L^{2}-index theorem for elliptic differential boundary problems

Thomas Schick^{∗} FB Mathematik Bunsenstr. 3 37073 Göttingen, Germany email: schick@uni-math.gwdg.de http://www.uni-math.gwdg.de/schick

Last edited: March 18, 2003 or later — last compiled: March 18, 2003

Abstract

Suppose M is a compact manifold with boundary ∂M. Let M̃ be a normal covering of M. Suppose (A, T) is an elliptic differential boundary value problem on M with lift (Ã, T̃) to M̃. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: These numbers are finite, and their difference, by definition the von Neumann index of (Ã, T̃), equals the index of (A, T). In this way, we extend the classical L^{2}-index theorem of Atiyah to elliptic differential boundary value problems.

MSC-classification: 58G12; 58G20 Keywords: index theory, L^{2}-index, boundary value problems, Boutet de Monvel calculus, covering manifolds,

1 Introduction

In this paper, we study elliptic differential boundary value problems on coverings of compact manifolds. Let M be a compact Riemannian manifold with boundary ∂M. Suppose E, F ↓M and Y ↓∂M are Riemannian vector bundles. Let A : C^{∞}(E) → C^{∞}(F) be a differential operator and T : C^{∞}(E) → C^{∞}(Y) a differential boundary operator so that the pair P := (A, T) is elliptic. The following definition will literally also be applied to non-compact spaces.

∗Partially funded by the DAAD

ker P := {f ∈ L^{2}(E); f ∈ C^{∞}, Af = 0 = T f}  and
coker P := {(F, f) ∈ L^{2}(F) ⊕ L^{2}(Y);
(F, Aφ)_{L^{2}(F)} + (f, T φ)_{L^{2}(Y)} = 0 ∀φ ∈ C_{0}^{∞}(E)}.

The classical theory of elliptic boundary problems states that the dimensions of kernel and cokernel are finite and studies ind(P) := dim ker P − dim coker P. The index theorem (recalled below) provides deep connections between topological, geometrical and analytical properties of the manifold.

Suppose M̃ ↓ M is a normal covering of M with deck transformation group Γ. Pull the bundles back to M̃ and lift the operators and metrics. We use the convention that corresponding objects on M̃ have the same notation decorated with an additional tilde. Note that Γ operates on the bundles, their sections and that P̃ = (Ã, T̃) is Γ-equivariant. Define the kernel and cokernel of P̃ literally in the same way as for P. They are in general infinite dimensional. But ker(P̃) and coker(P̃) have an additional structure: They are Hilbert modules over the group von Neumann algebra N (Γ). For these Hilbert modules, a normalized dimension dim_{Γ} with values in [0, ∞] is defined. It vanishes exactly if the module is trivial, it is additive under direct sums, and

|Γ| < ∞ ⟹ dim_{Γ} = \frac{1}{|Γ|} dim_{ℂ} .  (1.1)

The following is the main result of this paper:

1.2 Theorem. In the situation described above we have dim_{Γ} ker(P̃) < ∞, dim_{Γ} coker(P̃) < ∞ and

ind_{Γ}(P̃) := dim_{Γ} ker(P̃) − dim_{Γ} coker(P̃) = ind(P).

Remarkably, ind_{Γ}(P̃), the difference of two reals, is an integer.

The theorem is particularly interesting because for ind(P) on M a purely topological expression exists, compare Atiyah/Bott [2, Theorem 2]: Every elliptic boundary problem (A, p) defines a K-theoretic symbol class [σ(A, p)]. One assigns to this symbol the topological index, which equals the analytical index. Cohomologically,

L^{2}-index theorem for boundary problems 3

ind_{t}(A, p) = ∫_{S(M)} ch(σ(A))π^{∗}T (M) + ∫_{B(M)|_{∂M}} ch(σ(A, p))π^{∗}T (M), where π : T M → M is the projection, T (M) is a Todd class of M, ch the Chern character, and B(M) and S(M) are the disc and sphere bundle of T M.

1.3 Corollary (of Theorem 1.2). The index of elliptic differential boundary problems is multiplicative under finite coverings.

Proof. This follows from the multiplicativity (1.1) of dim_{Γ}.

In Theorem 1.2 we can replace coker(P̃) with the kernel of an adjoint boundary problem by Theorem 6.1. Sometimes it is easier to deal with kernels. As an application we compute the Euler characteristic of M in terms of L^{2}-harmonic forms on M̃ in Theorem 6.4. Dodziuk [5] and Donnelly/Xavier [6] have computed the sign of the Euler characteristic of closed negatively curved manifolds in this way. An extension to manifolds with boundary is given in [11, Section 6].

Our index theorem is the generalization of Atiyah’s L^{2}-index theorem [1] to manifolds with boundary. The proof is along the lines of Atiyah’s proof. In order to deal with boundary problems, we replace the calculus of pseudo-differential operators by the Boutet de Monvel calculus. As another foundation, in Section 2 we study traces for endomorphisms of Hilbert N (Γ)- modules. We use the theory of Sobolev spaces to simplify the work with regularizing operators and especially with their traces. An important result, which should be valuable also in other contexts, is:

1.4 Theorem. (compare Theorem 3.4) If r > dim M/2, then the inclusion of Sobolev spaces H^{s+r}(M̃) ↪ H^{s}(M̃) is a Γ-trace class operator.

The idea for the proof of the index theorem is: To P construct an inverse Q (modulo smoothing operators) in the BdM calculus which can be lifted to M̃, i.e. PQ = 1 − S_{1}, QP = 1 − S_{0} and P̃Q̃ = 1 − S̃_{1}, Q̃P̃ = 1 − S̃_{0}. Then the following two results prove the theorem:

• ind_{Γ}(P̃) = Sp_{Γ} S̃_{0}−Sp_{Γ} S̃_{1} (and the corresponding formula on the base with Γ = {1}).

• For lifts of smoothing operators, we have Sp_{Γ} S̃ = Sp S.

Note that our index theorem does not generalize the Atiyah-Patodi-Singer index theorem [3]. They deal with a specific non-local boundary condition. There is also an L^{2}-version of this type of index theorem, proved by Ramachandran [9]. He deals with Dirac type operators and the APSboundary conditions. Contrariwise, our result is valid for arbitrary elliptic differential boundary problems, but we only deal with local boundary conditions. In particular, we can not handle the signature.

This work is part of the Dissertation [11] of the author. I thank my advisor Prof. Wolfgang Lück for his constant support.

Throughout the paper, we use the following notation:

1.5 Definition. For c > 0 we define

a ≤^{c} b ⟺ a ≤ c · b

and similarly a <^{c} b, …. In a longer chain of inequalities, the same symbol (e.g., c) may be used for different constants.

If not stated otherwise, H is a Hilbert space, B(H_{1}, H_{2}) denotes the bounded operators from H_{1} to H_{2}, B(H) = B(H, H). M is a compact smooth manifold of dimension m with boundary ∂M and E, F↓M, Y ↓∂M are vector bundles.

2 Traces for N (Γ)-module morphisms

The Hilbert space l^{2}(Γ) = {∑_{g∈Γ} λ_{g} · g| ∑_{g} |λ_{g}|^{2} < ∞} obviously admits commuting unitary left and right Γ-actions. N (Γ) = B(l^{2}Γ)^{Γ} consists of all those operators which commute with the right action. A Hilbert N (Γ)- module is a Hilbert space V with left Γ-action so that an isometric embedding V ↪ l^{2}(Γ) ⊗ H exists which is compatible with the Γ-actions. Here H is an arbitrary Hilbert space with trivial Γ-action (in this paper, tensor products are always completed to Hilbert spaces). If V, W are two Hilbert N (Γ)- modules, a bounded linear map f : V → W which is compatible with the Γ-action is called an N (Γ)-module morphism.

On the N (Γ)-endomorphisms of l^{2}(Γ), we have the canonical finite trace tr_{Γ}(a) = (a(e), e)_{l^{2}(Γ)}. Moreover, on every Hilbert space H the trace Sp(A) = ∑_{i}(Ah_{i}, h_{i}) exists ((h_{i}) an orthonormal basis of H).

2.1 Definition. This yields a Γ-trace, called Sp_{Γ}, on the Γ-operators on l^{2}(Γ) ⊗ H which is defined by

Sp_{Γ}(a ⊗ A) = tr_{Γ}(a) · Sp(A).

This makes sense only for positive operators and for operators in the Γ-trace class ideal (defined as usual, see [4, chapter I]). We also have the Γ-Hilbert Schmidt (HS) operators defined by

f ∈ B(l^{2}Γ ⊗ H)^{Γ} is Γ-HS ⟺ Sp_{Γ}(f^{∗}f) < ∞.

Given an orthonormal base {u_{i}} of H, we define isometric embeddings U_{i} : l^{2}(Γ) → l^{2}(Γ) ⊗ H : x ↦ x ⊗ u_{i}. An explicit formula for the Γ-trace of a positive or Γ-trace class Γ-operator f on l^{2}(Γ) ⊗ H is then given by

Sp_{Γ}(f) = ∑_{i} tr_{Γ}(U_{i}^{∗}fU_{i}).

2.2 Definition. Let V_{k} be Hilbert N (Γ)-modules with isometric Γ-embed-dings i_{k} : V_{k} ↪ l^{2}(Γ) ⊗ H_{k}. Set p_{k} := i_{k}^{∗} (k = 1, 2). Let f : V_{1} → V_{2} be a Hilbert N (Γ)-module morphism.

We call f : V_{1} → V_{2} Γ-Hilbert Schmidt (Γ-HS) if Sp_{Γ}(i_{1}f^{∗}fp_{1}) < ∞, and we denote it Γ-trace class (Γ-tr) if Γ-HS morphisms f_{1} : V_{1} → V_{3} and f_{2} : V_{3} → V_{2} exist so that f = f_{2}f_{1} .

If V_{1} = V_{2} and f is Γ-tr we set

Sp_{Γ}(f) := Sp_{Γ}(i_{1}fp_{1}).

The following basic properties show in particular that this is well defined.

2.3 Theorem. Let f : V_{1} → V_{2}, g : V_{2} → V_{3}, e : V_{0} → V_{1} be Hilbert N (Γ)-module morphisms. Then:

(1) f Γ-tr ⟺ f^{∗} Γ-tr ⟺ |f| Γ-tr; f Γ-HS ⟺ f^{∗} Γ-HS.

(2) f Γ-HS ⟹ gf, fe Γ-HS.

(3) f Γ-tr ⟹ gf, fe Γ-tr.

(4) f Γ-tr and V_{1} = V_{3} ⟹ g ↦ Sp_{Γ}(gf) is ultra-weakly continuous.

(5) V_{1} = V_{3} and either f Γ-tr or f, g Γ-HS ⟹ Sp_{Γ}(gf) = Sp_{Γ}(fg).

(6) If V_{1,2} = l^{2}(Γ) ⊗ H, a is Γ-HS and B ∈ B(H) is HS, then f = a ⊗ B is Γ-HS. If a is Γ-tr and B is trace class, then f is Γ-tr with Sp_{Γ}(f) = tr_{Γ}(a) Sp(B).

Proof. These are rather straightforward consequences of the proofs of the corresponding well known properties of tr_{Γ} and Sp. (For a detailed proof compare [11, 9.13].) Note in particular that the statements are standard if V_{1} = V_{2} = V_{3} = l^{2}(Γ) ⊗ H. In view of Definition 2.2 and the polar decomposition, the general case is based on the following fact:

If u : l^{2}(Γ) ⊗ H_{2} → l^{2}(Γ) ⊗ H_{1} is a partial isometry and an N (Γ)- morphism, and if f : l^{2}(Γ) ⊗ H_{1} → l^{2}(Γ) ⊗ H_{2} is Γ-tr, then

Sp_{Γ}(uf) = Sp_{Γ}(fu).  (2.4)

First consider the case where u is injective (this implies u^{∗}u = 1). Then Sp_{Γ}(uf) = Sp_{Γ}(uu^{∗}uf) = Sp_{Γ}(u(fu)u^{∗}) by the trace property on l^{2}(Γ) ⊗ H_{1}. Since arbitrary trace class operators are linear combinations of positive operators, assume that g = fu is positive. Then

Sp_{Γ}(ugu^{∗}) = Sp_{Γ}(u\sqrt{g}(u\sqrt{g})^{∗}) =^{(∗)} Sp_{Γ}((u\sqrt{g})^{∗}u\sqrt{g})
= Sp_{Γ}(\sqrt{g}\sqrt{g}) = Sp_{Γ}(g).

It remains to establish for f as above

(∗) Sp_{Γ}(f^{∗}f) = Sp_{Γ}(ff^{∗}).

For this, choose an orthonormal basis {h_{i}}_{i∈I} of H_{1}. This gives rise to isometric embeddings U_{i} : l^{2}(Γ) → l^{2}(Γ)⊗H_{1} : x ↦ x⊗h_{i}. Similarly, choose an orthonormal basis {v_{j}} of H_{2} and construct V_{j} : l^{2}(Γ) → l^{2}(Γ)⊗H_{2}. Then

Sp_{Γ}(f^{∗}f) = ∑_{i∈I} tr_{Γ}(U_{i}^{∗}f^{∗}fU_{i}) = ∑_{i∈I} ∑_{j∈J} tr_{Γ}(U_{i}^{∗}f^{∗}V_{j}V_{j}^{∗}fU_{i})

= ∑_{i,j} tr_{Γ}(V_{j}^{∗}fU_{i}U_{i}^{∗}f^{∗}V_{j}) = ∑_{j} tr_{Γ}(V_{j}^{∗}ff^{∗}V_{j}) = Sp_{Γ}(ff^{∗}).

The fact that {h_{i}} is an orthonormal basis implies ∑_{i} U_{i}U_{i}^{∗} = 1 weakly. Moreover, we used the fact that tr_{Γ} is a trace and is normal. All summands are non-negative. Therefore, neither the order of summation nor convergence (allowing +∞ as possible value) are an issue.

Back to the the proof of (2.4). Suppose now u is surjective. Then u^{∗} is injective and

Sp_{Γ}(uf) =^{trace on l^{2}(Γ) ⊗ H_{1}} \overline{Sp_{Γ}(f^{∗}u^{∗})} = \overline{Sp_{Γ}(u^{∗}f^{∗})} = Sp_{Γ}(fu),

where \overline{z} is the complex conjugate of z.

If u is arbitrary, decompose u as follows:

u = p_{2} ∘ (1 ⊕ u) : X ⊗ H_{2} → (X ⊗ H_{2}) ⊕ (X ⊗ H_{2}) → X ⊗ H_{2}.

Sp_{Γ}(uf) = Sp_{Γ}(p_{2}(1 ⊕ u)f) =^{p_{2} surjective} Sp_{Γ}((1 ⊕ u)fp_{2})
=^{(1 ⊕ u) injective} Sp_{Γ}(fp_{2}(1 ⊕ u)) = Sp_{Γ}(fu).

To complete the proof one has to do (quite a lot of) computations of similar spirit and apply (2.4) and the trace properties for operators on l^{2}(Γ)⊗ H. This does not seem to be very enlightening and is left as an exercise.

As usual, armed with a Γ-trace we define the Γ-dimension:

2.5 Definition. Let V be a Hilbert N (Γ)-module. Then

dim_{Γ}(V) := Sp_{Γ}(id_{V}) ∈ [0, ∞].

We now come to an important result, which is essentially proved in Atiyah’s paper [1, p. 67]. He does not state it explicitly and in full generality, but his proof works nearly literally. (This proof can also be found in [11, 9.16].)

2.6 Proposition. Suppose V, W are Hilbert N (Γ)-modules. Let T_{0} : V → V and T_{1} : W → W be bounded Γ-morphisms which are Γ-tr. Let D : V → W be a closed operator with domain D(D) which commutes with the action of Γ. Especially, we require that D(D) is Γ-invariant and dense. Suppose

T_{1}D ⊂ DT_{0}; ker D ⊂ ker T_{0}; ker D^{∗} ⊂ ker T_{1}^{∗}.
Then Sp_{Γ}(T_{0}) = Sp_{Γ}(T_{1}).

3 L^{2}-Rellich lemma

Let M be a compact m-dimensional manifold with boundary ∂M (possibly empty). Let M̃ be a normal covering of M with covering group Γ (acting by isometries). Let E↓M be a vector bundle with pullback Ẽ↓M̃.

There is a natural way to define Sobolev spaces on M̃:

3.1 Definition. Choose a finite covering of M by charts κ_{i} with subordinate partition of unity φ_{i} so that E is trivial over the domain of κ_{i} with trivialization t_{i}. Lift charts, partition of unity and trivializations to M̃. Then we define the Sobolev norm |·|_{H^{s}} by

|σ|_{H^{s}}^{2} := ∑_{γ∈Γ} ∑_{i} |t̃_{i} ∘ (φ̃_{i} · γ^{∗}σ) ∘ κ̃_{i}^{−1}|_{H^{s}(ℝ^{m})}^{2} σ ∈ C_{0}^{∞}(Ẽ).

The Sobolev space H^{s}(Ẽ) is defined as the completion of C_{0}^{∞}(Ẽ) with respect to this norm. The inner product does depend on the choices, but not the topology.

We will show in this section that H^{s}(Ẽ) is a Hilbert N (Γ)-module and that the inclusion H^{s+r}(Ẽ) ↪ H^{s}(Ẽ) is Γ-HS for r > m/2.

Let W be the double of M with reflection fl : W → W. Let X↓W be the double of E. The reflection fl extends as a bundle map to X. Construct similarly W̃ and X̃. Then W̃ is a normal covering of W with covering group Γ. Again we denote the reflection fl.

3.2 Lemma. Fix s ∈ ℝ. There exists a bounded Γ-equivariant extension map e : H^{s}(M̃) → H^{s}(W̃), i.e. e(f)|M̃ = f ∀f ∈ H^{s}(M̃). The restricition map is also Γ-equivariant and bounded. The corresponding statement holds for Ẽ.

Proof. A straightforward exercise. One uses a Γ-invariant covering of M̃ by charts and the corresponding extension map on Euclidian space (Taylor [14, I.5.1]).

Suppose U ⊂ M̃ ⊂ W̃ is a fundamental domain for the covering p : M̃ → M. This means that U is open, p|_{U} is injective and M − p(U) is a set of measure zero. Choose U so that its closure is compact, and choose a compact submanifold with boundary T ⊂ W̃ of codimension zero, so that U ∪ fl(U) ⊂ T and so that the interior of T is mapped surjectively onto W.

3.3 Lemma. Suppose s ∈ ℝ. The map p defined by the composition

H^{s}(M̃) ⟶^{e} H^{s}(W̃) ⟶^{p̄} l^{2}(Γ) ⊗ H^{s}(T)
∈  ∈  ∈
f ⟶ ef ⟶ ∑_{g∈Γ} g ⊗ \underbrace{g^{−1}(ef)}_{=g^{∗}(ef)} |_{T}

is Γ-equivariant, and there exist C_{1,2} > 0 so that

|f|_{H^{s}(M̃)} ≤^{C_{1}} |pf|_{l^{2}(Γ)⊗H^{s}(T)} ≤^{C_{2}} |f|_{H^{s}(M̃)} .

In particular, H^{s}(M̃) (with the pull back norm under p) is a Hilbert N (Γ)- module. The corresponding statement holds for Ẽ.

Proof. By Lemma 3.2, e has the required properties. It remains to consider p̄. Obviously, p̄ is Γ-equivariant.

Because Γ is discrete and T is compact, it meets only finitely many, say N, of its translates {gT }_{g∈Γ}.

By definition, |∑ g ⊗ f_{g}|_{l^{2}(Γ)⊗H^{s}(T)}^{2} = ∑ |f_{g}|_{H^{s}(T)}^{2}. To show that p̄ is bounded let {U_{i}}_{i=1,...N} be open subsets of W̃ which cover T so that the covering projection maps each U_{i} injectively to W. Choose submanifold charts κ_{i} for (U_{i}, U_{i} ∩ T) and functions 0 ≤ φ_{i} ≤ 1 with compact support in U_{i} so that ∑_{i} φ_{i} = 1 on T. Recognize that for every single i we can extend (U_{i}, φ_{i}, κ_{i}) to a corresponding collection (U_{α,γ}^{i}, φ_{α,γ}^{i}, κ_{α,γ}^{i})_{α,γ} which can be used to compute Sobolev norms on W̃. The norm will depend on the data (hence on i), but all such norms are equivalent. Therefore for f ∈ H^{s}(W̃)

|p̄f|_{l^{2}(Γ)⊗H^{s}(T)}^{2} = ∑_{i=0}^{N} ∑_{γ∈Γ} |φ_{i}γ^{∗}f ∘ κ_{i}^{−1}|_{H^{s}(ℝ_{≥0}^{m})}^{2}

≤ ∑_{i} ∑_{γ} ∑_{α=1}^{N_{i}} |(φ_{α,γ}^{i}f) ∘ (κ_{α,γ}^{i})^{−1}|_{H^{s}(ℝ^{m})}^{2}

(since we have more and larger summands)

≤^{NC} |f|_{H^{s}(W̃)}^{2} .

On the other hand (fix i)

|f|_{H^{s}(W̃)}^{2} = ∑_{α=1}^{N_{i}} ∑_{γ} |(φ_{α,γ}^{i}f) ∘ (κ_{α,γ}^{i})^{−1}|_{H^{s}(ℝ^{m})}^{2}

(choose U_{α,γ}^{i} so small that each of them lies in the interior of some translate of T. Then we can for every fixed α add more positive summands to get (up to norm equivalence) |·|_{l^{2}(Γ)⊗H^{s}(T)}. Therefore:)

≤^{CN_{i}} |f|_{l^{2}(Γ)⊗H^{s}(T)} .

The computations for Ẽ are similar, but notationally more complicated.

3.4 Theorem. Suppose s, r ∈ ℝ. The inclusion ĩ : H^{s+r}(Ẽ) → H^{s}(Ẽ) is Γ-HS if r > m/2, and is Γ-tr if r > m.

Proof. Let X↓W be the double of E. The following diagram commutes by the geometric definition of p:

H^{s+r}(Ẽ) ⟶^{p_{s+r}} l^{2}(Γ) ⊗ H^{s+r}(X̃|_{T})
ĩ↓  ↓^{1⊗i}
H^{s}(Ẽ) ⟶^{p_{s}} l^{2}(Γ) ⊗ H^{s}(X̃|_{T}).

Remember that we have equipped H^{s}(Ẽ) with the Hilbert space structure which makes p an isometric embedding, therefore p^{∗}p = 1. This yields

ĩ = p_{s}^{∗}p_{s}ĩ = p_{s}^{∗}(1 ⊗ i)p_{s+r}.

Now we apply Properties (2) and (6) of Theorem 2.3, together with the classical result that for bundles over compact manifolds the inclusion H^{s+r} ↪ H^{s} is HS if r > m/2 and trace class if r > m.

4 Boutet de Monvel calculus

The Boutet de Monvel (BdM) calculus is a tool to deal with boundary value problems. It generalizes the calculus of pseudo-differential operators on manifolds without boundary. We will not go into the details but only give a reminder of those results which are essential for our applications. Detailed accounts can be found in [10] or [13] with proofs of the statements below. We will follow the notation of these sources, in particular [13].

The main point of the Boutet de Monvel calculus is the introduction of an algebra of operators which includes the boundary problems we want to study and also their inverses. To do this, we have to consider matrices of operators:

Let M be a manifold with boundary ∂M. Let E, F↓M be vector bundles over M, X, Y ↓∂M bundles over the boundary. A BdM operator P has the shape

P = (\begin{matrix}A + G & K \\ T & p \end{matrix}) : \begin{matrix}C_{0}^{∞}(E) \\ ⊕ \\ C_{0}^{∞}(X) \end{matrix} → \begin{matrix}C^{∞}(F) \\ ⊕ \\ C^{∞}(Y) \end{matrix},

where A and p are pseudo-differential operators on M and ∂M, respectively. A boundary value problem (A, T) will give typical entries in the matrix above.

Every BdM operator has an order μ ∈ [−∞, ∞) and a type d ∈ ℕ_{0}. The order is a generalization of the order of a (pseudo)differential operator, the type is determined by T and G and says “how much restriction to the boundary” is involved. It restricts the range of Sobolev spaces, to which P can be extended.

Up to smoothing operators, BdM operators are locally defined: P is BdM (of order ≤ μ and type ≤ d), if and only if for all cutoff functions φ and ψ (ψ = 1 on supp φ) the operator φPψ is BdM (of order ≤ μ and type ≤ d), and if φP(1 − ψ) is a smoothing operator of type zero.

By definition, P is a smoothing operator (i.e. of order −∞) of type d, if it has smooth integral kernels in the following sense: The pseudo-differential operators A and p have smooth integral kernels a(x, y) and p(x, y); and for F ∈ C_{0}^{∞}(E) and f ∈ C_{0}^{∞}(X) we have

\begin{aligned}GF (x) & = ∑_{i=1}^{d} ∫_{∂M} g_{i}(x, y′)(∂_{ν})^{i−1}F (y′)dy′ + ∫_{M} g_{0}(x, y)F (y)dy \\ Kf(x) & = ∫_{∂M} k(x, y′)f(y′)dy′ \\ T F (x′) & = ∑_{i=1}^{d} ∫_{∂M} t_{i}(x′, y′)(∂_{ν})^{i−1}F (y′)dy′ + ∫_{M} t_{0}(x′, y)F (y)dy, \end{aligned}

where ∂_{ν} denotes differentiation in inward unit normal direction.

Here g_{0} ∈ C^{∞}(Hom(p_{2}^{∗}E, p_{1}^{∗}F)↓M × M), and g_{i}, t_{i} and k are smooth sections of appropriate homomorphism bundles, too.

The following properties are basic extensions of corresponding properties of pseudo-differential operators. In compliance with our sources assume M is compact: Let P : C^{∞}(E) ⊕ C^{∞}(X) → C^{∞}(F) ⊕ C^{∞}(Y) and Q : C^{∞}(F) ⊕ C^{∞}(Y) → C^{∞}(G) ⊕ C^{∞}(Z) be BdM operators of order μ and type d and μ′, d′ respectively. Then the composition QP is a BdM operator of order μ + μ′ and type max{d′, d + μ′}. If s > d − 1/2, then P extends to a continuous operator

P : H^{s}(E) ⊕ H^{s}(X) → H^{s−μ}(F) ⊕ H^{s−μ}(Y).

We are interested in index problems. To do this, we have to define ellipticity: A BdM operator P of order μ ≥ 0 and type d ≤ μ is elliptic if and only if there exists a BdM operator Q : C^{∞}(F)⊕C^{∞}(Y) → C^{∞}(E)⊕C^{∞}(X) of order −μ and type zero so that

S_{0} := QP − 1  and  S_{1} := PQ − 1

are of order −∞ and S_{0} is of type μ, S_{1} of type zero. Q is called a parametrix of P (it is unique up to operators of order −∞).

As mentioned above, every differential boundary problem P = (A, T) : C_{0}^{∞}(E) → C_{0}^{∞}(F) ⊕ C_{0}^{∞}(Y) is a Boutet de Monvel operator. If it is elliptic in the Lopatinsky-Shapiro sense, it is also elliptic in the sense of the BdM algebra.

4.1 Definition. Equip M with a Riemannian metric. An operator P : C_{0}^{∞}(E) ⊕ C_{0}^{∞}(X) → C^{∞}(F) ⊕ C^{∞}(Y) is called ϵ-local (ϵ > 0), if

supp(Pf) ⊂ {x ∈ M; d(x, supp f) < ϵ} ∀f ∈ C_{0}^{∞}.

4.2 Proposition. Suppose M is a compact Riemannian manifold and ϵ > 0 is given. Every BdM operator P is the sum of an ϵ-local BdM operator (of unchanged order and type) and a smoothing operator of type zero.

Proof. Choose a finite covering of M by balls {U_{i}} of radius ϵ/2. Let {φ_{i}} be a subordinate partition of unity and ψ_{i} cutoff functions with ψ_{i} = 1 on supp φ_{i} and supp ψ_{i} ⊂ U_{i}. Set

P_{1} := ∑_{i} φ_{i}Pψ_{i}, P_{2} := P − P_{1} = ∑_{i} φ_{i}P(1 − ψ_{i}).

Then P_{2} is a smoothing BdM operator of type zero and P_{1} is ϵ-local.

4.3 Proposition. Let M̃↓M be a normal Riemannian covering of Riemannian manifolds with covering group Γ, where M is compact. Suppose the covering is trivial over balls of radius 2ϵ. Suppose

P : C^{∞}(E) ⊕ C^{∞}(X) → C^{∞}(F) ⊕ C^{∞}(Y)

is an ϵ-local operator which extends to a bounded operator

P : H^{s}(E) ⊕ H^{s}(X) → H^{s−μ}(F) ⊕ H^{s−μ}(Y).

Then P lifts to an operator

P̃ : C^{∞}(Ẽ) ⊕ C^{∞}(X̃) → C^{∞}(F̃) ⊕ C^{∞}(Ỹ),

which has a bounded extension

P̃ : H^{s}(Ẽ) ⊕ H^{s}(X̃) → H^{s−μ}(F̃) ⊕ H^{s−μ}(Ỹ).

Proof. Let {U_{i}}_{i=1,...,N} be a covering of M by balls of radius ϵ, let V_{i} be the corresponding balls of radius 2ϵ. Let φ_{i} be a subordinate covering of unity. This induces a Γ-invariant covering {U_{i,γ}}_{γ∈Γ} of M̃ with subordinate Γ-invariant partition of unity φ_{i,γ}. It is clear how to lift P. To check boundedness, let F = (F, f) ∈ C_{0}^{∞}(Ẽ) ⊕ C_{0}^{∞}(X̃) be given. Then (use |a + b|^{2} ≤ 3(|a|^{2} + |b|^{2}))

\begin{aligned}|P̃F|_{H^{s−μ}}^{2} & = |P̃ ∑_{i,γ} φ_{i,γ}F|_{H^{s−μ}}^{2} ≤^{3^{N}} ∑_{i=1}^{N} |P̃ ∑_{γ} φ_{i,γ}F|_{H^{s−μ}}^{2} \\ & =^{(∗)} ∑_{i,γ} |P̃φ_{i,γ}F|_{H^{s−μ}}^{2} ≤^{∥P∥^{2}} ∑_{i,γ} |φ_{i,γ}F|_{H^{s}}^{2} =^{Def} |F|_{H^{s}}^{2} . \end{aligned}

(∗) holds since supp(φ_{i,γ}) ∩ supp(φ_{i,γ′}) = ∅ if γ ≠ γ′.

Next we compute the trace of sufficiently regularizing BdM operators. Most important is the fact that the Γ-trace of a lift equals the trace of the operator on the base.

4.4 Theorem. Let P : C^{∞}(E) ⊕ C^{∞}(X) → C^{∞}(E) ⊕ C^{∞}(X) be a BdM operator of order −μ < −m = dim M and type d. For s > d − 1/2, P extends to a bounded trace class operator

P : H^{s}(E) ⊕ H^{s}(X) → H^{s}(E) ⊕ H^{s}(X).

The value of the trace is independent of s.

If P is ϵ-local then its lift P̃ : H^{s}(Ẽ)⊕H^{s}(X̃) → H^{s}(Ẽ)⊕H^{s}(X̃) (defined for s > d − 1/2) is Γ-tr and

Sp_{Γ}(P̃) = Sp(P).

If −μ = −∞ and P has integral kernels as on page 11 then explicitly

Sp(P) = ∫_{M} Sp_{E_{x}} a(x, x)dx + ∫_{∂M} Sp_{X′_{x}} p(x′, x′)dx′

+ ∫_{M} Sp_{E_{x}} g_{0}(x, x)dx + ∑_{i=1}^{d} ∫_{∂M} Sp_{E_{x′}} ∂_{ν,x}^{i−1}p_{i}(x, y)|_{x=x′=y}dx′

(Sp_{F} denotes the trace on the finite dimensional vector space F; ∂_{ν} is differentiation in normal direction).

Proof. The inclusion H^{s+μ} ↪ H^{s} is of trace class by Theorem 3.4. Therefore P : H^{s} →^{P} H^{s+μ} ↪ H^{s} is of trace class, being the composition of a bounded operator and a trace class operator. If μ − m > s′ − s > 0 then

Sp(P : H^{s′} → H^{s′}) = Sp(H^{s′} ↪ H^{s} →^{P} H^{s+μ} ↪ H^{s′})

= Sp(H^{s} →^{P} H^{s+μ} ↪ H^{s′} ↪ H^{s}) = Sp(P : H^{s} → H^{s}).

Here we used the trace property, noting that H^{s+μ} ↪ H^{s} is trace class. Inductively, the trace is independent of s for arbitrary s.

Identical arguments apply to the lift P̃, replacing trace by Γ-trace and using Theorem 3.4.

Now we come to the explicit computation, and μ = −∞. Observe (with P in the usual matrix form) Sp(P) = Sp(A) + Sp(G) + Sp(p). Note that A and p are actually defined on L^{2}. The above argument applies to show that Sp(A : H^{s} → H^{s}) = Sp(A : L^{2} → L^{2}). A is an integral operator with a smooth kernel and therefore with trace

Sp(A) = ∫_{M} Sp_{E_{x}} a(x, x)dx.

Similarly Sp(p) = ∫_{∂M} Sp_{X_{x′}} p(x′, x′)dx′. For the obvious splitting G = G_{0} + G_{1} + ⋯ + G_{d}, note that each summand is trace class. G_{0} behaves exactly as A does. For i > 0, the operator G_{i} is a composition

H^{s}(E) →^{∂_{ν}^{i−1}} H^{s−i+1}(E) →^{res} H^{s−i+1/2}(E|_{∂M}) →^{K_{i}} H^{∞}(E) ↪^{i} H^{s}(E).

Each of the operators is bounded and the inclusion is trace class (res denotes the restriction to the boundary and K_{i} is the obvious integral operator with smooth kernel from E|_{∂M} → E). Using the trace property and the fact that inclusions of Sobolev spaces commute with differentiation and restriction to the boundary, we see

Sp(G_{i}) = Sp(i ∘ \underbrace{res ∘∂_{ν}^{i−1} ∘ K_{i}}_{=:P_{i}}).

Now P_{i} is an integral operator with smooth kernel on ∂M, namely

P_{i}f(x′) = ∫_{∂M}(∂_{ν,x}^{i−1}g_{i})(x′, y′)f(y′)dy′.

Therefore it extends to a trace class operator on L^{2}(E|_{∂M}) with

Sp(G_{i}) = Sp(P_{i}) = ∫_{∂M} Sp_{E_{x′}} (∂_{ν,x}^{i−1}g_{i}(x, y))|_{x=x′=y}dx′.

This establishes the formula for Sp(P).

Identical arguments apply to the lift P̃ as far as follows:

Sp_{Γ}(P̃) = Sp_{Γ}(Ã) + Sp_{Γ}(p̃) + Sp_{Γ}(G̃_{0}) + ∑_{i=1}^{d} Sp_{Γ}(P̃_{i}),

where each summand is the lift of an integral operator with smooth kernel on L^{2}(E), L^{2}(X) and L^{2}(E|_{∂M}), respectively.

Therefore, it remains to show that for an ϵ-local trace class operator R on L^{2} the Γ-trace of the lift coincides with the trace on the base.

Let s_{i} be an orthonormal basis of L^{2}(E) such that the support of each s_{i} is contained in a set over which M̃| → M and E| → M are trvial. Choose for each s_{i} one lift s̃_{i} ∈ L^{2}(Ẽ). Then we have the standard formula for trace and Γ-trace (2.1)

Sp(R) = ∑_{i}(Rs_{i}, s_{i})_{L^{2}Ω^{∗}(X,V)}; Sp_{Γ}(R̃) = ∑_{i}(R̃s̃_{i}, s̃_{i})_{L^{2}Ω^{∗}(X̃,Ṽ)}.

The fact that R̃ is the lift of R and s̃_{i} the lift of s_{i} implies that the two expressions coincide, i.e.

Sp_{Γ}(R̃) = Sp(R).

This applies to all the above operators and completes the proof.

5 Proof of the L^{2}-index theorem

5.1 Situation. Let M̃↓M be a normal covering of a compact manifold with boundary with deck transformation group Γ. Let P = (A, T) : C_{0}^{∞}(E) → C_{0}^{∞}(F)⊕C_{0}^{∞}(Y) be an elliptic differential boundary problem on M. Denote its lift to M̃ with P̃ : C_{0}^{∞}(Ẽ) → C_{0}^{∞}(F̃) ⊕ C_{0}^{∞}(Ỹ). Suppose P has order μ ≥ 0 and type d ≤ μ.

We have the extension P : H^{μ}(E) → L^{2}(F) ⊕ L^{2}(Y).

Let H_{0} : L^{2}(E) → ker(P) be the orthogonal projection onto the kernel, H_{1} : L^{2}(F) ⊕ L^{2}(Y) → im(P)^{⊥} the orthogonal projection onto the cokernel of P. Similarly, let H̄_{0} and H̄_{1} be the projections onto kernel and cokernel of P̃.

We want to prove the L^{2}-index Theorem 1.2 for ∂-manifolds:

5.2 Theorem. dim_{Γ} ker P̃ = Sp_{Γ}(H̄_{0}) and dim_{Γ} coker P̃ = Sp_{Γ}(H̄_{1}) are finite, and

ind_{Γ}(P̃) := Sp_{Γ}(H̄_{0}) − Sp_{Γ}(H̄_{1}) = ind(P) = Sp(H_{0}) − Sp(H_{1}).

The idea of the proof is the following: H_{i} and H̄_{i} have in general nothing to do with each other. But suppose we could find a bounded liftable ”inverse” Q to P. Then the equations

PQ = 1 − H_{1} and QP = 1 − H_{0}

could be lifted and we could compare the trace of H_{i} and H̄_{i} directly. This is not possible. We use a parametrix instead:

Let Q be an ϵ-local parametrix of P (use Proposition 4.2) so that

\begin{aligned}& PQ = 1 − S_{1}, QP = 1 − S_{0} \\ ⟹ & P̃Q̃ = 1 − S̃_{1}, Q̃P̃ = 1 − S̃_{0}. \end{aligned}  (5.3)

Automatically, S_{0} = 1 − QP and S_{1} are ϵ-local since the right hand side is. Note that S_{0} and S̃_{0} are operators of order −∞ and type μ, whereas S_{1} and S̃_{1} have order −∞ and type zero.

We know already that Sp_{Γ} S̃_{i} = Sp S_{i} (Theorem 4.4). It remains to show that we can compute the index also in terms of the S_{i}, namely

Sp S_{0} − Sp S_{1} = Sp H_{0} − Sp H_{1}  (5.4)

(and similarly on M̃). This will be achieved using Theorem 2.6. We start with:

5.5 Proposition. The image of the projection H_{0} : L^{2}(E) → L^{2}(E) (i.e. the kernel of P) is contained in H^{∞}(E) and H_{0} restricts to a bounded operator H_{0} : H^{s}(E) → H^{s+t}(E) for arbitrary s, t ≥ 0. Especially H_{0} : H^{s} → H^{s} is trace class for every s ≥ 0 and the trace is independent of s.

The same holds for H̄_{0} if we replace tr by tr_{Γ}.

Proof. Elliptic regularity and the corresponding a priori estimates (the theory works as in the compact case, compare [12, 4.14] for a generalization) imply that the kernels of P and P̃ are contained in every Sobolev space H^{s}(E) and H^{s}(Ẽ) respectively, and that the Sobolev norms on this subspace are equivalent to the L^{2}-norm. This implies everything if we consider H_{0} as composition of the bounded operator H_{0} : H^{s} → H^{s+m+1} with the trace class operator i : H^{s+m+1} ↪ H^{s} (and similarly for H̄_{0}).

Now we can prove Equation 5.4. The following computations are formulated only for the lifted operators. They are valid also on the base with the obvious changes.

Multiplying the equations in (5.3) with H̄_{1} from the left and with H̄_{0} from the right, we get

H̄_{1} = H̄_{1}S̃_{1}  H̄_{0} = S̃_{0}H̄_{0},  (5.6)

where the equation for H̄_{0} is valid on H^{μ} and the one for H̄_{1} is valid on all of L^{2} . By multiplication of (5.3) with P̃ we get on H^{μ}

P̃S̃_{0} = S̃_{1}P̃.

Following Atiyah [1] we now define

T̄_{i} := (1 − H̄_{i})S̃_{i}(1 − H̄_{i}) (i = 0, 1).

Because of Theorem 2.3 (3) T̄_{0} is a Γ-tr operator on the Hilbert N (Γ)-module H^{μ} and T̄_{1} is a Γ-tr operator on the Hilbert N (Γ)-module L^{2}. Since H̄_{i} are projectors

Sp_{Γ} T̄_{0} = Sp_{Γ}(S̃_{0}(1 − H̄_{0})) = Sp_{Γ} S̃_{0} − Sp_{Γ} H̄_{0}  (use (5.6)),
Sp_{Γ} T̄_{1} = Sp_{Γ}((1 − H̄_{1})S̃_{1}) = Sp_{Γ} S̃_{1} − Sp_{Γ} H̄_{1}  (use (5.6)).

Therefore,

Sp_{Γ} T̄_{0} = Sp_{Γ} T̄_{1} ⟺ Sp_{Γ} S̃_{0} − Sp_{Γ} S̃_{1} = Sp_{Γ} H̄_{0} − Sp_{Γ} H̄_{1}.

Next observe

ker P̃ ⊂ ker T̄_{0}; ker P̃^{∗} ⊂ ker T̄_{1}^{∗};

P̃T̄_{0} = P̃S̃_{0} − \underbrace{P̃H̄_{0}}_{=0} S̃_{0} − P̃ \underbrace{S̃_{0}H̄_{0}}_{=H̄_{0}} + \underbrace{P̃H̄_{0}}_{=0} S̃_{0}H̄_{0} = S̃_{1}P̃ = ⋯ = T̄_{1}P̃.

Application of Proposition 2.6 with V = H^{μ}, W = L^{2} (then P : V → W is bounded) yields Sp_{Γ} T̄_{0} = Sp_{Γ} T̄_{1}, i.e. ind_{Γ} P̃ = Sp_{Γ} S̃_{0} − Sp_{Γ} S̃_{1}.

Similarly, ind P = Sp S_{0} − Sp S_{1}. Now Theorem 4.4 appplied to the ϵ-local smoothing operators S_{0}, S_{1} finishes the proof of Theorem 1.2.

6 Index and adjoint boundary value problems

The purpose of this section is to simplify the index formula by replacing the cokernel with the kernel of the adjoint.

6.1 Theorem. Let E, F ↓ M, X, Y ↓ ∂M be Riemannian vector bundles, P := (A, p) : C_{0}^{∞}(E) → C_{0}^{∞}(F) ⊕ C_{0}^{∞}(Y) an elliptic differential boundary problem. Suppose the differential boundary problem Q := (B, q) : C_{0}^{∞}(F) → C_{0}^{∞}(E) ⊕ C_{0}^{∞}(X) is adjoint to (A, p) with respect to the Greenian formula

(Ae, f)_{L^{2}(F)} − (e, Bf)_{L^{2}(E)} = (pe, sf)_{L^{2}(Y)} − (te, qf)_{L^{2}(X)}.  (6.2)

(Here t, s are auxiliary boundary differential operators, and adjointness means that the formula holds ∀e ∈ C_{0}^{∞}(E), ∀f ∈ C_{0}^{∞}(F).) Then

L^{2}(F) ⊕ L^{2}(Y) ⊃ im(P)^{⊥} →^{p_{1}} L^{2}(F) : (f, y) ↦ f

is an isomorphism onto ker(Q) with inverse

α : ker(Q) → im(P)^{⊥} : f ↦ (f, −sf).

Proof. First, we have to prove that the maps have range as stated. Take (f, y) ∈ im(P)^{⊥}. In particular, f ⊥ A({e; pe = 0}). Choosing e which are supported in the interior of M (these are dense in L^{2}) (6.2) implies Bf = 0. [12, Lemma 4.7] yields that the set {te| e ∈ C_{0}^{∞}(E) and pe = 0} is dense in L^{2}(Y) (observe that ellipticity implies that (p, t) is a Dirichlet system in the notion of [12, Lemma 4.7]). Then (6.2) also implies qf = 0. That α has the correct image follows immediately from the Greenian formula. It remains to check α ∘ p_{1} = 1_{im P^{⊥}}: If (f, y) ∈ im(P)^{⊥}, then for arbitrary e ∈ C_{0}^{∞}(E)

\begin{aligned}(pe, y) & =^{(f,y)⊥im(P)} −(Ae, f) \\ & =^{(6.2)} −(e, Bf) − (pe, sf) + (te, qf) =^{f∈ker Q} −(pe, sf). \end{aligned}

Again, [12, Lemma 4.7] implies that im(p) is dense in L^{2}(Y) and therefore y = −sf.

Being in the situation of the L^{2}-index Theorem 1.2, the isomorphism of Theorem 6.1 is equivariant under the group operation and coker(P̃) is Γisomorphic to ker(Q̃). Therefore the index theorem can be stated as follows:

6.3 Theorem. Suppose M is a compact boundary manifold with normal covering M̃ and covering group Γ. Let P := (A, T) be an elliptic differential boundary problem on M with lift P̃. Let Q := (B, S) be an adjoint with lift Q̃. Then

ind(P) = ind_{Γ}(P̃) = dim_{Γ}(ker P̃) − dim_{Γ}(ker Q̃).

We apply this to compute the Euler characteristic of a ∂-manifold. Lott/Lück [7] get the same result with other methods.

6.4 Theorem. Suppose M is a compact manifold with boundary ∂M = M_{1} ⨿ M_{2}. Let M̃ be a normal covering of M with covering group Γ. Then

χ(M, M_{1}) = ∑_{p}(−1)^{p} dim_{Γ} H_{(2)}^{p}(M̃, M̃_{1})

with H_{(2)}^{p}(M̃, M̃_{1}) = {ω ∈ C^{∞}(Λ^{p}T M̃); |ω|_{L^{2}} < ∞, dω = 0 = δω, b_{1}^{∗}(ω) = 0 = b_{2}^{∗}(∗ω)}. (b_{i} : M̃_{i} ↪ M̃ are the inclusions.)

Proof. To keep notation simple suppose M_{1} = ∅. We known χ(M) = ind(P^{ev}), where P^{ev/odd} are the boundary problems

(d + δ, b_{2}^{∗} ∘ ∗) : C^{∞}(Λ^{ev/odd}T M) → C^{∞}(Λ^{odd/ev}T M) ⊕ C^{∞}(Λ^{∗}T ∂M).

We have the following Greenian formula

((d + δ)ω, η)_{L^{2}(M)} =

= (ω, (δ + d)η)_{L^{2}(M)} ± ∫_{∂M} b^{∗}ω ∧ b^{∗}(∗η) ± ∫_{∂M} b^{∗}(η) ∧ b^{∗}(∗ω).

Theorems 6.1 and 6.3 yield then

χ(M) = ind(P̃^{ev}) = dim ker(P̃^{ev}) − dim ker(P̃^{odd}).

In view of elliptic regularity this is just the claim.

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