiabatic construction above induces a natural map

K_{Φ-c}^{∗} (ϕ) → K_{Φ-b}^{∗} (ϕ) .

This map restricts to

K_{Φ-c,−∞}^{∗} (ϕ) → K_{Φ-b,−∞}^{∗} (ϕ) ,

and fits into the following commutative diagram

\xymatrix{ & K_{Φ-c}^{∗} (ϕ) \ar[dl]_{_{ind_{a}}} \ar[d] \ar[dr]^{σ} \\ K^{∗} (B) & K_{Φ-b}^{∗} (ϕ) \ar[l]_{_{ind_{a}}} \ar[r]^{σ} & K_{c} (T^{∗} (X/B))}

Proof. Given a fibered cusp pseudo-differential operator, P ∈ Ψ_{Φ-c}^{0} (M/B; 𝔼) we can construct an element of Ψ_{εϕ}^{0} (M/B; 𝔼) by following the construction in [13, Proposition 8]. First the kernel of P gives us the normal operator of a putative P (ε) at the (lift of) ε = 0. This fixes the boundary value for the face 𝔉, which as we have just seen is fibered over the lifted variable

τ = \frac{x − ε}{x + ε} ∈ [−1, 1] ,

with fiber given by the front face in the fibered cusp calculus. Knowing the value at τ = 1, we extend to the rest of 𝔉 to be independent of τ. This fixes the kernel at the boundary of 𝔅 and we simply extend it as a conormal distribution to the diagonal and vanishing to infinite order at the other boundary faces. Having defined a kernel consistently on the boundary of M_{εϕ}^{2} we can choose an extension which is conormal to the diagonal in the interior, thus obtaining an element P (ε) ∈ Ψ_{εϕ}^{0} (M/B; 𝔼).

If P is Fredholm then the normal operators σ (P (ε)) and N_{𝔉} (P (ε)) are invertible as is N_{𝔅} (P (ε)) for small enough ε, say ε ∈ [0, δ]. Of course we can arrange for N_{𝔅} (P (ε)) to be invertible for all ε for instance by rescaling [0, δ] to [0, 1] or by perturbing P (ε) with a family A (ε) such that N_{𝔅} (A (ε)) is a finite rank smoothing perturbation vanishing at ε = 0 and making the family N_{𝔅} (P (ε)) invertible (as in [13, Remark 1.2]).

The invertibility of these normal operators allows us to apply Proposition 2.3 and conclude that R_{1} (P (E)) is fully elliptic. Thus this construction yields a map (2.1)

\{Ψ_{ϕ-c}^{0} (M/B; 𝔼) Fredholm\} ∋ P \underrightarrow{ãd} R_{1} (P (ε)) ∈ \{Ψ_{Φ-b}^{0} (M/B; 𝔼) Fredholm\},

albeit after making various choices. The choice of δ and extension to ε = 1 do not change the homotopy class of R_{1} (P (ε)). Furthermore, we can always extend a homotopy of P to a homotopy of P (ε) as well as the other equivalence relations defining the K-groups (stabilization and bundle isomorphism), so we have a map

K_{Φ-c}^{∗} (ϕ) ∋ [P] ⟶^{ad} [R_{1} (P (ε))] ∈ K_{Φ-b}^{∗} (ϕ) ,

well-defined independently of choices.

From the construction it is clear that perturbations of the identity by a smoothing operator are preserved as is the interior symbol, and from Proposition 2.3 so is the families index. □

3. Fredholm perturbations of the identity

In this section, we prove that the map

K_{Φ-c,−∞}^{∗} (ϕ) → K_{Φ-b,−∞}^{∗} (ϕ) ,

induced by the adiabatic construction in Theorem 2.4 is an isomorphism.

We first recall the excision argument from [13] and reduce to the case Φ = Id, i.e., the scattering and zero calculi. We start by replacing M with a collar neighborhood of its boundary, ∂M ×[0, 1]_{x}. The K-theory groups we are considering are made up of equivalence classes of normal operators, so there is no loss in restricting attention to the quantizations that are supported in this neighborhood, and identically the identity at x = 1.

The fibration extends off the boundary into the entire neighborhood ∂M ×[0, 1]_{x}, so we have the simpler situation

[figure: Z × [0, 1]; ∂X × [0, 1]; ∂M × [0, 1]; Y × [0, 1]; D × [0, 1]; ϕ̃; B]

where furthermore we have the following reduction.

Lemma 3.1. There are ‘excision’ isomorphisms

K_{Φ-c,−∞}^{∗} (ϕ) ≅ K_{sc,−∞}^{∗}(ϕ̃) and K_{Φ-b,−∞}^{∗} (ϕ) ≅ K_{0,−∞}^{∗}(ϕ̃)

Proof. Recall [14, Lemma 6.3], [13, proof of Prop. 3.1] that given

b ∈ Ψ_{Φ−sus}^{−∞} (∂M/D; E)

such that Id +b is invertible, we can assume that b acts on a finite-rank sub-bundle W of C^{∞} (∂M/D) pulled-back to T^{∗} (D/B)×ℝ and then think of b as the boundary family of a family in Ψ_{Φ-c}^{−∞} (M/B; E) or in Ψ_{sc}^{−∞} (D × [0, 1] /B; W). Doing the same to fibered boundary operators we obtain the isomorphisms above. □

Hence, it suffices to show that the adiabatic homomorphism between K_{sc,−∞}^{∗}(ϕ̃) and K_{0,−∞}^{∗}(ϕ̃) is an isomorphism. Both the kernel of an operator in Ψ_{sc}^{−∞} (M; E) and the kernel of an operator in Ψ_{0}^{−∞} (M; E) are, near their respective front faces, translation invariant in the ‘horizontal’ directions. We can apply the Fourier transform in these directions and restrict to the front face, and obtain in either case an element of

(3.1)  S (T^{∗}∂M) ˆ⊗_{π}Ċ^{∞} ([−1, 1]) ˆ⊗_{π} Hom (E) .

In the scattering case this is easily recognized as defining a class in K_{c}^{−1} (T^{∗}∂M × ℝ) ≅ K_{c}^{−2} (T^{∗}∂M) and respecting the product structure. We show in the following theorem that the zero case defines the same class in K^{0} (T^{∗}∂M). The coincidence of the classes defined by the commutative (scattering) product with that defined by the non-commutative (zero) product is a manifestation of Bott periodicity (cf. [3, Proposition 9.9]).

Theorem 3.2. The map K_{sc,−∞}^{∗}(ϕ̃) → K_{0,−∞}^{∗}(ϕ̃) induced by the adiabatic calculus is an isomorphism and fits into the commutative diagram

\xymatrix{K_{sc,−∞}^{∗} (ϕ) \ar[r]^{≅} \ar[d] & K_{c}^{∗} (T^{∗}∂M/B) \\ K_{0,−∞}^{∗} (ϕ) \ar[ur]^{≅}}

Before proving this theorem, we quickly review the relevant descriptions of the topological K-theory groups from [12]. Recall that for any manifold X we can describe K^{1} (X) as stable homotopy classes of maps

K^{1} (X) = lim_{→} [S; GL (N)] ,

so also [12, Proposition 3],

(3.2)  K^{1} (X) = [X; Ġ^{−∞} (V; E)] ,

where V is any compact manifold with boundary and

Ġ^{−∞}(V; E) = {Id +A : A ∈ Ψ̇^{−∞}(X; E) and (Id +A) is invertible}.

Similarly, we can define K^{0}(X) from K^{1}(X) by suspension, i.e.,

K^{0} (X) = lim_{→} [X; C^{∞} ((𝕊^{1}, 1); (GL (N) , Id))]

and this can also be though of as [12, Proposition 4]

(3.3)  K^{0} (X) = [X; G_{sus}^{−∞} (U; E)] ,

with U any closed manifold of positive dimension and, using S to denote Schwartz functions,

G_{sus}^{−∞} (U; E) = {Id +A : A ∈ S(ℝ; Ψ^{−∞}(U; E)) and (Id +A) is invertible}.

We define the K-theory of a non-compact manifold as the reduced K-theory of its one-point compactification,

K_{c}^{∗} (X) = K̃^{∗} (\overline{X}/\{∞\}) .

Proof (of Theorem 3.2). Recall that in [1] it is shown that the groups K_{0,−∞}^{q}(ϕ) are canonically isomorphic to the groups K_{c}^{q}(T^{∗}∂M/B). So, for each ε_{0} > 0, we have an isomorphism into K_{c}^{0} (T^{∗}∂M/B) consistent with the product on the ε = ε_{0} slice of the adiabatic calculus. The smoothness of the product in the adiabatic calculus shows that, for any Fredholm operator Id +A ∈ Ψ_{εϕ}^{−∞}, we obtain for each ε_{0} > 0 the same class in K_{c}^{0} (T^{∗}∂M).

Thus we have two homomorphisms from K_{sc,−∞}^{∗}(ϕ) into K_{c}^{0} (T^{∗}∂M). The first is the isomorphism given by identifying (3.1) with an element of K_{c}^{−1} (T^{∗}∂M × ℝ) ≅ K_{c}^{−2} (T^{∗}∂M). The second from extending the given Fredholm scattering operator into a Fredholm adiabatic operator and applying the isomorphism of [1]. The first can be represented by a map

\begin{aligned}[T^{∗}∂M × ℝ; GL_{N} (ℂ)] & = [S^{∗}∂M × ℝ_{+} × ℝ; GL_{N} (ℂ)]_{0} \\ & = [S^{∗}∂M; S (ℝ_{+} × ℝ) ⊗ GL_{N} (ℂ)]_{0} \end{aligned} ; (3.4)

with the zero subscript denoting that the restriction to zero in the ℝ_{+} factor depends only on the base variable in S^{∗}∂M. The second homomorphism comes from quantizing the product in (3.4) into the b, c-calculus with a parameter ε:

[S^{∗}∂M; S (ℝ_{+} × ℝ) ⊗ GL_{N} (ℂ)]_{0} ⟶^{q_{ε}} [S^{∗}∂M; Ψ_{b,c}^{−∞,−∞} ([0, 1] ; ℂ^{N})]_{0} .

When ε = 0 we have the commutative product and for positive ε, the noncommutative (cf. [3, Proposition 9.9, Theorem 9.5]). Hence the isomorphism between K_{0,−∞}^{0}(ϕ) and K_{c}^{0} (T^{∗}∂M) is taken into the isomorphism between K_{sc,−∞}^{0}(ϕ) and K_{c}^{0} (T^{∗}∂M) by the adiabatic limit. □

4. Six term exact sequence

The Atiyah-Singer Index Theorem induces natural maps

\xymatrix{K_{c}^{q}(T^{∗}M/B) \ar[r]^{I_{q}} & K_{c}^{q+1}(T^{∗}D/B) \\ [σ] \ar@{|->}[r] & [ind_{AS} r_{∂M}^{∗}σ]} ,

and, together with the identifications K_{c}^{q+1}(T^{∗}D/B) ≅ K_{Φ-b,−∞}^{q+1}(ϕ), these are maps K_{c}^{q}(T^{∗}M/B) → K_{Φ-b,−∞}^{q+1}(ϕ). Just as in [13, Thm. 6.3], they fit into a diagram

(4.1)  \xymatrix{K_{Φ-b,−∞}^{0} (ϕ) \ar[r]^{i_{0}} & K_{Φ-b}^{0} (ϕ) \ar[r]^{σ_{0}} & K_{c}^{0} (T^{∗}M/B) \ar[d]^{I_{0}} \\ K_{c}^{1} (T^{∗}M/B) \ar[u]^{I_{1}} & K_{Φ-b}^{1} (ϕ) \ar[l]^{σ_{1}} & K_{Φ-b,−∞}^{1} (ϕ) \ar[l]^{i_{1}}}

with the maps i_{∗} and σ_{∗} induced by inclusion and the principal interior symbol, which we now show is exact.

Proposition 4.1. The diagram (4.1) is exact.

Proof. While exactness at K_{Φ-b}^{∗}(ϕ) follows directly from the definitions, to establish exactness elsewhere we give a different description of the maps I_{q}.

Given a class [σ_{t}] ∈ K_{c}^{1}(T^{∗}M/B), it is shown in [13, Proposition 6.2] that there is a family of operators

P_{t} ∈ Ψ_{Φ-c}^{0}(M/B; ℂ^{N}) ∩ Fredholm, s.t.
σ(P_{t}) = σ_{t}, P_{0} = Id, P_{1} ∈ (Id +Ψ_{Φ-c}^{−∞}(M/B; ℂ^{N})) ∩ Fredholm

and the identification K_{Φ-c,−∞}^{0}(ϕ) ≅ K_{c}^{0}(T^{∗}D/B) takes [P_{1}] to [ind_{AS} r_{∂M}^{∗}σ]. Applying the map ãd from (2.1) we obtain a family

P̃_{t} ∈ Ψ_{Φ-b}^{0}(M/B; ℂ^{N}) ∩ Fredholm, s.t.
σ(P̃_{t}) = σ_{t}, P̃_{0} = Id, P̃_{1} ∈ (Id +Ψ_{Φ-b}^{−∞}(M/B; ℂ^{N})) ∩ Fredholm

and, from Theorem 3.2, the identification K_{Φ-b,−∞}(ϕ) ≅ K_{c}(T^{∗}D/B) takes [P̃_{1}] to [ind_{AS} r_{∂M}^{∗}σ]. Thus we can think of I_{1} as being [σ_{t}] ↦ [P̃_{1}].

An element [σ_{t}] is in the null space of I_{1} if P̃_{1} ∼ Id, i.e., precisely when [σ_{t}] is in the image of the map K_{Φ-b}^{1}(ϕ) ⟶^{σ_{1}} K_{c}^{1}(T^{∗}M/B). On the other hand the homotopy P̃_{t} shows that [P̃_{1}] is automatically trivial as an element K_{Φ-b}^{0}(ϕ), that is Im I_{1} ⊆ null i_{0}, and conversely, if [Q] ∈ null i_{0}, there is (after stabilization) a homotopy of operators from the identity to Q, the symbol of this homotopy defines an element in K_{c}^{1}(T^{∗}M/B) that is sent by I_{1} to [Q]. Thus we see that the diagram is exact at K_{c}^{1}(T^{∗}M/B) and at K_{Φ-b}^{1}(ϕ). Just as in [13, Theorem 6.3] the same arguments prove exactness at K_{c}^{1}(T^{∗}M/B) and K_{Φ-b}^{1}(ϕ) using Bott periodicity. □

As explained in the proof of this proposition, the diagrams

\xymatrix{K_{c}^{q}(T^{∗}M/B) \ar[r]^{I_{q}} \ar[dr]^{I_{q}} & K_{Φ-c}^{q+1}(ϕ) \ar[d]^{ad} \\  & K_{Φ-b}^{q+1}(ϕ)}

are commutative. Hence the six term exact sequences for the smooth K-theory of Φ-c and Φ-b operators make up a commutative diagram:

[figure: K_{Φ-c,−∞}^{0} (ϕ); ad; K_{Φ-c}^{0} (ϕ); ad; K_{c}^{0} (T^{∗}M/B); =; K_{Φ-b,−∞}^{0} (ϕ); K_{Φ-b}^{0} (ϕ); K_{c}^{0} (T^{∗}M/B); K_{c}^{1} (T^{∗}M/B); K_{Φ-b}^{1} (ϕ); K_{Φ-b,−∞}^{1} (ϕ); K_{c}^{1} (T^{∗}M); =; K_{Φ-c}^{1} (ϕ); ad; K_{Φ-c,−∞}^{1} (ϕ); ad]

and, since we have already shown that K_{Φ-c}^{q}(ϕ) ⟶^{ad} K_{Φ-b}^{q}(ϕ) is an isomorphism, the five-lemma implies the following.

Theorem 4.2. The adiabatic calculus induces an isomorphism of the smooth Ktheory groups of the Φ-c calculus and those of the Φ-b calculus,

K_{Φ-c}^{q}(ϕ) ⟶_{≅}^{ad} K_{Φ-b}^{q}(ϕ).

Appendix A. Triple adiabatic space

In this section we prove the composition formula for the adiabatic calculus. Roughly, the kernel of the composition is given by

K_{A∘B} (ζ, ζ′, ε) = ∫_{M_{εϕ}^{2}} K_{A} (ζ, ζ′′, ε) K_{B} (ζ′′, ζ′, ε) .

Our method is geometric, we construct a space M_{εϕ}^{3} with three b-fibrations ‘first’, ‘second’, and ‘composite’ down to M_{εϕ}^{2}. The kernel of the composite is then correctly given by

(π_{C})_{∗} (π_{F}^{∗} A · π_{S}^{∗}B) .

The basic triple space is M^{3} × [0, ϵ_{0}] where ϵ_{0} > 0 and M is a compact manifold with boundary and a specified fibration ϕ : ∂M ⟶ Y of the boundary. We first have to perform the ‘boundary adiabatic’ blow-ups to be able to map back to the single and double spaces

(A.1)  \begin{matrix}M_{(1)}^{3} = [M^{3} × [0, ϵ_{0}], A_{T} , A_{F} , A_{C}, A_{S}, A_{R}, A_{M}, A_{L}], \\ A_{T} = (∂M)^{3} × \{0\}, A_{F} = M × ∂M × ∂M × \{0\}, \\ A_{S} = ∂M × ∂M × M × \{0\}, A_{C} = ∂M × M × ∂M × \{0\}, \\ A_{R} = M × M × ∂M × \{0\}, A_{M} = M × ∂M × M × \{0\}, \\ A_{L} = ∂M × M × M × \{0\}. \end{matrix}

Then consider the lifts of the triple and double fibre diagonals, lying within the lifts of the faces (∂M)^{3} × [0, ϵ_{0}] and (∂M)^{2} × M × [0, ϵ_{0}] and its cyclic images. We can denote these as 𝔅_{T} , the triple boundary fibred diagonal and 𝔅_{S}, 𝔅_{C} and 𝔅_{F} , the double fibred diagonals. These are all p-submanifolds, meeting in the standard way for diagonals so we may define

(A.2)  M_{(2)}^{3} = [M_{(1)}^{3}, 𝔅_{T} , 𝔅_{S}, 𝔅_{C}, 𝔅_{F}].

The double and triple fibred-cusp diagonal faces are not p-submanifolds in M_{(1)} but lift to be p-submanifolds, which we denote 𝔉_{T} , 𝔉′_{F} , 𝔉′_{C}, 𝔉′_{S} and 𝔉′′_{F} , 𝔉′′_{C} and 𝔉′′_{F} . Here, 𝔉′_{O} is the lift of the corresponding double diagonal in the face produced by the blow-up of A_{O}, O = S, C, F and 𝔉′′_{O} is the lift of the intersection of this submanifold under the blow-up of A_{T} . The intersection properties of these submanifolds is essentially the same as for the fibred-cusp (or indeed the cusp) setting itself. Thus we complete the definition by blowing up in ‘the usual’ order

(A.3)  M_{ϵϕ}^{3} = [M_{(2)}^{3}, 𝔉_{T} , 𝔉′′_{S}, 𝔉′′_{C}, 𝔉′′_{F} , 𝔉′_{S}, 𝔉′_{C}, 𝔉′_{F}].

Proposition A.1. Each of the projections, dropping one or two factors of M, lift to b-fibrations

(A.4)  \begin{matrix}\xymatrix{(\begin{matrix}π_{F} \\ π_{C} \\ π_{S} \end{matrix} ) : M_{ϵϕ}^{3} \ar[r] \ar[r] \ar[r] & M_{ϵϕ}^{2}} , \\ (\begin{matrix}π_{L} \\ π_{M} \\ π_{R} \end{matrix} ) : M_{ϵϕ}^{3} & [figure] & [figure] & [figure] & M_{ϵ}. \end{matrix}

Proof. This is the usual commutativity of blow-ups argument. Rearranging the ‘fibred-cusp’ faces works just as for the (fibred-) cusp calculus. After these triple and two double faces have been blown down, the corresponding fibred-boundary faces can be blown down, as can the pure adiabitic faces. □

Proposition A.2.

Ψ_{εϕ}^{k} (M; G, H) ∘ Ψ_{εϕ}^{k′} (M; E, G) ⊂ Ψ_{εϕ}^{k+k′} (M; E, H)

Proof. This follows from Proposition A.1 via the push-forward and pull-back theorems of [9]. Indeed, given two operators (acting on half-densities for simplicity) A and B the kernel of their composition is given by

A ∘ B = (π_{C})_{∗} (π_{S}^{∗}A · π_{F}^{∗} B) .

As the maps are b-fibrations and transversal to the diagonals this yields a distribution conormal to the diagonal of the appropriate degree. Since the kernels of A and B vanish to infinite order at every face not meeting the diagonal, the product of their lifts will vanish to infinite order at every face not meeting the triple diagonal (as this is the intersection of the lift of the two diagonals). Hence only the faces coming from diag (M^{3}) × {0}, and the blow-ups of 𝔅_{T} and 𝔉_{T} potentially contribute to the push-forward. It follows that the resulting distribution is smooth down to the each of the boundary faces. □

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Department of Mathematics, Massachusetts Institute of Technology

E-mail address: pierre@math.mit.edu
E-mail address: rbm@math.mit.edu