DECOMPOSITION OF MANIFOLDS AND A SPLITTING FORMULA FOR A SPECTRAL FLOW

KENRO FURUTANI

1. Introduction

It is well known that the non-trivial component F̂_{∗} of the space F̂ = F̂_{−} ∪ F̂_{∗} ∪ F̂_{+} of bounded selfadjoint Fredholm operators (on a complex or a real Hilbert space) is a classifying space for the K-group, K^{−1} in the complex case and that of KO^{−7} in the real case ([AS]). Also we know, by Bott periodicity theorem, that the fundamental group of these spaces is isomorphic to ℤ. This isomorphism is given by the quantity “Spectral flow”, Sf : π_{1}(F̂_{∗}) → ℤ, although it is not stated explicitly in this paper [AS].

Since the Spectral flow was treated in the paper [APS], it appears in the various theory where it plays important roles, for example, in the study of spectral analysis of Dirac operators or in the theory of Floer homology for the study of low dimensional manifolds, and so on.

We have an intuitive understanding of the spectral flow by saying that it is the difference of the net numbers of the eigenvalues of the selfadjoint Fredholm operators in the family which change signs from minus to plus and from plus to minus, when the parameter of the family goes from 0 to 1. However in the paper [Ph] for the first time it was given a rigorous definition of the spectral flow for not only continuous loops in the space F̂_{∗} but also for arbitrary continuous paths in F̂_{∗} and was proved that the quantity is a homotopy invariant of continuous paths in the space F̂_{∗} with the fixed end points and it satisfies the additivity under catenation of the continuous paths. Thus the spectral flow is not only a spectral invariant but also a homotopy invariant, in so far as we treat it in the framework of the space F̂_{∗}.

Nowadays there are many types of formulas including various Spectral flows corresponding to families of Fredholm operators which are mostly not bounded operators, because simply they are families of differential operators. However in some cases the continuous family of such unbounded Fredholm operators can be interpreted as a continuous path in the space F̂_{∗} as treated in the papers [FO1], [BF1] and [CP]. Also the theorems in [Fl], [Yo] and [Ni] can be interpreted in the framework of the space F̂_{∗}. See also [Ta], [Ge], [CLM2], [OF] and [DK] and others. It is not clear for me whether any “continuous” families of Fredholm operators, especially a family of unbounded selfadjoint Fredholm operators with varying domains of definition, can be interpreted in the framework of the space F̂_{∗}.

The main purpose of my talk is to explain a General Splitting Formula for a Spectral Flow of a family {A + C_{t}}_{t∈[0, 1]} (C_{t} is of zeroth order) of first order selfadjoint elliptic differential operators on a closed manifold in the framework of the space F̂_{∗}.

2 KENRO FURUTANI

When we decompose the closed manifold M into two parts M_{±} by a hypersurface Σ, M = M_{−} ∪ M_{+}, M_{−} ∩ M_{+} = ∂M_{±} = Σ, then we also have a family of symmetric elliptic operators on each part M_{±} simply by the restriction, and at this level it is nothing more to have two families of symmetric operators defined on manifolds with boundary. This means there are no natural choices of boundary conditions among elliptic boundary conditions under which the family becomes a family of selfadjoint Fredholm operators. Under this circumstances it will not be apparent whether the spectral flow of the family {A + C_{t}} on the whole manifold is expressed as a sum of two spectral flows obtained by even imposing a suitable selfadjoint elliptic boundary condition on each part and adding a correction term which solely depends on the boundary manifold Σ, since partly by the reason that we have no local representations of the spectral flow.

To formulate our splitting formula in this situation, we shall take a following way, that is, firstly we construct a two-parameter family {A_{s,t}}_{s,t∈[0, 1]} of selfadjoint Fredholm operators on M with A_{t,t} = A + C_{t}. Then by noting the homotopy invariance and the additivity under catenation of the spectral flow, the sum of two spectral flows of {A_{s,0}}_{s∈[0, 1]} and {A_{1,t}}_{t∈[0, 1]} coincides with the spectral flow of the original family. From the definition, it is not true, but it looks like that the families {A_{s,0}}_{s∈[0, 1]} and {A_{1,t}}_{t∈[0, 1]} are defined on the each component respectively. In general it is not clear that each of these two spectral flows coincides with a spectral flow of a family obtained by imposing a suitable selfadjoint elliptic boundary condition on the restriction of {A + C_{t}} to each part. Then secondly we explain that if the operators in the family are a product form near the separating hypersurface, then there is a selfadjoint elliptic boundary condition on each part by which the restrictions of the operators to each part become a family of selfadjoint Fredholm operators and their spectral flows coincide with each of {A_{s,0}}_{s∈[0, 1]} and {A_{1,t}}_{t∈[0, 1]}. Moreover it turned out that these boundary conditions reflect the influence from one side to other side of M_{±} in a natural way. We verify these completely in the framework of the space F̂_{∗}. Our General Spectral Flow Formula([BF1]) and a Reduction Theorem([BFO]) of the Maslov index in the infinite dimension play a role as a bridge connecting these two spectral flows for the family of operators of a product form near the separating hypersurface Σ.

There are several similar formulas already treated in the papers [Ta], [DK], [CLM2] and others. Here I would like to emphasize that we can admit the non-invertible end points in the family {A + C_{t}} because we base on the rigorous definition as a homotopy invariant of the spectral flow given in [Ph] together with that of the Malsov index which is valid without any assumptions at the end points and our method will explain in some extent what kinds of conditions to the operators in the family we need to prove such a splitting formula. Moreover, since for a family of the operators with product form structure near the separating hypersurface, the space of our boundary values β = D_{max}/D_{min} is identical for any length of the “neck”, we expect this property also will give us a reduction of our splitting formula under taking adiabatic limit in within our framework.

Finally I remark that although there is a deep theory of pseudo-differential operators with transmition property including the theory of Calderón projector ([Ho2]), here I avoid the use of Calderón projector in the L_{2}-framework to treat with the Cauchy data space. Because it will make things confusing to use the Calderón projector in the L_{2}- framework from the beginning, and we miss the role of the product form assumption on

DECOMPOSITION OF MANIFOLDS AND A SPLITTING FORMULA FOR A SPECTRAL FLOW 3

the operators in our formula. However I must employ two facts without proofs from the pseudo-differential operator theory:

(a) the space of boundary values β = D_{max}/D_{min} of sections in the maximum domain in L_{2}-space of a first order selfadjoint elliptic differential operator on a manifold with boundary Σ is included in the Sobolev space of order −\frac{1}{2} on the boundary manifold ([Ho1]).

(b) if a first order symmetric elliptic differential operator is a product form near the boundary manifold (= separating hypersurface in our setting), then the closed extension defined by the Atiyah-Patodi-Singer boundary condition is selfadjoint and satisfy coercive estimate, which we need to determine the space β.

2. Notations and Main Theorem

2.1. Fredholm-Lagrangian-Grassmannian. Let (H, ⟨·, ·⟩ , ω) be a symplectic (real and separable) Hilbert space with the inner product ⟨·, ·⟩ and the symplectic form ω. In the theory below we do not replace the symplectic form ω after once it was introduced in the real Hilbert space H, but we may always assume that there exists a bounded operator J : H → H such that ω(x, y) = ⟨Jx, y⟩ for any x, y ∈ H and J^{2} = − Id by replacing the inner product by another one which defines an equivalent norm on H. So we can assume from the beginning the following “compatible” conditions among ω, the inner product and the almost complex structure J:

⟨Jx, Jy⟩ = ⟨x, y⟩
ω(Jx, Jy) = ω(x, y) for all x, y ∈ H.

Let Λ(H) denotes the set of all Lagrangian subspaces of H. Each Lagrangian subspace is closed, and the topology of Λ(H) is given by embedding it into the space B(H) of bounded linear operators on H as the orthogonal projection operator onto the Lagrangian subspace.

Definition 2.1. Let λ ∈ Λ(H), then the Fredholm-Lagrangian-Grassmannian of H with respect to λ is defined as

FΛ_{λ}(H) := {μ ∈ Λ | (μ, λ) is a Fredholm pair}.

The Maslov index in the infinite dimensional case, Mas({c(t)}, λ) ∈ ℤ, for arbitrary continuous curves in FΛ_{λ}(H) with respect to the Maslov cycle 𝔐_{λ}(H) = {μ ∈ FΛ_{λ}(H) | μ ∩ λ ≠ {0}} is defined by the similar arguments in [Ph](see [BF1] and [FO2]).

2.2. Symmetric Operators and Boundary Value Space. Let H be a real separable Hilbert space and A a densely defined closed symmetric operator with the domain D_{m}. We denote the domain of the adjoint operator A^{∗} by D_{M}.

Let β be the factor space of D_{M} by D_{m}, β = D_{M}/D_{m}, and let γ : D_{M} → β (γ(x) = [x]) be the projection map. The space β becomes a symplectic Hilbert space with the inner product induced by the graph norm

⟨x, y⟩_{G} := ⟨x, y⟩ + ⟨A^{∗}x, A^{∗}y⟩

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on D_{M} and the symplectic form given by “Green’s form”

ω([x], [y]) := ⟨A^{∗}x, y⟩ − ⟨x, A^{∗}y⟩ for x, y ∈ β.

For D_{m} ⊂ D ⊂ D_{M}, we denote

A_{D} := A^{∗}|_{D},

then

A_{D} is closed symmetric ⇔ γ(D) is closed isotropic
A_{D} is self-adjoint ⇔ γ(D) is Lagrangian

Definition 2.2. We call γ(Ker A^{∗}) the Cauchy data space.

Now we make two Assumptions (A) and (B).

Assumption (A). We assume that A admits at least one self-adjoint Fredholm extension A_{D} with the domain of definition D ⊂ D_{M} and A_{D} has the compact resolvent.

Then γ(Ker(A^{∗})) is a Lagrangian subspace of β and (γ(Ker A^{∗}), γ(D)) is a Fredholm pair.

Assumption (B). We assume that there exists a continuous curve {C_{t}}_{t∈[0,1]} in the space of bounded self-adjoint operators on H and that the operators A^{∗} + C_{t} − s for small s satisfy the condition that for any t

Ker(A^{∗} + C_{t} − s) ∩ D_{m} = {0} for|s| ≪ 1.

Then the family of Lagrangian subspaces {γ(Ker(A^{∗} + C_{t}))} is a continuous curve in FΛ_{γ(D)}(β).

Theorem 2.3 (General Spectral Flow Formula). Let {A_{D} + C_{t}} be a family of selfadjoint operators satisfying Assumption (A) and (B) above. Then we have

Sf({A_{D} + C_{t}}) = Mas({γ(S_{t})}, γ(D)).

2.3. A Reduction Theorem for the Maslov Index. Let β and L be two symplectic Hilbert spaces with the symplectic form ω_{β} and ω_{L}. We assume that each space β and L is decomposed into a direct sum of two Lagrangian subspaces in the following way:

β = θ_{−} ⊕ θ_{+}

and

L = L_{−} ⊕ L_{+}.

We also assume that there are injective maps with dense images

i_{−} : θ_{−} → L_{−}
i_{+} : L_{+} → θ_{+}

such that ω_{L}(i_{−}(x), y) = ω_{β}(x, i_{+}(y)) for x ∈ θ_{−} and y ∈ L_{+}.

Then

Theorem 2.4 (Reduction Theorem). There is a natural continuous map

ℓ : FΛ_{θ_{−}}(β) → FΛ_{L_{−}}(H)

preserving the Maslov index:

Mas({μ_{t}}, θ_{−}) = Mas({ℓ(μ_{+})}, L_{−}).

DECOMPOSITION OF MANIFOLDS AND A SPLITTING FORMULA FOR A SPECTRAL FLOW 5

2.4. Decomposition of manifolds and first order differential operators. Let A be an elliptic and selfadjoint first order differential operator acting on a real vector bundle 𝔼 on a closed manifold M. The selfadjointness of the operator A will be meant by fixing an L_{2}-inner product on the space of smooth sections C^{∞}(M, 𝔼). Let {C_{t}} be a continuous family of selfadjoint bundle maps of 𝔼, and we regard them with the same notation as a continuous family of zeroth order operators on L_{2}(M, 𝔼). Let Σ be a hypersurface, which separates the manifold M into two parts M_{±} with the commom boundary Σ = ∂M_{−} = ∂M_{+}. We assume that the operators A + C_{t} + s (|s| << 1) satisfy Assumption (B) above and we assume that these operators A + C_{s} are all of the “product form” near the hypersurface Σ, that is, the operator A + C_{s} has the following form on a cylindrical neighborhood U(Σ) ≅ (−1, 1) × Σ of Σ:

A + C_{s} = σ (\frac{∂}{∂τ} + B_{s})

where τ ∈ (−1, 1), σ is a bundle map of 𝔼|_{Σ}, B_{s} is a selfadjoint elliptic operator on 𝔼|_{Σ}, π^{∗}(𝔼|_{Σ}) ≅ 𝔼|_{U(Σ)}, and π : U(Σ) → Σ is defined through a fixed identification (−1, 1) × Σ ≅ U(Σ). σ and B_{s} do not depend on the normal variable τ.

Let D_{min} be a subspace in H^{1}(M, 𝔼) (the first order Sobolev space with values in 𝔼) consisting of sections which vanish on Σ. Then the operator A defined on D_{min}, we denote it by A_{0}, is a closed symmetric operator. We will denote the domain of the adjoint operator (A_{0})^{∗} by D_{max}. Here we are assuming that the hypersurface Σ separates the manifold M into two parts M_{±} with common boundary Σ = ∂M_{±}, so D_{min} and D_{max} also decompose into two components D_{min}^{±} and D_{max}^{±} corresponding to submanifolds M_{±}:

D_{m} = D_{−} ⊕ D_{+},
D_{max} = D_{max}^{−} ⊕ D_{max}^{+}
A_{0} = A_{0}^{−} ⊕ A_{0}^{+},

and D_{max}^{±} is the domain of the each adjoint operator of A_{0}^{±}.

Although graph inner products on D_{max} by operator A + C_{t} is not identical, but the norms are all equivalent and the symplectic forms ω_{±} on β^{±} = D_{max}^{±}/D_{min}^{±} does not depend on the parameter.

Remark 2.5. If Σ is only orientable and does not separate M, that is, M\Σ is connected, then D_{min}(and also D_{max}) does not decompose in the above way, but β = D_{max}/D_{min} is a sum of two spaces β^{±} of boundary values taken from each side. However in this case the Cauchy data space γ(Ker(A_{0})^{∗}) is not a sum of two Lagrangian subspaces in β^{±}.

Proposition 2.6 ([Ho1]). β^{±} is a subspace in H^{−1/2}(Σ, 𝔼|_{Σ}), the Sobolev space on Σ of order −1/2. Also we have β^{−} ∩ β^{+} = H^{1/2}(Σ, 𝔼|_{Σ}).

Let γ_{±} be the map from D_{max}^{±} to β^{±}. The families {Λ_{t}^{±}} = {γ_{±}(Ker((A_{0}^{±} + C_{t})^{∗}))} are continuous curves in Λ(β^{±}).

Theorem 2.7. Λ_{t}^{−} ⊕ Λ_{t}^{+} ∈ FΛ_{δ}(β), and Sf{A + C_{t}} = Mas({Λ_{t}^{−} ⊕ Λ_{t}^{+}}, δ) where

δ = {(φ, φ) ∈ β^{−}⊕β^{+} | ^{∃}f ∈ H^{1}(M, 𝔼), γ_{±}(f|_{M_{±}}) = φ} ≅ diagonal of H^{1/2}(Σ)⊕H^{1/2}(Σ).

Next let us consider the family of operators {T_{s}^{−}} on M_{−} with the domain D_{0},

(2.1)  D_{0} = {f ∈ H^{1}(M_{−}, 𝔼|_{M_{−}}) | ^{∃}φ ∈ Λ_{0}^{+}, γ_{−}(f) = φ}

and

T_{s}^{−}(f) = (A + C_{s})(f) on M_{−}.

From the “product form” assumption and the Reduction Theorem 2.4, we know that γ_{−}(D_{0}) is a Lagrangian subspace in β^{−}. Hence we have

Proposition 2.8. Each operator T_{s}^{−} on D_{0} is a selfadjoint operator. So this implies that

∥u∥_{1} ≤ c′(∥T_{s}^{−}u∥_{0} + ∥u∥) for u ∈ D_{0}

with a uniform constant c′ > 0.

Now we have the well-defined spectral flow for the continuous family \{T_{s}^{−}∘\sqrt{1 + (T_{s}^{−})^{2}}^{−1}\}:

Proposition 2.9.

Sf(\{T_{s}^{−}\}) ≡ Sf(\{T_{s}^{−} ∘ \sqrt{1 + (T_{s}^{−})^{2}}^{−1}\}) = Mas(\{Λ_{s}^{−}\}, γ_{−}(D_{0}))
= Mas({Λ_{s}^{−} ∩ L_{2}(Σ)}, Λ_{0}^{+} ∩ L_{2}(−Σ)).

Similarly when we define the family of operators {T_{t}^{+}} on M_{+} with the domain D_{1} given by

D_{1} = {v ∈ H^{1}(M_{+}, 𝔼|_{M_{+}}) | ^{∃}φ ∈ Λ_{1}^{−}, γ_{+}(v) = φ}
T_{t}^{+}(v) = (A + C_{t})v on M_{+}.

then

Proposition 2.10.

Sf({T_{t}^{+}}) = Mas({Λ_{t}^{+}}, γ_{+}(D_{1}))
= Mas({Λ_{t}^{+} ∩ L_{2}(Σ)}, Λ_{1}^{−} ∩ L_{2}(−Σ)).

Summing up these results we can now state our main Theorem.

Theorem 2.11 (Splitting Formula for a Spectral Flow).

Sf{A + C_{t}} = Sf{T_{t}^{−}} + Sf{T_{t}^{+}}.

Of course this is not a unique way. For example, if we change the definition of the domains D_{0} and D_{1} by an obvious way, then we have a similar splitting formula.

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DECOMPOSITION OF MANIFOLDS AND A SPLITTING FORMULA FOR A SPECTRAL FLOW 7

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Kenro Furutani

Department of Mathematics

Faculty of Science and Technology Science University of Tokyo 2641 Noda, Chiba 278-8510 Japan.

  E-mail address: furutani@ma.noda.sut.ac.jp