A K-Theoretic Relative Index Theorem and Callias-Type Dirac Operators

Ulrich Bunke^{∗}

Abstract

We prove a relative index theorem for Dirac operators acting on sections of an A − C^{∗}-Clifford bundle, where A is some C^{∗}-algebra. We develop the index theory of complex and real Callias-type Dirac operators.

Contents

1 A K-theoretic relative index theorem 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Cutting and pasting invariance . . . . . . . . . . . . . . . . . . . . . 13 1.4 Cobordisms of positive scalar curvature metrics . . . . . . . . . . . . 14

2 Index theory for Callias type operators 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 The Higson corona and admissible endomorphisms . . . . . . . . . . . 20 2.3 The odd-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 The map u . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Computation of u . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.3 Interpretation as KK-intersection product . . . . . . . . . . . 25 2.3.4 The spectral flow pairing s . . . . . . . . . . . . . . . . . . . . 27

2.4 The even-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.1 The construction of g . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 Computation of g . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.3 Interpretation as KK-intersection product . . . . . . . . . . . 30 2.4.4 The relative index pairing r . . . . . . . . . . . . . . . . . . . 32

2.5 The boundary map in K-homology . . . . . . . . . . . . . . . . . . . 33 2.6 Real Callias-type operators . . . . . . . . . . . . . . . . . . . . . . . . 35

∗Institut für Reine Mathematik, Humboldt-Universität zu Berlin, Ziegelstr.13a, Berlin 10099, Germany, E-mail: ubunke@mathematik.hu-berlin.de

This paper consists of two fairly independent sections. In the first section we consider the index theory of Dirac operators acting on A − C^{∗}-Clifford bundles over complete Riemannian manifolds. Assuming invertibility at infinity we develop the analytical details for producing a Kasparov module representing the index of the Dirac operator. Our main result is the K-theoretic relative index theorem. We provide applications related to the positive scalar curvature question.

In the second section we develop the index theory of Callias-type Dirac operators on odd as well as on even-dimensional manifolds M in a systematic way. We show, how the construction of Callias type operators is related to the boundary map in the long K-theory sequence of (M̄, ∂_{h}M). Here M is a complete Riemannian manifold and M̄ is a certain compactification by the Higson corona ∂_{h}M. We also consider real Dirac operators of Callias type being equivariant with respect to the action of some Clifford algebra. The resulting index theorem is again applied to obstructions against positive scalar curvature metrics.

This work was partly done at the Max Planck Institut für Mathematik, Bonn. The author wants to thank Stephan Stolz for the very stimulating discussion. He initiated this work by posing the problems, which are solved using the K-theoretic relative index theorem, and suggested many simplifications. There were also profitable discussions with S. Hurder. The author thanks the referees for forcing him to rewrite a miserable first version of that paper. In the meantime [2] appeared, it contains the index theorem for Callias-type operators on odd-dimensional manifolds.

1 A K-theoretic relative index theorem

1.1 Introduction

Let D : C^{∞}(M, E) → C^{∞}(M, E) be a generalized Dirac operator acting on sections of a Z_{2}-graded Clifford bundle E over a complete Riemannian manifold. If zero is not in the essential spectrum σ_{ess}(D) of D, then the index

ind D = dim ker D^{+} − dim ker D^{−}

is well-defined. We say that D is positive at infinity if there is a constant c > 0 and a compact set K ⊂ M such that R_{|M\K} ≥ c, where R := D^{2} − Δ is the endomorphism occurring in the Weitzenböck formula. If D is positive at infinity, then 0 ∉ σ_{ess}(D).

The original version of the relative index theorem due to Gromov/Lawson [16] computes ind D_{1} − ind D_{2} for two Dirac operators that are positive at infinity and coincide outside of compact sets. Thus, the D_{i} live on manifolds M_{i}, i = 1, 2, and there are open cocompact sets U_{i} ⊂ M_{i} with smooth boundary ∂Ū_{1} ≅ ∂Ū_{2} such that D_{1|U_{1}} ≅ D_{2|U_{2}} and R_{i|U_{i}} ≥ c > 0. The equality D_{1|U_{1}} ≅ D_{2|U_{2}} means that there is an isometry γ : U_{1} → U_{2} covered by an isomorphism of Hermitian vector bundles Γ : E_{1|U_{1}} → E_{2|U_{2}} such that D_{1}Γ^{∗} = Γ^{∗}D_{2}. We deform the metric of M_{1} and the Clifford bundle structure of E_{i} near ∂Ū such that they become compatible with a product structure on a collar neighbourhood of ∂Ū. Moreover, we consider E_{2|M_{2}\U_{2}} with the opposite Clifford bundle structure, where X ∈ T M_{2} acts by −X and with the opposite grading. Let M^{♯} := M_{1} \ U_{1} ∪_{∂Ū} M_{2} \ U_{2} and glue the bundles using the morphism given by Clifford multiplication with the unit normal vector at ∂Ū_{1} (a nice reference for the relative index theorem, such glueing procedures and related topics is Booß/Wojciechowski [5]). Let D^{♯} be the resulting Dirac operator over M^{♯}.

Theorem 1.1 (Gromov/Lawson)

ind D_{1} − ind D_{2} = ind D^{♯}.

Another way to look upon this theorem is as follows. Consider M = M_{1}∪M_{2} and the opposite Clifford bundle structure and grading on E_{2}. Let D be the Dirac operator over M. Obviously ind D = ind D_{1} − ind D_{2}. We can cut M at ∂Ū_{1} ∪ ∂Ū_{2} and glue it together again using the diffeomorphism interchanging the two boundary components, obtaining M̃ together with a new Z_{2}-graded Clifford bundle and a Dirac operator D̃. Then M̃ = M^{♯} ∪ (U_{1} ∪_{∂U} U_{2}) and D̃ is invertible over U_{1} ∪_{∂U} U_{2}. Hence ind D̃ = ind D^{♯}. The relative index theorem states that cutting and glueing as described above does not change the index:

ind D = ind D̃.

There are several generalizations of the relative index theorem [8],[10], Donnelly [15], Anghel [1],[2], Brüning/Moscovici [7], Borisov/Müller/Schrader [6], Roe [26]. Applied to compact manifolds, the relative index theorem is equivalent to the excision property of the index. The aim of the present section is to give a K-theoretic variant of this theorem, which applies also for operators acting on A − C^{∗}-Clifford bundles, where A is a C^{∗}-algebra over the base field k = R, C. Such operators were considered first by Mis̆c̆enko/Fomenko [24].

Let M be a complete Riemannian manifold and A be a Z_{2}-graded C^{∗}-algebra. An A − C^{∗}-Clifford bundle E is a bundle of projective graded A-Hilbert-C^{∗}-modules together with a metric connection and a Clifford multiplication satisfying the Leibnitz rule and the compatibility with the scalar products of the fibres. If the C^{∗}-algebra A is not unital, then, by convention, a module V over A is called projective if 1 ∈ B(V) is compact (see Blackadar [4] for the definition of compact operators over A-Hilbert-C^{∗}-modules). Let D be the associated Dirac operator. We define Sobolev spaces H^{l}, l ≥ 0, (see Fomenko/Mis̆c̆enko [24] and (2)) which are A-Hilbert-C^{∗}-modules. We have D ∈ B(H^{l}, H^{l−1}), l ≥ 1. Our basic assumption is

Assumption 1 (invertibility at infinity) There is a function f ∈ C_{c}^{∞}(M) such that f ≥ 0 and (D^{2} + f)^{−1} ∈ B(H^{0}, H^{2}) exists.

The index ”_{c}” stands for compactly supported. We can then construct a Kasparov module (see Blackadar [4] for definitions) representing the index of D. In Lemma 1.8 we will give sense to the operator F := D(D^{2} + f)^{−1/2} ∈ B(H^{0})^{odd}.

Let C(M) be the C^{∗}-algebra of bounded continuous functions on M with the sup-norm (we consider real or complex functions depending on the context). Let C_{0}(M) be the closure of C_{c}^{∞}(M) in C(M). Define

C_{g}^{∞}(M) := {f ∈ C^{∞}(M) ∩ C(M) | |df| ∈ C_{0}(M, T^{∗}M)}

and let C_{g}(M) be the closure of C_{g}^{∞}(M) in C(M). There is a ∗-homomorphism C_{g}(M) → B(H^{0}) such that the function h ∈ C_{g}(M) acts as a multiplication operator. (H^{0}, F) is a Kasparov module over the pair of C^{∗}-algebras (C_{g}(M), A) by Proposition 1.13. Let the symbol M stand for all structures over M and [M] ∈ KK(C_{g}(M), A) be the class represented by (H^{0}, F). By Lemma 1.10 the class [M] does not depend on the choice of the function f ∈ C_{c}^{∞}(M).

The reason for using the rather complicated notions of Kasparovs KK-Theory is that we want to apply the relative index theorem in the case of graded C^{∗}-algebras. Until now, a reasonable extension of the index theory of Mis̆c̆enko/Fomenko to graded algebras has not been developed. In this section we will use only the definition of the KK-groups and the equivalence relations ’compact perturbation’ and ’operator homotopy’. Thus, nothing really deep is involved.

We will prove the following relative index theorem. Let E_{i} → M_{i}, i = 1, 2, be two A-C^{∗}-Clifford bundles with Dirac operators D_{i} being invertible at infinity. Let W_{i} ∪_{N_{i}} V_{i} be partitions of M_{i} where N_{i} are compact hypersurfaces. Assume that there is a commutative diagram of isomorphisms of all structures

Γ : E_{1|U(N_{1})} → E_{2|U(N_{2})}
↓  ↓
γ : U(N_{1}) → U(N_{2})
↑  ↑
γ_{|N_{1}} : N_{1} → N_{2}

where the U(N_{i}) are tubular neighbourhoods of N_{i}, i = 1, 2. We cut M_{i} at N_{i}, glue the pieces together interchanging the boundary components and obtain M_{3} := W_{1} ∪_{N} V_{2} and M_{4} := W_{2} ∪_{N} V_{1}. Moreover, we glue the bundles using Γ, which yields A-C^{∗}-Clifford bundles E_{3} → M_{3} and E_{4} → M_{4} and we assume that D_{i}, i = 3, 4 are again invertible at infinity. We define [M_{3}] and [M_{4}] as above.

The algebra C_{g}(M) is unital. Hence, there is an embedding i : k → C_{g}(M) and an induced map

i^{∗} : KK(C_{g}(M), A) → KK(k, A).

Set {M_{i}} := i^{∗}[M_{i}] ∈ KK(k, A) for i = 1, …, 4.

Theorem 1.2 (K-theoretic relative index theorem)

{M_{1}} + {M_{2}} − {M_{3}} − {M_{4}} = 0  (1)

In the present section we consider those applications which were our motivation for proving a K-theoretic relative index theorem. Let k := R, M^{n} be complete, Riemannian and spin, E be the real Clifford bundle with fibres isomorphic to the Clifford algebra C^{n,0} admitting a right C^{n,0}-action and V be a flat bundle of A-Hilbert-C^{∗}-modules over M. Define the C^{n,0} ⊗ A − C^{∗}-Clifford bundle S := E ⊗ V. Assume that there is a compact set K ⊂ M and a constant c > 0 such that for the scalar curvature s we have the estimate s_{|M\K} ≥ c. We show that under some condition of bounded geometry on V the operator D is invertible at infinity (see Lemma 1.18).

We solve two problems suggested to the author by S. Stolz. The first asserts that cutting and pasting at compact hypersurfaces with positive scalar curvature does not change the index of D (Theorem 1.16). The second problem is to construct, for any discrete group π, a group homomorphism

R_{n}(π) → KK_{n}(R, C_{r}^{∗}(π)),

where R_{n}(π) is a group of n-dimensional B = Bπ×BSpin manifolds with prescribed positive scalar curvature metric at the boundary ∂M.

In view of the applications in the second section note that all the results of the present section remain true if one replaces the Dirac operator associated to the A − C^{∗}-Clifford bundle by an A-equivariant Callias-type operator.

1.2 Analytical results

Let M be a complete Riemannian manifold and E be an A − C^{∗}-Clifford bundle with associated Dirac operator D. We form the completions H^{l}, l ≥ 0, of C_{c}^{∞}(M, E) with respect to the norms

∥ϕ∥_{l}^{2} = ∑_{k=0}^{l} ∫_{M} ∥D^{k}ϕ(x)∥^{2}, ϕ ∈ C_{c}^{∞}(M, E),  (2)

where the norm on the right-hand side is the pointwise norm coming from the A-Hilbert-C^{∗}-module structure of the fibres. The H^{l} are A-Hilbert-C^{∗}-modules with the A-valued scalar product

< ϕ, ψ >_{l}= ∑_{k=0}^{l} ∫_{M} < D^{k}ϕ(x), D^{k}ψ(x) > .

There is an analogue of Rellich’s theorem.

Lemma 1.3 (Mis̆c̆enko/Fomenko,[24]) For any f ∈ C_{c}^{∞}(M) the multiplication operator f : H^{l} → H^{k} is compact for k < l.

Since the space of compact operators is closed in the norm topology the compactness of the multiplication operators with functions will extend to larger classes of functions. D extends to an operator D ∈ B(H^{l}, H^{l−1}) for each l ≥ 1. The symmetry property of D extends to H^{1}:

< ψ, Dϕ >=< Dψ, ϕ >, ψ, ϕ ∈ H^{1}.

In the present subsection we carry out certain calculations which, at a first glance, may appear as trivial applications of the spectral theory of self-adjoint operators. However, one should be careful since no such theory has been developed for unbounded self-adjoint operators over an A-Hilbert-C^{∗} module. We will give fairly detailed arguments for the first few lemmas. The estimates and compactness claimed in the later proofs follow by similar arguments using suitable factorizations.

Let D be invertible at infinity and set B := (D^{2} + f)^{−1}.

Lemma 1.4 We have

0 < d := inf_{ψ∈H^{2},∥ψ∥_{0}=1}(∥Dψ∥_{0}^{2} + ∥\sqrt{f}ψ∥_{0}^{2}).

Proof: Let ψ ∈ H^{2} with ∥ψ∥_{0} = 1. Then

1 = ∥ < ψ, ψ >_{0} ∥
= ∥ < (D^{2} + f)Bψ, ψ >_{0} ∥
= ∥ < DBψ, Dψ >_{0} + < \sqrt{f}Bψ, \sqrt{f}ψ >_{0} ∥
≤ ∥DB∥_{B(H^{0})}∥Dψ∥_{0} + sup_{x∈M}|\sqrt{f(x)}|∥B∥_{B(H^{0})}∥\sqrt{f}ψ∥_{0}
≤ C(∥Dψ∥_{0}^{2} + ∥\sqrt{f}ψ∥_{0}^{2})^{1/2}

for some C < ∞. The lemma follows with d ≥ C^{−2}. □

Lemma 1.5 For λ ≥ 0 the operator

R(λ) := (D^{2} + f + λ^{2})^{−1} ∈ B(H^{0}, H^{2})

exists and satisfies the estimates

∥R(λ)∥_{B(H^{0})} ≤ (d + λ^{2})^{−1}  (3)
∥D^{2}R(λ)∥_{B(H^{0})} ≤ C  (4)

with C < ∞ independent of λ.

Proof: Assume that the lemma holds for all 0 ≤ λ ≤ Λ. This is certainly true for Λ = 0. Then, for all λ with |λ^{2} − Λ^{2}| < (d + Λ^{2}),

R(λ) = ∑_{i=0}^{∞} R(Λ)^{i}(Λ^{2} − λ^{2})^{i}R(Λ) ∈ B(H^{0}, H^{2}).

The estimate (3) follows from

∥(D^{2} + f + λ^{2})ψ∥_{0} ≥ (d + λ^{2})∥ψ∥_{0}, ∀ψ ∈ H^{2}

(see Lemma 1.4) since H^{2} ⊂ H^{0} is dense. (4) follows from (3) by

∥D^{2}R(λ)ϕ∥_{0} ≤ ∥ϕ∥_{0} + ∥(f + λ^{2})R(λ)ϕ∥_{0} ≤ C∥ϕ∥_{0}, ∀ϕ ∈ H^{0}.

□ Lemma 1.6 We have DR(λ) ∈ B(H^{0}) and the estimate

∥DR(λ)∥_{B(H^{0})} ≤ C(d + λ^{2})^{−1/2}

holds for some C < ∞ independent of λ ≥ 0.

Proof: There is a C < ∞ such that for all ϕ ∈ H^{0} with ∥ϕ∥_{0} = 1

∥DR(λ)ϕ∥_{H^{0}}^{2} = ∥ < DR(λ)ϕ, DR(λ)ϕ > ∥_{0}
= ∥ < R(λ)ϕ, D^{2}R(λ)ϕ > ∥_{0}
≤ ∥R(λ)ϕ∥_{0}∥D^{2}R(λ)ϕ∥_{0}
≤ C^{2}(d + λ)^{−1}

holds by (3),(4). Thus, DR(λ) is bounded and satisfies the desired estimate. We have to verify that it has an adjoint. Formally, (DR(λ))^{∗} = R(λ)D, but this does not make sense since D is not defined on all H^{0}. First we check that R(λ)^{∗} = R(λ). For all ϕ ∈ H^{2}, ψ ∈ H^{0}

< R(λ)(D^{2} + f + λ^{2})ϕ, ψ >_{0} − < (D^{2} + f + λ^{2})ϕ, R(λ)ψ >_{0}
= < ϕ, ψ >_{0} − < ϕ, (D^{2} + f + λ^{2})R(λ)ψ >_{0}
= 0

using the symmetry of D. Any ρ ∈ H^{0} has the representation ρ = (D^{2} + f + λ^{2})ϕ for some ϕ ∈ H^{2}. Thus, for any ρ, ψ

< R(λ)ρ, ψ >_{0}=< ρ, R(λ)ψ >_{0}

and hence R(λ)^{∗} = R(λ). Next we show

(DR(λ))^{∗} = DR(λ) + R(λ)grad(f)R(λ),

where grad(f) is the Clifford multiplication by the gradient vector field of f. For all ψ, ϕ ∈ H^{0} we have

< ψ, DR(λ)ϕ >_{0} = < (D^{2} + f + λ^{2})R(λ)ψ, DR(λ)ϕ >_{0}
= < D^{2}R(λ)ψ, DR(λ)ϕ >_{0} + < (f + λ^{2})R(λ)ψ, DR(λ)ϕ >_{0}
= < DR(λ)ψ, D^{2}R(λ)ϕ >_{0} + < R(λ)ψ, (f + λ^{2})DR(λ)ϕ >_{0}
= < DR(λ)ψ, ϕ >_{0} − < DR(λ)ψ, (f + λ^{2})R(λ)ϕ >_{0}
+ < R(λ)ψ, (f + λ^{2})DR(λ)ϕ >_{0}
= < DR(λ)ψ, ϕ >_{0} − < R(λ)ψ, [D, (f + λ^{2})]R(λ)ϕ >_{0}
= < (DR(λ) + R(λ)grad(f)R(λ))ψ, ϕ >_{0} .

This shows the claim. □

Lemma 1.7 For any ψ ∈ H^{1}

[D, R(λ)]ψ = −R(λ)grad(f)R(λ)ψ.

Proof: For all ψ, ϕ ∈ H^{1} we have by Lemma 1.6

< ψ, R(λ)Dϕ >_{0} = < DR(λ)ψ, ϕ >_{0}
= < ψ, (DR(λ) + R(λ)grad(λ)R(λ))ϕ >_{0} .

□

Lemma 1.8 For any ψ ∈ H^{1} the integral

F ψ := \frac{2}{π} ∫_{0}^{∞} DR(λ)ψdλ  (5)

converges in norm and defines an operator F ∈ B(H^{0})^{odd}.

Proof: For λ ≥ 0

R(λ) = \frac{1}{λ^{2}} B(B + 1/λ^{2})^{−1},

as it can be seen by composing with D^{2} + f + λ^{2}. Applying the functional calculus for bounded self-adjoint operators in the C^{∗}-algebra B(H^{0}) we have

\frac{2}{π} ∫_{0}^{∞} R(λ)dλ = \frac{2}{π} ∫_{0}^{∞} B(B + 1/λ^{2})^{−1} \frac{dλ}{λ^{2}}
= \frac{2}{π} ∫_{0}^{∞} B(B + z^{2})dz
= B^{1/2},

where the integrals converge in norm. There is a C < ∞ such that, for any ψ ∈ H^{1}, it holds

∥B^{1/2}Dψ∥_{0}^{2} = ∥ < B^{1/2}Dψ, B^{1/2}Dψ >_{0} ∥
= ∥ < BDψ, Dψ >_{0} ∥
= ∥ < (DB + Bgrad(f)B)ψ, Dψ >_{0} ∥
= ∥ < D^{2}Bψ, ψ >_{0} + < DBgrad(f)Bψ, ψ >_{0} ∥
≤ C∥ψ∥_{0}^{2}.

B^{1/2}D extends by continuity to a bounded operator L : H^{0} → H^{0}. By Lemma 1.7 we can write

F ψ := Lψ − \frac{2}{π} ∫_{0}^{∞} R(λ)grad(f)R(λ)dλψ.  (6)

The integral on the r.h.s is operator norm convergent by (3) and defines a bounded operator in B(H^{0}). It remains to show that L ∈ B(H^{0}), i.e. that L admits an adjoint. For any ϕ ∈ H^{1}, ψ ∈ H^{2} it holds

< Lϕ, ψ >_{0} = < B^{1/2}Dϕ, ψ >_{0}
= < Dϕ, BB^{1/2}(D^{2} + f)ψ >_{0}
= < ϕ, DBB^{1/2}(D^{2} + f)ψ >_{0} .

The operator DBB^{1/2}(D^{2} + f) : H^{2} → H^{0} satisfies

∥(DBB^{1/2}(D^{2} + f)ψ∥_{0} ≤ ∥L∥_{B(H^{0})}∥ψ∥_{0}

for all ψ ∈ H^{2} and extends by continuity to L^{∗} : H^{0} → H^{0}. Hence, L ∈ B(H^{0}). □

Lemma 1.9 F − F^{∗} ∈ K(H^{0})

Proof: By Lemma 1.8 and 1.7, for all ψ ∈ H^{1},

(F^{∗} − F)ψ = \frac{2}{π} ∫_{0}^{∞} R(λ)grad(f)R(λ)ψdλ.

By Lemma 1.5

∥R(λ)grad(f)R(λ)∥_{B(H^{0})} ≤ C(d + λ^{2})^{−2}

for some C < ∞ independent of λ. In order to see that the integrand is compact use Lemma 1.3 and the factorization.

H^{0} ⟶^{R(λ)} H^{2} ⟶^{grad(f)} H^{0} ⟶^{R(λ)} H^{0},

where the middle arrow is compact. A norm convergent integral of compact operators is compact by the closedness of the ideal of compact operators. □

Lemma 1.10 Let f, f_{1} ∈ C_{c}^{∞}(M), f, f_{1} ≥ 0, be such that

(D^{2} + f)^{−1}, (D^{2} + f_{1})^{−1} ∈ B(H^{0}, H^{2})

exist. Define F, F_{1} using f, f_{1}. Then F − F_{1} ∈ K(H^{0}).

Proof: We claim

DR(λ) − DR_{1}(λ) ∈ K(H^{0})  (7)

and

∥DR(λ) − DR_{1}(λ)∥_{B(H^{0})} ≤ C(d + λ^{2})^{−3/2}  (8)

for some C < ∞ independent of λ. The Lemma now follows by integration.

We have

R(λ) − R_{1}(λ) = R(λ)(f_{1} − f)R_{1}(λ)

as it can be seen by composing with D^{2} + f + λ^{2}. Property (7) follows from Lemma 1.3 and the factorization

H^{0} ⟶^{R_{1}(λ)} H^{2} ⟶^{f_{1}−f} H^{0} ⟶^{DR(λ)} H^{0},

where the middle arrow is compact. By (3) and (4) we have

∥DR(λ) − DR_{1}(λ)∥_{B(H^{0})} ≤ ∥DR(λ)∥_{B(H^{0})}sup_{x∈M}|f_{1}(x) − f(x)|∥R_{1}(λ)∥_{B(H^{0})}
≤ C(d + λ^{2})^{−3/2}

for some C < ∞ independent of λ. □

Lemma 1.11 F^{2} − 1 ∈ K(H^{0})

Proof: By (5), the compactness of the operator defined by the integral on the r.h.s of (6) and Lemma 1.9 it is enough to show that LL^{∗} − 1 ∈ K(H^{0}). For ψ ∈ H^{2} we have

(LL^{∗} − 1)ψ = B^{1/2}D^{2}BB^{1/2}(D^{2} + f)ψ − ψ
= B(D^{2} + f)ψ − ψ − B^{1/2}fBB^{1/2}(D^{2} + f)ψ
= −B^{1/2}fB^{1/2}ψ.

Let ”∼” denote equality modulo compact operators.

fB^{1/2} = \frac{2}{π} ∫_{0}^{∞} fR(λ)dλ
= \frac{2}{π} ∫_{0}^{∞}(R(λ)f + R(λ)(Dgrad(f) + grad(f)D)R(λ))dλ
∼ B^{1/2}f

Hence, LL^{∗} − 1 ∼ −fB ∈ K(H^{0}) by Lemma 1.3. □

Lemma 1.12 Let h ∈ C_{g}(M). Then [F, h] ∈ K(H^{0}).

Proof: h can be approximated in the supremum norm by functions h_{n} ∈ C_{g}^{∞}(M). Thus, it is enough to show the Lemma for h ∈ C_{g}^{∞}(M). Using Lemma 1.8 for any ψ ∈ H^{1}

[F, h]ψ = \frac{2}{π} ∫_{0}^{∞}[DR(λ), h]ψdλ
= \frac{2}{π} ∫_{0}^{∞} (D[R(λ), h]ψ + grad(h)R(λ)ψ) dλ,  (9)

where we employ that h : H^{1} → H^{1} is bounded. It takes some trouble to write down expressions which, in order to be well-defined, do not involve bounds on the second derivatives of h. It holds

[R(λ), h] = R(λ)h − hR(λ)
= R(λ)h(D^{2} + f + λ^{2})R(λ) − D^{2}R(λ)hR(λ) − (f + λ^{2})R(λ)hR(λ)
D^{2}R(λ)hR(λ) = DR(λ)DhR(λ) − DR(λ)grad(f)R(λ)hR(λ)
D[R(λ), h] = DR(λ)h(D^{2} + f + λ^{2})R(λ) − D^{2}R(λ)DhR(λ)
−D^{2}R(λ)grad(f)R(λ)hR(λ) − D(f + λ^{2})R(λ)hR(λ).

We drop all terms that are integrable and compact and obtain the remainder

S(λ) := DR(λ)hD^{2}R(λ) − D^{2}R(λ)DhR(λ)
= DR(λ)hD^{2}R(λ) − D^{2}R(λ)hDR(λ) − D^{2}R(λ)grad(h)R(λ)
= DR(λ)hD^{2}R(λ) − DR(λ)DhDR(λ)
+DR(λ)grad(f)R(λ)hDR(λ) − D^{2}R(λ)grad(h)R(λ)
= −DR(λ)grad(h)DR(λ) + DR(λ)grad(f)R(λ)hDR(λ) − D^{2}R(λ)grad(h)R(λ).

All terms are compact and integrable in norm. The second integrand in (9) is compact and integrable, too. □

Proposition 1.13 (H^{0}, F) is a Kasparov module over the pair of C^{∗}-algebras (C_{g}(M), A). The class [H^{0}, F] ∈ KK(C_{g}(M), A)is independent of the choice of f.

Proof: We have shown

F ∈ B(H^{0})^{odd} Lemma 1.8
F − F^{∗} ∈ K(H^{0}) Lemma 1.9
F^{2} − 1 ∈ K(H^{0}) Lemma 1.11
[F, h] ∈ K(H^{0}) ∀h ∈ C_{g}(M) Lemma 1.12
F − F_{1} ∈ K(H^{0}) F, F_{1} defined with f, f_{1} ∈ C_{c}^{∞}(M) Lemma 1.10

□ Recall the notation and geometric situation of the K-theoretic relative index theorem.

Theorem 1.14 (K-theoretic relative index theorem)

{M_{1}} + {M_{2}} − {M_{3}} − {M_{4}} = 0  (10)

Proof: The idea of the proof is as follows. The l.h.s. of (10) is represented by (H, F), where

H = H_{1} ⊕ H_{2} ⊕ H_{3}^{op} ⊕ H_{4}^{op}

and

F = F_{1} ⊕ F_{2} ⊕ F_{3} ⊕ F_{4}.

We will construct an operator X ∈ B(H) with degX = 1, X^{2} = 1, X^{∗} = X, [X, F] := XF + FX ∈ K(H). Thus

[H, F] ∈ Im(β^{∗}) : KK(C^{0,1}, A) → KK(k, A)

induced by the embedding β : k → C^{0,1}. We show β^{∗} = 0 in Lemma 1.15. We choose smooth cut-off functions χ_{W_{i}}, χ_{V_{i}}, i = 1, 2, with

supp χ_{W_{i}} ⊂ W_{i} ∪ U(N_{i})
supp χ_{V_{i}} ⊂ V_{i} ∪ U(N_{i})
χ_{V_{i}}^{2} + χ_{W_{i}}^{2} = 1
χ_{W_{1}|U(N_{1})} = γ^{∗}χ_{W_{2}|U(N_{2})}
χ_{V_{1}|U(N_{1})} = γ^{∗}χ_{V_{2}|U(N_{2})}.

Note that L^{2}(W_{1} ∪ U(N_{1}), E) is contained in H_{1} as well as in H_{3}. We can define a : H_{1} → H_{3} by first multiplying by χ_{W_{1}} and then applying this identification of subspaces. Then a^{∗}a and aa^{∗} are simply the multiplications by χ_{W_{1}}^{2} on H_{1} and H_{3}, respectively. We define b : H_{1} → H_{4}, c : H_{2} → H_{3} and d : H_{2} → H_{4} analogously. Let z ∈ B(H) be the Z_{2}-grading and let

X := z ( \begin{matrix}0 & 0 & −a^{∗} & −b^{∗} \\ 0 & 0 & −c^{∗} & d^{∗} \\ a & c & 0 & 0 \\ b & −d & 0 & 0 \end{matrix} ) ∈ B(H).

Then degX = 1, X^{2} = 1, X^{∗} = X. It remains to show

XF + FX ∈ K(H).  (11)

It is enough to verify the compactness of operators of the type

a^{∗}F_{3} − F_{1}a^{∗} ∈ B(H^{3}, H^{1}).

Let

h := L^{2}(W_{1}, E_{1|W_{1}}) ⊕ L^{2}(V_{1}, E_{1|V_{1}}) ⊕ L^{2}(V_{2}, E_{2|V_{2}}).

There are canonical embeddings H_{1}, H_{3} → h. F_{1}, F_{3} act naturally on h after extension by zero to the subspace where they were not defined. Let χ := χ_{W_{1}} and ρ ∈ C_{c}^{∞}(W_{1} ∪ U(N_{1})) be such that ρχ = χ. The non-negative functions that were used to define F_{i} are denoted by f_{i}, i = 1, 2, 3, 4. We can assume that χf_{1} = χf_{3}.

We have

χF_{3} − F_{1}χ = \frac{2}{π} ∫_{0}^{∞}(χD_{3}R_{3}(λ) − D_{1}R_{1}(λ)χ)dλ
= \frac{2}{π} ∫_{0}^{∞}(χD_{3}R_{3}(λ) − D_{1}R_{1}(λ)χ(D_{3}^{2} + f_{3} + λ^{2})R_{3}(λ))dλ
= \frac{2}{π} ∫_{0}^{∞}(χD_{3}R_{3}(λ) − D_{1}R_{1}(λ)χ(D_{1}^{2} + f_{1} + λ^{2})ρR_{3}(λ))dλ
= \frac{2}{π} ∫_{0}^{∞}(χD_{3}R_{3}(λ) − D_{1}χR_{3}(λ) − D_{1}R_{1}(λ)[χ, D_{1}^{2}]ρR_{3}(λ))dλ
= \frac{2}{π} ∫_{0}^{∞}(−grad(χ)R_{3}(λ) + D_{1}R_{1}(λ)(D_{1}grad(χ) + grad(χ)D_{1})ρR_{3}(λ))dλ.

The first term of the integrand is compact and integrable in the norm by Lemma 1.5. The second term can be written as

D_{1}^{2}R_{1}(λ)grad(χ)R_{3}(λ)+D_{1}R_{1}(λ)grad(f_{1})R_{1}(λ)grad(χ)R_{3}(λ)+D_{1}R_{1}(λ)grad(χ)D_{3}R_{3}(λ)

and every term is integrable and compact, too. This proves the desired compactness in (11). The theorem follows now from the following lemma.

Lemma 1.15 The embedding β : k → C^{0,1} induces the zero-map β^{∗} : KK(C^{0,1}, A) → KK(k, A).

Proof: Let (H, F) represent an element in KK(C^{0,1}, A) and let X ∈ C^{0,1} be the odd generator. Then X ∈ B(H). Consider the operator homotopy

F_{t} = cos(t)F + sin(t)X.

We have F_{0} = F, F_{π/2} = X and hence [H, F] = [H, X] = 0 ∈ KK(k, A). □

1.3 Cutting and pasting invariance

Let M^{n} be a closed Riemannian spin manifold and V → M be a flat bundle of projective modules over some real C^{∗}- algebra A. We consider the twisted Dirac operator D : C^{∞}(M, S) → C^{∞}(M, S) with S := E ⊗ V, where E is the real Clifford bundle associated to the spin structure of M admitting a right C^{n,0}-action. Let {M} ∈ KK(R, C^{n,0} ⊗ A) be the index of D.

Let N ⊂ M be a compact two-sided hypersurface. We assume a tubular neighbourhood U of N with a product metric. Let

Γ : V_{|N} → V_{|N}
↓  ↓
γ : N → N

be a commutative diagram of isomorphisms of all structures, in particular γ is an isometry. Then we can cut M at N and glue together again using γ in order to obtain M_{1}. Moreover, we can glue the bundle S_{|M\N} using Γ to obtain a bundle S_{1} over M_{1} with Dirac operator D_{1} giving {M_{1}}.

Theorem 1.16 If the induced Riemannian metric on N has positive scalar curvature, then {M} = {M_{1}} ∈ KK(R, A ⊗ C^{n,0}).

Proof: We add to M the manifold R × N, which has a Dirac operator induced from the restriction S_{|N}. This Dirac operator is invertible (Lemma 1.18) and hence {M} = {M} + {R × N}. U has two boundary components N_{±}. We partition M into M \ U and U, and R × N into (−∞, −1] × N ∪ [1, ∞) × N and [−1, 1] × N. There are identifications of the cutting hypersurfaces id : N_{−} → {−1} × N and γ : N_{+} → {1} × N. Analogously we identify the bundles using Γ at the positive boundary part. We cut at these hypersurfaces and glue together, interchanging U and [−1, 1] × N and using the identifications given above, and obtain M_{1} ∪ R × N together with the corresponding Dirac operators. Again the cylinder does not contribute to the index. Applying the K-theoretic relative index theorem we obtain {M} = {M_{1}}. □

Since {M}, {M_{1}}, do not depend on the choice of the Riemannian metric on M, M_{1} it is enough to require that N can carry a metric of positive scalar curvature such that γ is an isometry. The compactness of M is not important. It is enough to have invertibility of the Dirac operators at infinity. In the complex case a similar theorem is true for A = C without the assumption that N carries a positive scalar curvature metric. This can be seen by the local index theorem. In the real case this theorem is not true even for A = R without the assumption as an example on S^{1} shows. The author thanks K. Wojciechowski for pointing out an error in an earlier version of that paper.

1.4 Cobordisms of positive scalar curvature metrics

Fix a finitely generated group π. Any spin manifold N with π_{1}(N) = π gives rise to a B := BSpin × Bπ-manifold (see [30]). The B structure

f : N → B

is given by the product of the classifying maps of the spin structure and of the universal cover of N. Consider the set S_{n}(π) of tuples (M, N, F, h) where (M, N, F) is an n-dimensional B-manifold with B-structure F, with boundary N = ∂M and h is a positive scalar curvature metric on N. S is a semigroup under disjoint union. Let ∼ be the equivalence relation given by B-bordism. A B-bordism of (M, N, F, h) and (M_{1}, N_{1}, F_{1}, h_{1}) consists of a B-manifold (W, N, N_{1}, Φ) with boundaries (N, F_{|N}) and (N_{1}, F_{1|N_{1}}), a positive scalar curvature metric g on W which is a product metric near ∂W and restricts to h, h_{1} at N, N_{1}, and a B-manifold (V, Ψ) with boundary (M ∪_{N} W ∪_{N_{1}} M_{1}, (F, Φ, F_{1})). Note that R_{n}(π) := S_{n}(π)/ ∼ is a group. A similar group has been considered by B. Hajduk. It is a special case of a construction due to S. Stolz [29]. Let C_{r}^{∗}(π) be the reduced real group C^{∗}-algebra of π.

Theorem 1.17 There is a canonical homomorphism R_{n}(π) → KK_{n}(R, C_{r}^{∗}(π)).

Proof: Let (M, N, F, h) ∈ S_{n}(π). Choose a metric on M such that it is a product metric near N and restricts to h. Glue a metric cylinder [0, ∞) × N along the boundary of M and obtain the complete manifold M̄. Extend F constantly over the cylinder. F^{∗}Eπ is a π-principal fibre bundle. Associate C_{r}^{∗}(π) and obtain a flat bundle V with fibre C_{r}^{∗}(π) using the canonical action of π on C_{r}^{∗}(π) from the left. Let E be the real Clifford bundle with fibre C^{n,0} associated to the spin structure of M̄ and form S := E ⊗ V. S is a (C_{r}^{∗}(π) ⊗ C^{n,0}) − C^{∗}-Clifford bundle. Let D be the associated Dirac operator. The scalar curvature is positive at infinity.

Lemma 1.18 D is invertible at infinity

We can form {M̄} ∈ KK(R, C^{n,0} ⊗ C_{r}^{∗}(π)). Clearly, the map associating the class {M̄} to (M, N, F, h) ∈ S_{n}(π) is additive. We must show that it factors through R_{n}(π). Let (W, N, Φ) be a B-manifold with boundary (N, F_{|N}), g be a positive scalar curvature metric on W which is a product metric near ∂W and restricts to h on N, and (V, Ψ) be a B-manifold with boundary (M ∪_{N} W, (F, Φ)). Let L := M̄ ∪W̄ and L̃ := W ∪_{N} M ∪ R × N. Then {L} = {M̄} and {L̃} = {W ∪_{N} M} since, on the remaining components, there are positive scalar curvature metrics and the Dirac operators are invertible there. By the relative index theorem {L} = {L̃}. But {L̃} = 0 since the Dirac operator on W ∪_{N} M is zero-bordant. Hence {M̄} = 0. This proves the theorem. The idea for this construction was suggested by S. Stolz. □

Proof of 1.18: (see also [28]). Because of the presence of the C^{∗}-algebra we cannot simply apply the standard theory of self-adjoint operators. The main point is to verify that the image of D^{2} + f : H^{2} → H^{0} is dense for a suitable f ∈ C_{c}^{∞}(M̄) .

As a first step we show that D + λϵ is invertible for λ ≠ 0. We realize V as a direct summand of a bundle W whose complexification is trivial. Consider the embedding of Hilbert-C^{∗}-modules

C_{r}^{∗}(π) → C_{r}^{∗}(π) ⊗ lR^{2}

given by f → f ⊗ e_{1}, where e_{1} is the first vector of an orthonormal base of lR^{2}. Let W := F^{∗}Eπ ×_{π} C_{r}^{∗}(π) ⊗ lR^{2}. Then V is a direct summand in W. Let W^{c} := W ⊗_{R} C be the complexification of W and V^{c} be the complexification of V. These complexifications carry real structures generally denoted by σ. By the result of [23] W^{c} is trivial. Choose an isomorphism

Φ : W_{|M}^{c} → M × C_{r}^{∗}(π) ⊗_{R} lC^{2} =: U

and extend it constantly over the cylindrical end. Since M is compact, we can choose Φ smooth and C^{1}-bounded, i.e.

∇^{W} Φ − Φ∇^{U} ∈ C_{b}(M̄, T^{∗}M̄ ⊗ B(W^{c}, U^{c})).  (12)

Let D_{U} and D_{W} be the complex Dirac operators associated to E^{c} ⊗ U and E^{c} ⊗ W^{c}. D_{W} commutes with the real structure σ of E^{c} ⊗ W^{c}. We have (D_{α} + λϵ)^{2} = D_{α}^{2} + λ^{2} and for any ϕ ∈ H^{2} it holds

∥(D_{α} + λϵ)ϕ∥^{2} = ∥D_{α}ϕ∥^{2} + λ^{2}∥ϕ∥^{2} ≥ λ^{2}∥ϕ∥^{2},  (13)

where α = U, W, E^{c}, ∅. Since U is trivial, D_{U} + λϵ is invertible for any λ ≠ 0, (D_{U} + λϵ)^{−1} = (D_{E^{c}} + λϵ)^{−1} ⊗ 1_{U}. We have a Neumann series

(Φ(D_{W} + λϵ)Φ^{−1}))^{−1} = (D_{U} + λϵ + (ΦD_{W} Φ^{−1} − D_{U}))^{−1}
= ∑_{i=0}^{∞}(−1)^{i} ((D_{U} + λϵ)^{−1}(ΦD_{W} Φ^{−1} − D_{U}))^{i} (D_{U} + λϵ)^{−1},

where the sum converges for |λ| large enough because of (12) and (13). Thus, D_{W} + λϵ, is invertible for large λ and by the estimate (13) we can conclude that D_{W} +λϵ is invertible for any λ ≠ 0 using again an argument with a Neumann series. Since D_{W} + λϵ commutes with the real structure, also its restriction to the real subspace is invertible. As D + λϵ is a direct summand it is invertible, too.

We show now that D is invertible at infinity. D^{2} + λ^{2} is invertible for λ ≠ 0. Let f ∈ C_{c}^{∞}(M) be positive and such that f + τ/4 ≥ c > 0 where τ is the scalar curvature of M. We must show that D^{2} + f is invertible too. We have the estimate

∥(D^{2} + λ^{2} + f)ϕ∥ = ∥(Δ + τ/4 + λ^{2} + f)ϕ∥ ≥ c∥ϕ∥

uniformly in λ. Again using a Neumann series we can produce an inverse of D^{2} + f ∈ B(H^{0}, H^{2}). □

2 Index theory for Callias type operators

2.1 Introduction

Let S → M be a Clifford bundle (see Lawson/Michelsohn [21] for a definition) over a complete Riemannian manifold M. The associated Dirac operator with the domain C_{c}^{∞}(M, S) is essentially self-adjoint on the Hilbert space H^{0} := L^{2}(M, S) (Chernoff [13]). In order to apply the Fredholm index theory one has to assume 0 ∉ σ_{ess}(D) (σ_{ess}(D) denotes the essential spectrum of D). This condition is not satisfied in general.

There is a way of obtaining another essentially self-adjoint operator B := D + Φ by adding a suitable self-adjoint bundle endomorphism Φ ∈ C^{∞}(M, End(S)) such that 0 ∉ σ_{ess}(B). B is called a Callias-type operator (first examples were considered by Callias [12]). In this part of the paper we are concerned with the index theory of Callias-type operators.

We will deal with the question of finding suitable Φ and with the computation of ind B. The definition of the index of B involves an auxiliary Z_{2}-grading (B is self-adjoint). There is quite an effective and surprisingly easy method for computing indB. All the information on the index of B is contained in a tubular neighbourhood of a compact hypersurface N ⊂ M cutting off the part where Φ is invertible. The computation can be reduced to an index computation on a compact manifold. In the example of Callias, M = R^{n}, n odd and ind B is expressed as the index of a Dirac operator on a large sphere S^{n−1} ⊂ R^{n} twisted with a bundle of the positive eigenvectors of Φ on S^{n−1}. The result of Callias was generalized by Anghel [1],[2] to general odd-dimensional manifolds. Another example is Roe’s index theorem for partitioned manifolds [25], which has a reformulation in terms of operators of Callias-type due to Higson [18].

All examples mentioned above start initially with a Dirac operator on an ungraded Clifford bundle E → M. The grading enters later by forming matrices. They give non-trivial results on odd-dimensional manifolds.

There is an even-dimensional counterpart of the theory which starts with Z_{2}- graded Dirac operators. The index problem for Callias-type operators on evendimensional manifolds has not been considered before.

In the present paper we adopt a more structural point of view (in contrast to [2], which focuses on the computation of the index). We will construct the endomorphisms Φ out of K-classes on the boundary ∂_{h}M of a certain compactification M̄ of M. This compactification was introduced by Higson [17] and described in [27]. ∂_{h}M is called the Higson corona. It is the spectrum of the C^{∗}-algebra C_{g}(M)/C_{0}(M). Taking the index of B := D + Φ we obtain a homomorphism u : K^{0}(∂_{h}M) → Z if D is ungraded and g : K^{1}(∂_{h}M) → Z if D is graded.

The maps u and g are closely related with two other constructions, namely the spectral flow pairing s : K_{c}^{1}(M) → Z (constructed with an ungraded operator) and the relative index pairing r : K_{c}^{0}(M) → Z (constructed with a graded Dirac operator). All these maps fit into the following commutative diagram which consists of the long exact sequence of complex K-theory associated to the short exact sequence of C^{∗}-algebras

0 → C_{0}(M) → C_{g}(M) → C(∂_{h}M) → 0

completed with the maps s,r,u,g to Z.

[figure: Z; Z; K^{1}(∂_{h}M); K_{g}^{1}(M); K_{c}^{1}(M); K_{c}^{0}(M); K_{g}^{0}(M); K^{0}(∂_{h}M); exp; ∂; r; s; u; g]

In order to save notations we have employed the identifications K_{g}^{∗}(M) := K_{∗}(C_{g}(M)), K_{c}^{∗}(M) = K_{∗}(C_{0}(M)) and K^{∗}(∂_{h}M) := K_{∗}(C(∂_{h}M)).

It is very convenient to use the language of KK-theory. Thus, we associate to B := D + Φ a Kasparov module which represents a class [M] ∈ KK(C_{g}(M), C) and the restriction {M} ∈ KK(C, C) = Z ( we again use the convention that the symbol M stands for all structures over M). We use the most general equivalence relation given by homotopy (see Blackadar [4], Jensen/Thomsen [20] for definitions). We will only need the definition of the KK-theory and the characterization (not the existence) of the intersection product by connections. Note, that C_{g}(M) is not separable.

The Kasparov module associated to B := D + Φ is an explicit example of a Kasparov intersection product [Φ] ⊗ [D], where [D] is a K-homology class given by D and [Φ] is a K-class defined by Φ. We will show this in the corresponding places.

We will compute the maps g and u in terms of the index of Dirac operators on certain compact manifolds. Our main computational tools are the deformation invariance of {M}, vanishing theorems for {M} (e.g. if Φ is invertible or if D is itself Fredholm) and the relative index theorem.

The idea of the index computation for a Callias-type operator is to use cutting and pasting and the vanishing theorems in order to reduce the problem to a cylinder over some compact hypersurface N. There, one can use separation of variables. This method was previously used by Roe [25], Higson [18] and Anghel [1], [2].

There is a close relation between Callias-type operators and the boundary map in K-homology considered by Baum/Douglas/Taylor [3]. We can reprove the results of [3] modulo torsion. In a similar fashion the exotic index theory of Roe [27] is related to the index theory of Callias-type operators.

A large part of our methods also work in case of real operators and of operators being equivariant with respect to the action of some C^{∗}-algebra A. The index of B is then an element of K_{∗}(A) (or of KO_{∗}(A) in the real case). Difficulties arise in the case when A is infinite-dimensional. There are two critical points that have to be resolved. The first one is to verify that D^{2} + 1 is invertible. This is not obvious (at least to the author) and equivalent to the density of the image of D^{2} + 1. We prove a special case in Lemma 1.18. The second problem arises in our computations on the cylinder. Here, we explicitly identify kernels of two operators living on different spaces and want to conclude that their indices coincide. This is true if the kernels are complemented projective modules. However, in general this is not the case and one has to interpret the index as in Fomenko/Mis̆c̆enko [24] (assuming A ungraded to avoid even more complications). We hope to solve these problems (at least partially) in a forthcoming paper.

For the purpose of the present paper we consider examples of Callias-type operators being equivariant with respect to some real Clifford algebra A := C^{k,0}. Recall that C^{k,l} is the real or complex Clifford algebra (depending in the context) generated by e_{1}, …, e_{k+l} subject to the relations

e_{i}e_{j} + e_{j}e_{i} = \{ \begin{matrix}0 & i ≠ j \\ −2 & i = j ≤ k \\ 2 & l + k ≥ i = j > k \end{matrix} ,

e_{j}^{∗} = −e_{j}, j ≤ k and e_{j}^{∗} = e_{j}, l + k ≥ j > k.

In order to keep the paper in a reasonable size we have decided to be somewhat sketchy with the analytical details of the proofs. While, in the first section, we have to overcome serious analytical problems due to the presence of infinite-dimensional C^{∗}-algebras, in the second part the arguments are much more standard. In several places we claim the compactness of a certain operator, that can be verified using the same methods as in the first section. We freely use product structures for Dirac operators near hypersurfaces, where we perform cutting and pasting. The details can be found e.g in Booß/Wojciechowski [5].

2.2 The Higson corona and admissible endomorphisms

Let M be a complete Riemannian manifold with Dirac operator D acting on sections of a complex, graded Clifford bundle S → M. We want to construct admissible Φ ∈ Γ(M, End(S)) out of K-classes living on some compactification M̄ of M. An inspection of the constructions below shows that the largest compactification we can work with is the compactification by the Higson corona ∂_{h}M (introduced by Higson [17], see also Roe [27]). K-classes living on smaller compactifications can be pulled back to ∂_{h}M and we can apply our constructions to them, too.

Recall the definition of the C^{∗}-algebras C_{0}(M) and C_{g}(M) from Section 1.1. There is an exact sequence

0 → C_{0}(M) → C_{g}(M) → C(∂_{h}M) → 0,

where C(∂_{h}M) := C_{g}(M)/C_{0}(M). The topological space ∂_{h}M := spec C(∂_{h}M) maximal ideal space of the C^{∗}-algebra C(∂_{h}M). M̄ := spec C_{g}(M) = M ∪ ∂_{h}M is the smallest compactification to which all functions of C_{g}(M) extend continuously.

The Higson corona is rather large. It maps to any smaller compactification defined by a subalgebra of C_{g}(M), in particular to the one-point compactification and the end-compactification given by the closure of the algebra of functions being constant outside of a compact set.

Any element of K^{∗}(∂_{h}M) := K_{∗}(C(∂_{h}M)) can be represented by a projection (∗ = 0) or a unitary (∗ = 1) in some matrix algebra Mat(l)⊗C(∂_{h}M) for some l ≥ 1. Two such matrices represent the same element in K-theory if they are homotopic as projections or unitaries in a possibly larger matrix algebra. We use the standard embeddings into the left upper corner

Mat(l) ⊂ Mat(l + 1),  ∗ = 0
U(l) ⊂ U(l + 1),  ∗ = 1

(often referred to as stabilization).

Definition 2.1 An endomorphism Φ ∈ C^{1}(M, End(S)) is called admissible if

1. deg Φ = 1 (with respect to the grading of S) and Φ^{∗} = Φ,

2. ΦD + DΦ is bounded of zero order and

3. there is a compact set K ⊂ M such that ΦD + DΦ + Φ^{2} ≥ c > 0 on M \ K for some constant c.

If Φ is admissible, then in particular Φ ∈ B(H^{l}), l = 0, 1. A compact set K ⊂ M with Φ^{2} ≥ c > 0 on M \ K is called an essential support of Φ. Later we will show that all the information on the index of B := D + Φ is contained in an essential support of Φ. Note that by replacing Φ by Φ_{t} := tΦ one can have also DΦ_{t} + Φ_{t}D + Φ_{t}^{2} = t(DΦ + ΦD) + t^{2}Φ^{2} ≥ c > 0 outside of a given essential support for large t. B is essentially self-adjoint with the domain C_{c}^{∞}(M, S) for admissible Φ.

Lemma 2.2 If Φ ∈ C^{1}(M, End(S)) is admissible, then B := D + Φ is invertible at infinity (see Assumption 1).

Proof: Let K ⊂ M such that ΦD + DΦ + Φ^{2} ≥ c > 0 on M \ K for some constant c. Let s ∈ C_{c}^{∞}(M) be a non-negative function such that s ≥ c on K. Then B^{2} + λs is self-adjoint and positive for large λ. □

Let s ∈ C_{c}^{∞}(M), s ≥ 0, be such that 0 ≤ (B^{2} + s) ∈ B(H^{0}, H^{2}) exists. Set F := B(B^{2} + s)^{−1/2}. We call (H^{0}, F) the Kasparov module associated to B and it’s class [H^{0}, F] ∈ KK(C_{g}(M), A) is independent of the choice of s by Lemma 1.10, where A is C or some C^{k,0} in the real examples (B is equivariant with respect to A).

Let Φ be an admissible endomorphism and B := D + Φ. Consider a continuous perturbation B_{t} = D_{t} + Φ_{t} of B in B(H^{1}, H^{0}) (where we define H^{l} with D_{0}). Let [M_{t}] ∈ KK(C_{g}(M), A) be associated to B_{t}.

Lemma 2.3 [M_{t}] = [M_{0}] for small t.

Proof: Let s ∈ C_{c}^{∞}(M), s ≥ 0, such that (B^{2} + s)^{−1} ∈ B(H^{0}, H^{2}) exists. Then (B_{t}^{2} + s)^{−1} ∈ B(H^{0}, H^{2}) for small t. The family F_{t} is strictly continuous and the compact remainder terms F_{t} − F_{t}^{∗}, F_{t}^{2} − 1 and [F_{t}, f], f ∈ C_{g}(M) are normcontinuous. This can be seen from the integral formulas for these operators given in the first section. Thus, F_{t} defines a homotopy. □

Corollary 2.4 Let {B_{t}}_{t∈I} be a continuous family in B(H^{1}, H^{0}) such that B_{t} is invertible at infinity for any t ∈ I. Then [M_{t}] = [M].

Proof: One finds a suitable finite partition of the interval I and applies Lemma 2.3. □

A continuous path Φ_{t} ∈ C(M, End(S)) of admissible endomorphisms leads to a continuous family B_{t} ∈ B(H^{1}, H^{0}). Further, a smooth deformation of D over compact subsets does not affect the class [M]. Let {M} ∈ KK(k, A), k = C, R, be the restriction of [M].

Corollary 2.5 If Φ is admissible and has an empty essential support, then {M} = 0.

Proof: Consider the family B_{t} := D + tΦ. If Φ is admissible, so is tΦ for t ≥ 1. Moreover, B_{t} is invertible for large t > 0. Hence {M} = {M_{t}} = 0. □

2.3 The odd-dimensional case

2.3.1 The map u

Let M be a complete Riemannian manifold and E → M be a (complex) Clifford bundle over M with a Dirac operator D_{E} that is ungraded. Let [p] ∈ K^{0}(∂_{h}M) be represented by a projection p ∈ C(∂_{h}M) ⊗ Mat(l) for some large l. We can find a ”lift” P ∈ C_{g}^{∞}(M) ⊗ Mat(l) of p such that P = P^{∗}, P^{2} − P ∈ C_{0}^{∞}(M) ⊗ Mat(l) and P (mod C_{0}(M) ⊗ Mat(l)) represents [p]. Stabilization and homotopies of p can be lifted to stabilization and homotopies of P. Form the Z_{2}-graded Clifford bundle S := E ⊗ (C^{l} ⊕ (C^{l})^{op}) with the associated Dirac operator D given by the matrix

D := ( \begin{matrix}0 & D_{E} ⊗ 1 \\ D_{E} ⊗ 1 & 0 \end{matrix} ) .

Here C^{l} is trivially graded (i.e. all elements have degree 0) and (C^{l})^{op} consists of elements of degree 1. Let

Φ := ı ⊗ ( \begin{matrix}0 & 1 − 2P \\ 2P − 1 & 0 \end{matrix} ) ∈ C^{1}(M, End(S)).

Lemma 2.6 Φ is admissible.

Proof: We have degΦ = 1 and Φ = Φ^{∗}. Moreover,

DΦ + ΦD = 2ı ( \begin{matrix}−gradP & 0 \\ 0 & gradP \end{matrix} )

holds. gradP tends to zero at infinity while Φ^{2} tends to 1. Thus, there is a compact set K ⊂ M such that DΦ + ΦD + Φ^{2} ≥ c > 0 on M \ K for some constant c. □

Let {M} ∈ KK_{0}(C, C) = Z be the class associated to the Callias-type operator

B := D + Φ.  (14)

Then {M} only depends on the class of p in K-theory. In fact, two different choices of the lift P can be deformed into each other such that we get a deformation of the corresponding admissible Φ’s. Stabilization results in forming a direct sum of B with some invertible operator of Dirac type. Homotopies of p inside finite matrices can be lifted to homotopies of the P and thus of the Φ inside the admissible endomorphisms. All this does not affect the class {M} (Lemma 2.4).

Definition 2.7 The map u : K^{0}(∂_{h}M) → Z is defined by u : [p] → {M}.

It is easy to see that u is a group-homomorphism.

Since D commutes with the action of C^{0,1} given by

ϵ → ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

where ϵ generates C^{0,1}, we have the following fact.

Proposition 2.8 If there is a gap in the spectrum of D_{E}, then u = 0.

Proof: Let D and Φ be as in the construction above. Φ anticommutes with ϵ. Let λ ∈ res(D) ∩ R. Consider the two-parameter family

B_{t,τ} := D − τλϵ + tΦ.

We can deform B_{1,0} to B_{0,1} inside the operators of Callias-type that are invertible at infinity. Let first τ go from zero to one and then t go from one to zero. B_{0,1} and B_{1,0} define the same KK-klass. B_{0,1} commutes with ϵ and, hence, defines the zero-element in KK-theory. □

This fits well into the results of Roe and others stating that the odd index constructed out of an ungraded Dirac operator vanishes if there is a gap in the spectrum.

2.3.2 Computation of u

Let N ⊂ M be a compact hypersurface cutting M = M_{−} ∪_{N} M_{+} such that M_{−} is compact and contains an essential support of Φ. Deform the metric, the Clifford bundle structure and Φ in a neighbourhood of N such that there is a product collar near N and Φ is constant in normal direction. Let M_{1} := M_{−} ∪_{N} [0, ∞) × N. There is a Clifford bundle S_{1} → M_{1} respecting the product structure over the cylindrical end restricting to S_{|M_{−}} on M_{−} and S_{1|[0,∞)×N} = pr_{N}^{∗}S_{|N}. One can extend −Φ_{|M_{−}} constantly along the cylinder obtaining Φ_{1}. By scaling Φ and Φ_{1} we can assume that Φ_{1} is admissible. Form M_{2} := M ∪ −M_{1}^{op}, where op means that we choose the opposite grading on the Clifford bundle over M_{1} and − stands for redefining the Clifford bundle structure such that X ∈ T M_{1} acts as −X (note that we compress all structures over the manifolds into the symbols M, M_{1}, …). Then {M_{2}} = {M} − {M_{1}}.

Now apply the relative index theorem. Cut the manifold M_{2} at the two copies of N, i.e. N∪{0}×N, and glue them together interchanging the boundary components, thus obtaining M_{3} := M_{−} ∪_{N} −M_{−}^{op} ∪ M_{+} ∪_{N} −(R_{+} × N)^{op}. In order to glue the Clifford bundles we use an isomorphism

Γ_{1} := ( \begin{matrix}0 & n \\ n & 0 \end{matrix} ) ,

where n is the Clifford multiplication by the unit normal vector field at N pointing into the positive direction. It holds that Γ_{1}Φ = −ΦΓ_{1} and the admissible endomorphisms glue nicely. By the relative index theorem {M_{2}} = {M_{3}}, but {M_{3}} = 0. On the one hand, {M_{−} ∪_{N} −M_{−}^{op}} = 0 since this manifold is compact and, hence, the Dirac operator has a gap in the spectrum such that we can apply Proposition 2.8. On the other hand, an essential support of the admissible endomorphism over M_{+} ∪_{N} −(R_{+} × N)^{op} is empty. Thus {M_{+} ∪_{N} −(R_{+} × N)^{op}} = 0 by Corollary 2.5 and, hence, {M} = {M_{1}}. We have replaced one half of the manifold by a cylinder. In the next step we do the same with the other half.

Consider M_{4} := M_{1} ∪ −M_{1}, where we choose the admissible endomorphism Φ_{4} as −Φ_{1} on the first component and as Φ_{1} on the second. Then {M_{4}} = 2{M_{1}}. Now deform Φ_{4} on a compact set such that it is zero near the two copies {0} × N of N in M_{4}. Cut M_{4} at these two copies and glue them together again interchanging the boundary components, thus obtaining M_{5} := M_{−} ∪_{N} −M_{−} ∪ M_{6} where M_{6} = R × N. For glueing the Clifford bundle we use the even isomorphism of S_{|_{N}} given by multiplication by

Γ_{2} := diag(n, n).

By the relative index theorem 2{M_{1}} = {M_{5}}, but {M_{−} ∪_{N} −M_{−}} = 0 since this manifold is compact and therefore we can apply Proposition 2.8. Hence 2{M} = {R × N}. Thus we have reduced the index computation to a cylinder.

Let Φ_{c} be the admissible endomorphism over the cylinder. By a further deformation over compact sets we can assume that Φ_{c} = ψΦ_{|N}, where ψ is a smooth function on R × N depending only on the first coordinate such that ψ(r, n) = sign(r) for |r| ≥ 2 and |∂/∂rψ(r, n)| ≤ 1/2. Moreover, we can deform Φ_{|N} into an involution or, what is the same, P_{|N} into a projection. This amounts to an index-preserving deformation of Φ_{c} by Lemma 2.4. Then

\{R × N\} = index (n\frac{∂}{∂r} − ψ(r)ı(1 − 2P) + D_{N} )
= index (\frac{∂}{∂r} + ψ(r)nı(1 − 2P) − n(1 − 2P)D_{N}(1 − 2P))

where D_{N} is the Dirac operator on S_{|N}. We have

(1 − 2P)D_{N}(1 − 2P) = (1 − P)D_{N}(1 − P) + P D_{N}P − P D_{N}(1 − P) − (1 − P)D_{N}P

and (1 − P)D_{N}P and P D_{N}(1 − P) are of zero order. By an index-preserving deformation

{R × N} = index T ∈ KK(C, C) = Z

with

T := (\frac{∂}{∂r} + ψ(r)nı(1 − 2P) − [n(1 − P)D_{N}(1 − P) + nP D_{N}P]) .

By separation of variables (similar to the calculation of Higson [18]) one obtains that the kernel of T and T^{∗} consists of elements of the form

f_{v}^{−}(r) = exp (− ∫_{0}^{r} ψ(s)ın(1 − 2P)) v
f_{v}^{+}(r) = exp (∫_{0}^{r} ψ(s)ın(1 − 2P)) v

with v ∈ kern[(1−P)D_{N}(1−P)+P D_{N}P]. The involution ın(1−2P) anticommutes with n[(1 − P)D_{N}(1 − P) + P D_{N}P]. We can split

ker n[(1 − P)D_{N}(1 − P) + P D_{N}P] = W^{+} ⊕ W^{−}

into the eigenspaces of ın(1 − 2P) to the eigenvalues ±1. f_{v}^{+} is in L^{2} iff v ∈ W^{−} and f_{v}^{−} is in L^{2} iff v ∈ W^{+}. Thus index T = dim W^{+} − dim W^{−}. Let V_{−} → N be the vector bundle over N given by P and V_{+} be the bundle given by (1 − P). Then n[(1 − P)D_{N}(1 − P) + P D_{N}P] is the sum of the two twisted Dirac operators D_{N,V_{+}} and D_{N,V_{−}} and dim W^{+} − dim W^{−} = ind_{ın} D_{N,V_{+}} − ind_{ın} D_{N,V_{−}}, where the grading is given by the involution ın. By cobordism invariance ind_{ın} D_{N,V_{+}} = −ind_{ın} D_{N,V_{−}}. Thus,

u([p]) = \{M\} = \frac{1}{2}\{R × N\} = \frac{1}{2}index T = ind_{ın}D_{N,V_{+}}.

Let us formulate this result as a theorem.

Theorem 2.9 Let [p] ∈ K^{0}(∂_{h}M) and P ∈ C_{g}^{∞}(M) ⊗ Mat(l) be a self-adjoint representative of p. Let N ⊂ M be a compact hypersurface cutting M = M_{−} ∪_{N} M_{+}, where M_{−} is compact and (1 − 2P)^{2} ≥ c > 0 on M_{+} for some constant c. Let V_{+} be the sub-bundle of N × C^{l} given by the eigenvalues < \frac{1}{2} of P_{|N}. Then u([p]) = ind_{ın}D_{N,V_{+}}, where D_{N,V_{+}} is the Dirac operator of E_{|N} ⊗ V_{+} and the grading is given by the multiplication by ın. n is the unit normal vector field at N pointing into the direction of M_{+}.

This theorem reproves and generalizes the theorems of Callias [12], Anghel [1] and was also proved in [2]. It contains the index theorem for partitioned manifolds of Roe [25] in the formulation of Higson [18] as a special case. In fact, a partitioned manifold M is given by a complete Riemannian manifold M together with a real function ψ ∈ C^{∞}(M) such that |ψ| = 1 outside of a compact set and ψ^{−1}(0) is a smooth compact hypersurface. Then ψ induces an element [(1 + ψ)/2] ∈ K^{0}(∂_{h}M). Let N_{+} ∪ N_{−} be the boundary of a small tubular neighbourhood M_{−} of N such that ψ_{|N_{−}} < 0. By Theorem 2.9 u([ψ]) = ind_{ın}D_{N_{−}} = ind_{ın}D_{N}, where n is the unit normal vector field pointing into the region of M, where ψ is negative.

Since the index of a Dirac operator on an odd-dimensional manifold vanishes, we have u = 0 if M is even-dimensional.

2.3.3 Interpretation as KK-intersection product

The operator D := D_{E} ⊗zX on E ⊗C^{1,0} defines a class [D] ∈ KK(C_{0}(M)⊗C^{1,0}, C). We use the convention (a ⊗ b)(c ⊗ d) = ac ⊗ bd independent of the degree of the elements. This differs from the usual convention, where there is a sign (−1)^{deg(b)deg(c)}. X is the odd generator of C^{1,0} with X^{2} = −1, X^{∗} = −X and z denotes the Z_{2}- grading. Let h := L^{2}(M, E ⊗ C^{1,0}) and F_{2} := D(D^{2} + 1)^{−1/2} ∈ B(h). Then [f ⊗ X, F_{2}] = (f ⊗ X)F_{2} + F_{2}(f ⊗ X) ∈ K(h) for f ∈ C_{0}(M). One can show this using an integral representation for (D^{2} + 1)^{−1/2} (similar as in Lemma 1.8) and the fact that [f ⊗ X, D] is an endomorphism of E ⊗ C^{1,0} vanishing at infinity. Moreover, F_{2}^{∗} = F_{2} and f(F_{2}^{2} − 1) ∈ K(h), ∀f ∈ C_{0}(M), and (h, F_{2}) is a Kasparov module over (C_{0}(M) ⊗ C^{1,0}, C) representing [D] := [h, F_{2}] ∈ KK(C_{0}(M) ⊗ C^{1,0}, C).

Consider L := C_{0}(M) ⊗ C^{l} ⊗ C^{1,0} as a graded C_{0}(M) ⊗ C^{1,0}- Hilbert-C^{∗}-module. The endomorphism Φ := ı(1 − 2P) ⊗ X defines a bounded operator in B(L). Moreover, Φ^{2} − 1 = 4(P^{2} − P) ∈ C_{0}(M) ⊗ Mat(l) ⊗ C^{1,0} is in K(L), Φ^{∗} = Φ, [f, Φ] = 0, ∀f ∈ C_{g}(M) and (L, Φ) is a Kasparov module over the pair (C_{g}(M), C_{0}(M) ⊗ C^{1,0}) representing [Φ] := [L, Φ] ∈ KK(C_{g}(M), C_{0}(M) ⊗ C^{1,0}).

Let {Φ} ∈ KK(C, C_{0}(M) ⊗ C^{1,0}) be the restriction of [Φ]. Note that

KK(C, C_{0}(M) ⊗ C^{1,0}) ≅ K_{c}^{1}(M)  (15)

(see Blackadar [4] 17.5.6.). Following [4] 17.5.4 the class {Φ} is represented by the self-adjoint unitary (mod K(C_{0}(M) ⊗ C^{l})) T := 1 − 2P. By [4] 17.5.6 the unitary U := exp(2πıP) corresponds to {Φ} under (15) and represents the class exp([p]) ∈ K_{c}^{1}(M).

Lemma 2.10 The class {Φ} ∈ KK(C, C_{0}(M) ⊗ C^{1,0}) corresponds to exp([p]) ∈ K_{c}^{1}(M) under the identification (15).

Let B be the operator of Callias-type (14) and [M] ∈ KK(C_{g}(M), C) be associated to B. Let τ_{C^{1,0}} : KK(C_{g}(M), C) →^{∼} KK(C_{g}(M) ⊗ C^{1,0}, C^{1,0}) be the isomorphism defined in Blackadar, [4] 17.8.5.

Proposition 2.11

τ_{C^{1,0}}[M] = [Φ] ⊗_{C_{0}(M)} [D] ∈ KK(C_{g}(M) ⊗ C^{1,0}, C^{1,0}).

Proof: Consider H̃ := L ⊗_{C_{0}(M)} h. Then

H̃ ≅ L^{2}(M, E) ⊗ C^{l} ⊗ C^{1,0} ⊗ C^{1,0}.

The first C^{1,0}-factor comes from L and carries the C^{1,0}-right action while the second C^{1,0}-factor comes from h and carries the C^{1,0} left action. We define the operator of Dirac type

B̃ := D_{E} ⊗ 1_{l} ⊗ z ⊗ zX + ı(1 − 2P) ⊗ X ⊗ 1.

Here (1 − 2P) acts on L^{2}(M, E) ⊗ C^{l} and z denotes the Z_{2}-grading. B̃ is invertible at infinity and defines some F̃ ∈ B(H̃). The same analysis as in the proof of Proposition 2.18 shows that (H̃, F̃) represents [Φ] ⊗_{C_{0}(M)} [D].

[M] is represented by (H, F), where F was a bounded version of B. Moreover, τ_{C^{1,0}}[M] is represented by (H ⊗ C^{1,0}, F ⊗ 1). The proof is finished by constructing a unitary A : H̃ → H ⊗ C^{1,0} intertwining with the left and right C^{1,0}-actions such that (B ⊗ 1)A = AB̃. Then (F ⊗ 1)A − AF̃ ∈ K(H̃, H ⊗ C^{1,0}) follows.

We have H = L^{2}(M, E) ⊗ C^{l} ⊗ C^{1,0} and B = D_{E} ⊗ 1_{l} ⊗ zX + ı(1 − 2P) ⊗ X. We define a graded C^{1,0}-bimodule isomorphism

a : C^{1,0} ⊗ C^{1,0} → C^{1,0} ⊗ C^{1,0}

with respect to the actions • left multiplication by c ∈ C^{1,0} on the l.h.s: c → 1⊗c^{0}+z⊗c^{1}, where c = c^{0}+c^{1}

is the decomposition into homogeneous components

• right multiplication on the l.h.s.: c → c^{0} ⊗ 1 + c^{1} ⊗ z

• left multiplication on the r.h.s.: c → 1 ⊗ c^{0} + z ⊗ c^{1}

• right multiplication on the r.h.s.: c → 1 ⊗ c.

Define a as the left multiplication by

\frac{1}{\sqrt{2}}(1 ⊗ 1 − X ⊗ Xz)

and let

A := 1_{E} ⊗ 1_{l} ⊗ a : H̃ → H ⊗ C^{1,0}.

An easy calculation shows that A has the required properties. □

2.3.4 The spectral flow pairing s

Let M be a complete Riemannian manifold and E → M be an ungraded Clifford bundle over M. A cycle [W] ∈ K_{c}^{1}(M) is represented by a unitary matrix W ∈ C^{∞}(M)⊗Mat(l) with W = 1 outside of a compact set K ⊂ M. Let S := E⊗C^{l} and D be the associated Dirac operator. It is self-adjoint and defines a spectral projection Q ∈ B(H^{0}) onto the positive spectral subspace of D. Consider A := (1 − Q) + W Q, where W acts on H^{0} as a unitary multiplication operator. It is shown in [11] that A is a Fredholm operator.

Definition 2.12 The spectral flow pairing s : K_{c}^{1}(M) → Z is defined by s([W]) := indexA.

In fact, indexA depends only on the stable homotopy class of W and is additive under direct sums. One can compute this index by reducing to a compact manifold containing K. If M were compact, then s([W]) would be given by the spectral flow of a family between D and W^{∗}DW.

We will show the commutativity of the lower part of the diagram of the introduction. Let exp : K^{0}(∂_{h}M) → K_{c}^{1}(M) be the exponential map.

Theorem 2.13 u = s ∘ exp

Proof: Let [p] ∈ K^{0}(∂_{h}M) be represented by a self-adjoint P ∈ C_{g}^{∞}(M) ⊗ Mat(l). We can choose P such that P is a projection outside of a compact set K ⊂ M. Then exp([p]) is represented by the unitary W := exp(2πıP). In fact, W = 1 on M \ K. Choose a compact hypersurface N ⊂ M which splits M = M_{−} ∪_{N} M_{+} such that K ⊂ M_{−}. Let I × N be a tubular neighbourhood of N in M. We can choose another P_{1} representing p such that P_{1} = P on the plus-side of I × N and P_{1} = 0 on the minus-side. Let W_{1} = exp(2πıP_{1}). Then W_{1} = 1 outside of the tubular neighbourhood of N and W_{1} represents the same class as W. Consider P_{1}, W_{1} instead of P, W. As it has been shown in [11] one can compute s([W]) by reduction to M_{1} := S^{1} × N, a manifold which naturally containes our tubular neighbourhood I × N. Deform the metric and the Clifford bundle structure near N such that they respect the product structure of the tubular neighbourhood. Then glue S_{|I×N} at the boundaries of I × N, thus obtaining a Clifford bundle S_{1} → M_{1}. Since W = 1 on these boundaries, it extends to W_{1} over M_{1}. Now form A_{1} := (1 − Q_{1}) + W_{1}Q_{1}, where Q_{1} is the positive spectral projection of the Dirac operator on S_{1}. By the results of [11] index A_{1} = index A and index A_{1} is the spectral flow of a family between D_{1} and W_{1}^{∗}D_{1}W_{1}. Since W_{1} is the image of a projection P_{1} := P_{|1×N} under the suspension map, this spectral flow equals ind_{ın}P_{1}D_{N}P_{1}. A comparison with Theorem 2.9 finishes the proof. □

The spectral flow pairing does not necessarily vanish if D has a gap in the spectrum. It rather gives the spectral flow of the familiy between D and W^{∗}DW in this gap [9]. If the class represented by W is in the image of the exponential map and D has a gap in the spectrum, then s([W]) = 0.

2.4 The even-dimensional case

2.4.1 The construction of g

Let M be a complete Riemannian spin manifold and E → M be a Z_{2}-graded Clifford bundle over M with grading z. An element [u] ∈ K^{1}(∂_{h}M) is represented by a unitary matrix u ∈ C(∂_{h}M) ⊗ Mat(l) for some large l. Let U ∈ C_{g}^{∞}(M) ⊗ Mat(l) be a ”lift” of u such that U^{∗}U − 1 ∈ C_{0}(M) ⊗ Mat(l) and U (mod C_{0}(M) ⊗ Mat(l)) represents [u]. Stabilization and homotopies of u can be lifted to stabilization and homotopies of U. Form the Z_{2}-graded bundle V := M × (C^{l} ⊕ (C^{l})^{op}) and the graded tensor product S := E ⊗ V. S is a Clifford bundle with Dirac operator D. Let

Φ := z ⊗ ( \begin{matrix}0 & U^{∗} \\ U & 0 \end{matrix} )

and

B := D + Φ.  (16)

Lemma 2.14 Φ is admissible.

Proof: We have deg Φ = 1, Φ^{∗} = Φ and

DΦ + ΦD = z ⊗ ( \begin{matrix}0 & −grad U^{∗} \\ −grad U & 0 \end{matrix} )

vanishes at infinity while Φ^{2} tends to 1 there. There is some compact set K ⊂ M such that DΦ + ΦD + Φ^{2} ≥ c > 0 on M \ K for some constant c. □

Let {M} ∈ KK(C, C) = Z be the class associated to the Callias-type operator (16.) It depends only on the stable homotopy class of U and therefore only on [u]. Moreover, it is additive under the direct sums of [u]’s.

Definition 2.15 The map g : K^{1}(∂_{h}M) → Z is defined by g([u]) := {M}.

2.4.2 Computation of g

Let N ⊂ M be a compact hypersurface cutting M = M_{−} ∪_{N} M_{+} such that M_{−} is compact and contains an essential support of Φ. We can deform U to a unitary matrix on M_{+} using the polar decomposition. Thus, let us assume this. We further assume that (after deformation near N) the metric and the Clifford bundle structure near N respect the product structure of a tubular neighbourhood I × N ⊂ M of N. Consider M_{1} := M ∪ R × N, where the Clifford bundle on R × N comes from the restriction S_{|N}. We extend Φ_{|N} constantly along the R-direction obtaining Φ_{1} over M_{1}. Then {M} = {M_{1}} since an essential support of Φ_{1} over R × N is empty. We apply now the relative index theorem. Cut at the two copies N ∪ {0} × N and glue the pieces interchanging the boundary components, thus obtaining M_{2} = M_{−} ∪_{N} R_{+} × N ∪ R_{−} × N ∪_{N} M_{+}. The Clifford bundles are glued using the natural isomorphism. The admissible endomorphisms glue nicely to Φ_{2}. Then {M} = {M_{1}} = {M_{2}}. However, over R_{−} × N ∪ M_{+} an essential support of Φ_{2} is empty and hence {M} = {M_{3}}, where M_{3} := M_{−} ∪_{N} R_{+} × N. We have replaced half of the manifold by a cylinder.

Consider the matrix

W := ( \begin{matrix}U & 0 \\ 0 & 1 \end{matrix} ) ∈ C^{∞}(M_{3}, End(S_{3})).

W is even, unitary and constant in the R-direction over the cylindrical end. Deform the connection ∇ on S_{3} near {1} × N such that it is W ∇W^{∗} in a neighbourhood of {1} × N and deform Φ_{3} there such that

( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) = W Φ_{|N}W^{∗}

holds there. Now we cut M_{−} ∪_{N} R_{+} × N at {1} × N ∪ {2} × N and glue the pieces together such that we obtain M_{4} := M_{−} ∪_{N} [0, 1] × N ∪_{N} [2, ∞] × N ∪ S^{1} × N. We use W^{∗} in order to glue the Clifford bundles

W^{∗} : S_{|{1}×N} → S_{|{2}×N}.

The admissible endomorphisms glue nicely to Φ_{4}. By the relative index theorem {M} = {M_{4}}. Since W Φ_{|N}W^{∗} can be extended invertibly over M_{−} ∪_{N} [0, 1] × N, we have {M} = {S^{1} × N}. Since this manifold is compact, we can deform Φ_{4} to zero there. Let us formulate this result as a theorem.

Theorem 2.16 Let [u] ∈ K^{1}(∂_{h}M) be represented by U ∈ C_{g}^{∞}(M) ⊗ Mat(l). Let N ⊂ M be a compact hypersurface of M splitting M = M_{−} ∪_{N} M_{+} such that M_{−} is compact and U is invertible over M_{+}. Then g([u]) is given by the index of the twisted Dirac operator D_{L} over S^{1} × N where L → S^{1} × N is constructed from the trivial bundle I × N × C^{l} by glueing with

U_{|N}^{∗} : {0} × N × C^{l} → {1} × N × C^{l}.

g([u]) is also the spectral flow of a family between D_{N} and U_{|N}^{∗}D_{N}U_{|N}, where D_{N} is a ”half” of the Dirac operator on S_{|N}.

2.4.3 Interpretation as KK-intersection product

The operator D_{E} on E defines a class [D] ∈ KK(C_{0}(M), C). Let h := L^{2}(M, E) and F_{2} := D_{E}(D_{E}^{2} + 1)^{−1/2}. Then f(F_{2}^{2} − 1) ∈ K(h), [F_{2}, f] ∈ K(h) for any f ∈ C_{0}(M) and F_{2}^{∗} = F_{2}. Thus (h, F_{2}) is a Kasparov module over (C_{0}(M), C) and represents [D] := [h, F_{2}] ∈ KK(C_{0}(M), C).

Consider L := C_{0}(M) ⊗ C^{l} ⊗ C^{2} as a graded C_{0}(M)-Hilbert-C^{∗}-module (the grading is induced by C^{2} = C ⊕ (C)^{op}). The endomorphism

Φ := ( \begin{matrix}0 & U^{∗} \\ U & 0 \end{matrix} )

acts as a bounded operator in B(L). Moreover,

Φ^{2} − 1 ∈ C_{0}(M) ⊗ Mat(2l) ⊂ K(L),

Φ^{∗} = Φ, [f, Φ] = 0 for all f ∈ C_{g}(M) and (L, Φ) is a Kasparov module over the pair (C_{g}(M), C_{0}(M)) representing [Φ] := [L, Φ] ∈ KK(C_{g}(M), C_{0}(M)).

Let {Φ} ∈ KK(C, C_{0}(M)) be the restriction of [Φ]. Note that

KK(C, C_{0}(M)) ≅ K_{c}^{0}(M)  (17)

(see Blackadar [4] 17.5.5). The unitary U (mod K(C_{0}(M) ⊗ C^{l})) defines an element [U] ∈ K_{1}(Q^{s}C_{0}(M)) where Q^{s}C_{0}(M) is the stable outer multiplier algebra of C_{0}(M). Then ∂[U] ∈ K_{0}(C_{0}(M)) = K_{c}^{0}(M) is the element corresponding to {Φ} under the identification (17). This ∂ is associated to

0 → C_{0}(M) ⊗ K → M^{s}C_{0}(M) → Q^{s}C_{0}(M) → 0.

Since the index map ∂ is natural, we have

Lemma 2.17 The class {Φ} ∈ KK(C, C_{0}(M)) corresponds to ∂[u] ∈ K_{c}^{0}(M) under the identification (17).

Let B = D_{E} ⊗ z + Φ be the operator of Dirac type (16) (up to a non-important factor z) and [M] := [H, F] ∈ KK(C_{g}(M), C) be the class associated to B.

Proposition 2.18 [M] = [Φ] ⊗_{C_{0}(M)} [D]

Proof: We have

H = L ⊗_{C_{0}(M)} h = L^{2}(M, E) ⊗ C^{l} ⊗ C^{2}.

We use the characterization of the intersection product in terms of connections (see Connes/Skandalis [14], Blackadar [4], Jensen/Thomsen [20]). It is enough to verify that F is a F_{2}-connection and that [F, Φ] ≥ 0 (mod K(H)). Let f ⊗ v ∈ L, f ∈ C_{0}(M), v ∈ C^{2l}. Then T_{f⊗v} ∈ B(h, H) is defined by

T_{f⊗v}ϕ = fϕ ⊗ v ϕ ∈ h.

Its adjoint T_{f⊗v}^{∗} ∈ B(H, h) is given by

T_{f⊗v}^{∗}(ψ ⊗ w) = fψ < v, w >, ψ ∈ L^{2}(M, E), w ∈ C^{2l}.

By definition F is and F_{2}-connection if for any f ∈ C_{0}(M) and homogeneous v ∈ C^{2l}

T_{f⊗v}F_{2} − (−1)^{deg(v)}F T_{f⊗v} ∈ K(h, H)
F_{2}T_{f⊗v}^{∗} − (−1)^{deg(v)}T_{f⊗v}^{∗}F ∈ K(H, h).

Let ∼ denote equality modulo compact operators. Then

F T_{f⊗v} ∼ (B|B^{2} + s|^{−1/2}f(1 ⊗ v)
∼ (D_{E} ⊗ z)(D_{E}^{2} ⊗ 1_{l} + 1)^{−1/2}f(1 ⊗ v)

because of f ∈ C_{0}(M). Since [f, F_{2}] ∈ K(h), we obtain

(−1)^{deg(v)}T_{f⊗v}F_{2} = (−1)^{deg(v)}fD_{E}(D_{E}^{2} + 1)^{−1/2} ⊗ v
∼ D_{E}(D_{E}^{2} + 1)^{−1/2}f ⊗ zv

and

F T_{f⊗v} − (−1)^{deg(v)}T_{f⊗v}F_{2} ∈ K(h, H).

Next we consider

F_{2}T_{f⊗v}^{∗} = D_{E}(D_{E}^{2} + 1)^{−1/2}f ⊗ v^{v},

where v^{v} :=< v, . > is the dual vector of v. It holds

(−1)^{deg(v)}T_{f⊗v}^{∗}F ∼ (−1)^{deg(v)}(f ⊗ v^{v})B|B^{2} + s|^{−1/2}
∼ (−1)^{deg(v)}(f ⊗ v^{v})(D_{E} ⊗ z)(D^{2} ⊗ 1_{l} + 1)^{−1/2}
∼ D_{E}(D_{E}^{2} + 1)^{−1/2}f ⊗ v^{v}.

Thus,

F_{2}T_{f⊗v}^{∗} − (−1)^{deg(v)}T_{f⊗v}F ∈ K(H, h)

and F is an F_{2}-connection. We show the positivity condition.

[Φ, F] ∼ [Φ, B|B^{2} + s|^{−1/2}]
= [Φ, (D ⊗ z + Φ)]|B^{2} + s|^{−1/2} − B[Φ, |B^{2} + s|^{−1/2}]
∼ 2Φ^{2}|B^{2} + s|^{−1/2}
∼ 2|B^{2} + s|^{−1/2}
≥ 0.

We have employed that Φ has entries in C_{g}^{∞}(M) and that Φ^{2}−1 ∈ C_{0}(M)⊗Mat(2l). □

The algebra C_{g}(M) is not separable. Therefore, we use the strongest equivalence relation generated by homotopy. Other equivalence relations may give different KKgroups. In the literature, in order to construct the product it is assumed in general that the algebra in the first place of the KK-group is separable. We do not use an abstract existence theorem, we rather write down an explicit module representing the product. In order to avoid non-separable algebras one could also consider the modules over C instead of C_{g}(M) at the cost of losing some pieces of information.

2.4.4 The relative index pairing r

Let M be a complete Riemannian manifold and E → M be a Z_{2}-graded Clifford bundle over M. An element τ ∈ K_{c}^{0}(M) is represented by a tuple (V_{1}, V_{2}, W), where V_{1}, V_{2} are vector bundles over M and W ∈ C^{∞}(M, Hom(V_{1}, V_{2})) is invertible outside of a compact subset K ⊂ M. Let N ⊂ M be a hypersurface cutting M = M_{−}∪_{N} M_{+} such that M_{−} is compact and contains K. Deform the metric and the Clifford bundle in order to have a product structure near N. Take M̃ = M_{−}∪_{N} −M_{−}^{op}, where we glue the Clifford bundle obtaining Ẽ using the odd morphism given by the multiplication by the unit normal vector field n pointing into M_{+}. We glue V_{1|M_{−}} with V_{2|−M_{−}^{op}} using the identification

W_{|N} : V_{1|M_{−}} → V_{2|−M_{−}^{op}},

obtaining V → M̃. Let D̃_{V} be the twisted Dirac operator on Ẽ ⊗ V.

Definition 2.19 The relative index pairing r : K_{c}^{0}(M) → Z is defined by r(τ) := ind D̃_{V}.

This definition does not depend on the choice of the representative of τ and the hypersurface N (for the latter see [16]).

We will show the commutativity of the upper part of the diagram of the introduction.

Theorem 2.20 r ∘ ∂ = g.

Proof: Let [u] ∈ K^{1}(∂_{h}M) be represented by the unitary matrix u ∈ C(∂_{h}M) ⊗ Mat(l) and let U ∈ C_{g}^{∞}(M) ⊗ Mat(l) be a ”lift” of u. We can assume that U is unitary outside of a compact set K ⊂ M. Then ∂([u]) is given by τ = [(M ×C^{l}, M × C^{l}, U)]. Let N ⊂ M be a compact hypersurface cutting M = M_{−} ∪_{N} M_{+} such that K ⊂ M_{−}. Let I × N be a tubular neighbourhood of N. We can deform U on M_{−} such that U_{M_{−}\I×N} = 1. Hence we can construct r(τ) as follows. We glue

M̃ = S^{1} × N = I × N ∪ −(I × N)^{op}.

We have to glue the trivial bundle I ×N ×C^{l} with the trivial bundle −(I ×N)^{op}×C^{l} using the identifications

1 : {0} × N × C^{l} → −({0} × N)^{op} × C^{l}

and

U_{|{1}×N} : {1} × N × C^{l} → −({1} × N)^{op} × C^{l},

thus obtaining V → S^{1} × N. Then r(τ) = ind D̃_{V}. A comparison with Theorem 2.16 gives the desired result. □

2.5 The boundary map in K-homology

Let M̄ be a compact Riemannian manifold with boundary ∂M̄ = N and assume a metric product collar near N. We can make M̄ complete by glueing a cylinder R_{+}×N along N and we obtain M. We choose a warped product metric dr^{2}+f(r)g^{N} on the cylinder with f(r) → 0 for r → ∞. We can compactify M by adjoining a copy of N at infinity and obtain M_{N}. This compactification is associated with the algebra C(M_{N}), which is the closure in C(M) of the smooth bounded functions that are independent of the radial variable near infinity. Due to the choice of the metric C(M_{N}) ⊂ C_{g}(M). Hence M_{N} is a corona compactification (see Roe [27]) and all the results of the preceding subsections remain valid in this case. There is a canonical map q : ∂_{h}M → N. One can lift K-classes from N to ∂_{h}M and perform the constructions of Callias-type operators.

Let E → M be a graded (∗ = 0) or ungraded (∗ = 1) Clifford bundle over M with associated Dirac operator D. It defines a class [D] ∈ K_{∗}^{c}(M). The exact sequence

0 → C_{0}(M) → C(M_{N}) → C(N) → 0

induces a long exact sequence in K-homology

→ K_{∗}(M_{N}) → K_{∗}^{c}(M) →^{∂} K_{∗+1}(N) →,

where the grading of the K-groups has values in Z_{2}. The results of the preceding subsections can be used to characterize the boundary ∂[D] ∈ K_{∗+1}(N) up to torsion.

For every x ∈ K^{∗+1}(N) we have already computed < ∂x, [D] >∈ Z. Thus, we know the pairings

< x, ∂[D] >=< ∂x, [D] >∈ Z

for all x ∈ K^{∗+1}(N). This characterizes ∂[D] up to torsion.

Let E be ungraded and ∗ = 1. In Section 2.3.2 we have constructed a Dirac operator D_{N} on N . It is associated to the Clifford bundle E_{|N}, which is now graded by the multiplication by ın, where n is the outward-pointing unit normal vector field at N. By Theorem 2.9, Lemma 2.10 and Proposition 2.11 we have for any [p] ∈ K^{0}(N) represented by the projection p ∈ Mat(l) ⊗ C(N) for some l > 0

< ∂[p], [D] >= u([p]) = −indD_{N,V} ,

where V is the vector bundle on N defined by p and D_{N,V} is the twisted Dirac operator. Thus for all [p] ∈ K^{0}(N)

< [p], ∂[D] >= − < [p], [D_{N}] > .

Here [D_{N}] ∈ K_{0}(N) is the K-homology class defined by the graded Dirac operator D_{N} and we have used the fact that the pairing between K-theory and K-homology of N is given by taking the index of twisted Dirac operators.

Theorem 2.21 ∂[D] = −[D_{N}] ∈ K_{0}(M) (mod torsion)

This in agrees with the result of Baum/Douglas/Taylor [3], Prop.5.1. The different sign in that reference is due to the use of the inner normal for grading the bundle E_{|N}.

Let now E be graded and ∗ = 0. We have mentioned a Dirac operator D_{N} acting on the ungraded bundle E_{|N}^{+}. From Theorem 2.16, Lemma 2.17 and Proposition 2.18 for any [u] ∈ K^{1}(N) represented by the unitary u ∈ Mat(l) ⊗ C(N) it holds

< ∂[u], [D] >= g([u]) = sf{D_{N}, u^{∗}D_{N}u},

where sf{D_{N}, u^{∗}D_{N}u} is the spectral flow of a straight line between D_{N} and u^{∗}D_{N}u. Let Q be the positive spectral projection of D_{N}. Then

sf{D_{N}, u^{∗}D_{N}u} = ind(1 − Q + uQ) =< [u], [D_{N}] > .

[D_{N}] ∈ K_{1}(N) is the class defined by D_{N} and we have used the fact that the spectral flow pairing is the pairing between the odd K-theory and odd K-homology of N.

Theorem 2.22 ∂[D] = [D] ∈ K_{1}(N) (mod torsion).

In order to compare this with the result of Baum/Douglas/Taylor we have to translate [D_{N}] to the Cuntz-picture. Note that [D_{N}] = [H, F] ∈ KK(C(N), C^{0,1}) with H := L^{2}(N, E) ⊗ C^{0,1} and F := D_{N}(D_{N}^{2} + 1)^{−1/2} ⊗ X, where C(N) acts by left-multiplication on H. In the Cuntz-picture used in [3] (see Blackadar [4], 17.6.6) [D_{N}] is given by the projection (mod K)

P := \frac{1}{2}(D_{N}(D_{N}^{2} + 1)^{−1/2} + 1),

which commutes (mod K) with the action of C(N). However, P = Q (mod K), where Q is the Calderon projection of D, since both operators are pseudo-differential and have the same principal symbol. Thus ∂[D] is also represented (mod torsion) by the Calderon projector of D and this is in nice agreement with [3], Prop. 4.3.

Let M be a complete Riemannian manifold and ∂M be a reasonable corona in the sense of Roe, [27], 5.27, such that the transgression map T : H^{∗}(∂M) → HX^{∗+1}(M) is defined. HX^{∗}(M) is the exotic cohomology introduced in [27]. Let E → M be a graded (∗ = 0) or ungraded (∗ = 1) Clifford bundle with Dirac operator D. Roe defined an index element Ind(D) ∈ K_{∗}(B_{H}), where B_{H} is the algebra of locally tracable operators with bounded propagation on H := L^{2}(M, E) (see [27], 4.8 for B_{H} and sections 4.3 and 4.6. for Ind(D)). Let χ : HX^{∗}(M) → HC^{∗}(B_{H}) be the character map from exotic to cyclic cohomology. Then, from the intermediate steps of the proof of Proposition [27], 5.2.9, we can obtain for any x ∈ K_{∗+1}(∂M)

< χ ∘ T ∘ ch(x), Ind(D) >=< ∂x, [D] >= ( \begin{matrix}u(x) : & ∗ = 1 \\ g(x) : & ∗ = 0 \end{matrix} ) .  (18)

Thus, we have interpreted the pairing on the l.h.s. of (18) as the index of a Calliastype operator. The author was informed by S. Hurder that G. Yu has a more satisfactory proof (unpublished) of the above fact solely using K-theoretic arguments.

2.6 Real Callias-type operators

In this section we consider KK-theory for real C^{∗}-algebras and C_{g}(M), C_{0}(M) etc. consist of real functions.

Let M be a complete Riemannian manifold and E → M be a real Z_{2}-graded Clifford bundle over M admitting a right action of the real Clifford algebra C^{n,0}. Let z be the grading operator. The Dirac operator D_{E} is C^{n,0}-equivariant and provides an element [D] ∈ KK(C_{0}(M), C^{n,0}). [D] is represented by (H, F) with H := L^{2}(M, E) and F := D_{E}(D_{E}^{2} + 1)^{−1/2}. For later purpose it is useful to employ [D]_{k,n−k} ∈ KK(C_{0}(M) ⊗ C^{0,k}, C^{n−k,0}). For k ≤ n there is a map

β_{k} : KK(C_{0}(M), C^{n,0}) → KK(C_{0}(M) ⊗ C^{0,k}, C^{n−k,0}).

Let (H, F) represent some element [H, F] ∈ KK(C_{0}(M), C^{n,0}). Then β_{k}[H, F] is represented by the same (H, F). The C_{0}(M) ⊗ C^{0,k}-left action on H is given by f ⊗ e → fze, where e ∈ R^{k}, f ∈ C_{0}(M) and z is the Z_{2}-grading. We assume that R^{k} ⊂ R^{n} generates C^{0,k} as well as C^{k,0} ⊂ C^{n,0} and e acts by the right multiplication. Let R^{n−k} = (R^{k})^{⊥} ⊂ R^{n} generate C^{n−k,0}, then C^{n−k,0} commutes with fze, e ∈ R^{k}. If we restrict the right multiplication to C^{n−k,0}, then we get a Kasparov module (H, F) over the pair (C_{0}(M) ⊗ C^{0,k}, C^{n−k,0}). Set [D]_{k,n−k} := β_{k}[D].

We want to pair [D]_{k,n−k} with certain elements [Φ] ∈ KK(C_{g}(M), C_{0}(M)⊗C^{0,k}). We consider a [Φ] represented by a matrix Φ ∈ Mat(l) ⊗ C_{g}^{∞}(M) ⊗ C^{0,k} for some l > 0 with the properties Φ^{∗} = Φ, deg Φ = 1 and that Φ^{2} − 1 vanishes at infinity. We construct an operator of Callias-type representing

[M] = [Φ] ⊗_{C_{0}(M)⊗C^{0,k}} [D]_{k,n−k} ∈ KK(C_{g}(M), C^{n−k,0}).

Form the graded tensor product S := (C^{0,k})^{⊕l} ⊗_{C^{0,k}} E = C^{l} ⊗ E. This is a real Z_{2}- graded Clifford bundle with right action of C^{n−k,0}. Let D = 1_{l} ⊗ D_{E} be the Dirac operator of S and form B := D + Φ. The Callias-type operator B is equivariant with respect to C^{n−k,0}.

Lemma 2.23 Φ is admissible.

The same analysis as in Section 2.4.3 shows

Proposition 2.24

[Φ] ⊗_{C_{0}(M)⊗C^{0,k}} [D]_{k,n−k} = [M] ∈ KK(C_{g}(M), C^{n−k,0})

is the class associated to B.

Let {M} ∈ KK(R, C^{0,n−k}) be the restriction of [M].

In order to compute {M} we will make a special choice of Φ. Let f ∈ C(∂_{h}M) ⊗ R^{k} be given such that f^{2} = −1, i.e. f has values in the unit sphere. Let F ∈ C_{g}^{∞}(M) ⊗ R^{k} be a lift of f and set Φ := zF.

We can assume that 0 is a regular value of F. Then N := F^{−1}(0) is a compact manifold of dimension n−k. We use F in order to identify a small neighbourhood of N with D^{k} ×N, where D^{k} is the unit disk in R^{k}. After deformation of the metric we can assume that F is an isometry on this neighbourhood. Let F̃ be a deformation of F in a neighbourhood of N such that F̃(x, n) = x/|x| in a neighbourhood of S^{k−1} × N = ∂(D^{k} × N). Let M_{1} := R^{k} × N with F_{1}(x, n) = F̃(x, n) on D^{k} × N and x/|x| for |x| ≥ 1. M_{1} has a Clifford bundle S_{1} induced from S respecting the product structure. Let Φ_{1} := −zF_{1} and M_{2} := M ∪−M_{1}^{op}. Cut M_{2} at the two copies of S^{k−1} × N and glue the pieces together interchanging the boundary components, thus obtaining M_{3} = M_{4} ∪ M_{5} with

M_{4} := M \ D^{k} × N ∪_{S^{k−1}×N} −(R^{k} \ D^{k} × N)^{op}

and

M_{5} := D^{k} × N ∪_{S^{k−1}×N} −(D^{k} × N)^{op}.

We glue the Clifford bundles using the odd morphism given by left multiplication with the unit normal vector field at the boundary S^{k−1} × N. By the relative index theorem {M_{2}} = {M_{3}}, but {M_{3}} = 0. In fact, M_{5} is compact and Φ_{5} can be deformed to zero. The Dirac operator itself, without perturbation, is equivariant with respect to a larger Clifford algebra. On M_{4} an essential support of the admissible endomorphism is empty. Hence {M} = {M_{1}}. Thus, we have reduced the problem to a product M_{1} = R^{k} × N, where Φ_{1}(x, n) = zx/|x| for |x| ≥ 1.

Let M := M_{1} and Φ := Φ_{1} for a moment. The Clifford bundle S → R^{k} × N is the pull back of a bundle S^{k} over N admitting a right action of C^{n,0} and a left action of C^{k,0} (which commutes with the action of the Clifford algebra bundle of N in a graded way). This left action is used to define the Clifford multiplication with T R^{k} on pr_{N}^{∗}S^{k} in order to obtain the Clifford bundle S over R^{k} × N. We want to use induction in k. In order to employ the product structure R^{k} = R × R^{k−1} we have to deform Φ such that it respects this product structure. For 1/2 > ϵ > 0 let ψ ∈ C^{∞}(R) be such that ψ(r) = sign(r) for |r| ≥ 2/ϵ, ψ(r) ∈ [−1, 1] for |r| ≤ 2/ϵ and |∂_{r}ψ| ≤ 2ϵ. Define Φ_{1}(x, n) := z(ψ(x^{1}), …, ψ(x^{k})) ∈ R^{k}.

Lemma 2.25 Φ_{1} is admissible.

Proof: We have deg Φ_{1} = 1 and Φ_{1}^{∗} = Φ_{1}. Moreover, Φ_{1}D + DΦ_{1} + Φ_{1}^{2} = −z grad(zΦ_{1}) + Φ_{1}^{2}. But |grad(zΦ_{1})| ≤ 2kϵ and for |x| ≥ 2k/ϵ we have Φ_{1}^{2} ≥ 1. □

Let {M_{1}} be the class associated to B_{1} := D + Φ_{1}.

Lemma 2.26 For ϵ > 0 small enough it holds that {M} = {M_{1}}.

Proof: We can deform Φ to Φ_{1} inside the admissible endomorphisms. Let Φ_{t} = tΦ + (1 − t)Φ_{1}. Then, for |x| ≥ 2k/ϵ,

Φ_{t}D + DΦ_{t} + Φ_{t}^{2}
= −tz grad(zΦ) − (1 − t)z grad(zΦ_{1}) + t^{2}Φ^{2} + (1 − t)^{2} ∑_{i=1}^{k} ψ(x^{i})^{2}

+2t(1 − t)\frac{1}{|x|} ∑_{i=1}^{k} x^{i}ψ(x^{i})

≥ t^{2} + (1 − t)^{2} − t|grad(zΦ)| − 2(1 − t)kϵ
≥ 1/8

for ϵ small enough. □

Let now R^{k} = R × R^{k−1} and n be the unit normal vector in the R-direction. n generates C^{1,0} and we can split S^{k} = S^{k−1} ⊗ C^{1,0} by the following procedure. Let e := zlr, where l is the action of n from the left and r is the action of n from the right. e is an even involution of S^{k}. Let S^{k−1} be the positive eigenspace of e. Then S^{k−1} admits a C^{n−1,0}-action from the right, where C^{n−1,} is generated by R^{⊥} ⊂ R^{n}, is a Clifford bundle over N and admits a C^{k−1,0} action from the left commuting in the graded sense with the action of the Clifford algebra bundle of N. In fact, the generators of C^{k−1,0} acting from the left and of C^{n−1,0} acting from the right commute with the involution e.

For 0 ≤ i ≤ k let M^{(i)} := R^{i} × N with the Clifford bundle induced from S^{i}. Construct the admissible endomorphism Φ_{1}^{(i)} as above and form {M^{(i)}}.

Proposition 2.27 {M^{(i)}} = {M^{(i−1)}}

Proof: We identify ker(D^{(i)} + Φ^{(i)}) with ker(D^{(i−1)} + Φ^{(i−1)}) as a C^{n−k,0}-module. With respect to S^{i} = S^{i−1} ⊗ C^{1,0} we have

D^{(i)} + Φ^{(i)} = l\frac{∂}{∂x^{i}} + zrψ(x^{i}) + [D^{(i−1)} + Φ^{(i−1)}] ⊗ 1.

Multiplying by −l we obtain

A := \frac{∂}{∂x^{i}} + zlrψ(x^{i}) − l[D^{(i−1)} + Φ^{(i−1)}] ⊗ 1.

A respects the splitting

L^{2}(R^{i}×N, S^{i}) = L^{2}(R)⊗ker([D^{(i−1)}+Φ^{(i−1)}]⊗1)⊕L^{2}(R)⊗ker([D^{(i−1)}+Φ^{(i−1)}]⊗1)^{⊥}

and is invertible on the second summand. The elements of ker A are of the form

f_{v}(x^{i}) = exp(−zlr ∫_{0}^{x^{i}} ψ(t)dt)v.

From the L^{2}-condition we see that ker A is the 1-eigenspace of zlr in ker([D^{(i−1)} + Φ^{(i−1)}]⊗1), but this is exactly ker(D^{(i−1)} +Φ^{(i−1)}). All identifications we have made respect the structure of graded C^{n−k,0}-modules. This proves the Proposition since the class {M} is determined by the kernel of the Callias-type operator as a graded C^{n−k,0}-module. □

We come now back to the general situation. Let M be a complete Riemannian manifold and S → M be a real Z_{2}-graded real Clifford bundle over M admitting a right C^{n,0}-action with associated Dirac operator D. Let F ∈ C_{g}^{∞}(M) ⊗ R^{k} for 0 < k ≤ n as above and N := F^{−1}(0). We have constructed a Clifford bundle S^{0} → N with Dirac operator D_{N}^{(0)} admitting a right C^{n−k,0}-action. Putting the above results together we have

Theorem 2.28 {M} = ind D_{N}^{(0)}.

We will apply this result to obstructions against the existence of positive scalar curvature metrics. Let (M^{n}, g) be a complete Riemannian spin manifold of dimension n. There is a natural real Clifford bundle S → M admitting a C^{n,0}-action S := P_{Spin} ×_{Spin(n)} C^{n,0}. Let F ∈ C_{g}^{∞}(M) ⊗ R^{k}, n ≥ k ≥ 1, with |F | → 1 at infinity. Assume that 0 ∈ R^{k} is a regular value of F and define N := F^{−1}(0). Then N is a compact spin manifold and S^{0} is the natural Clifford bundle associated to the spin structure of N. In this case ind D_{N}^{(0)} = α(N), where α(N) is the alpha invariant of N (see e.g. Hitchin [19] and Lawson/Michelson [21]). Let D be the Dirac operator of S. Let τ denote the scalar curvature of M. By the Weitzenböck formula, if τ is uniformly positive at infinity, then D^{2} = Δ + τ/4 D is invertible at infinity. In this case we can deform the admissible endomorphism to zero without changing the index. However, D is equivariant with respect to a larger Clifford algebra C^{n,0} and hence the class of its kernel vanishes. It follows α(N) = 0. Recall that two Riemannian metrics g_{0}, g_{1} are called quasi-isometric if there is a constant C < ∞ such that

\frac{1}{C} g_{0} ≤ g_{1} ≤ Cg_{0}.

Since the conditions on F only depend on the quasi-isometry class of the Riemannian metric on M, we obtain

Corollary 2.29 Let (M^{n}, g) be a complete Riemannian spin manifold and F ∈ C_{g}^{∞}(M)⊗R^{k} be such that |F | → 1 at infinity and 0 is a regular value. If α(F^{−1}(0)) ≠ 0, then there is no metric in the quasi-isometry class of g that has uniform positive scalar curvature at infinity.

This generalizes similar results of Roe [27] to dimensions n−k = 1, 2 (mod 8), where the index has values in Z_{2}. A special case is:

Corollary 2.30 If α(N) ≠ 0, then there is no metric with uniform positive scalar curvature at infinity on the product R^{k} × N for any k in the quasi-isometry class of the product metric.

Corollary 2.31 If α(N) ≠ 0 ,then there is no metric of non-negative scalar curvature τ on R^{k} × N with lower bound τ(x, n) ≥ \frac{c}{|x|+1} for some c > 0 in the quasiisometry class of the product.

Proof: Assume that M := R^{k}×N has a metric with non-negative scalar curvature in the quasi-isometry class of the product satisfying τ(x, n) ≥ \frac{c}{|x|+1} for some constant c > 0. Consider B_{t} := D + tzF with F ∈ C^{∞}(M, R^{k}) depending only on the first coordinate and satisfying F^{−1}(0) = {0} × N and F (x, n) = x/|x| for |x| > 1. Then B_{t}^{2} = Δ + τ/4 − zt grad F + t^{2}F^{2}. But |grad F (x, n)| ≤ \frac{c_{1}}{|x|+1} for some c_{1} > 0. Thus

τ/4 − zt grad F + t^{2}F^{2} ≥ \frac{c/4 − c_{1}t}{|x| + 1} + t^{2}F^{2}.

If we chose t < \frac{c}{4c_{1}} , then B_{t} is positive and hence invertible. It follows that {M} = {M_{t}} = 0 and α(N) = 0. □

Corollary 2.32 Let M be a complete Riemannian spin manifold with non-negative scalar curvature. Let F ∈ C^{∞}(M) ⊗ R^{k} have 0 ∈ R^{k} as a regular value and be constant of unit length outside of a tubular neighbourhood U of N = F^{−1}(0). Assume that the scalar curvature is positive on U. Then α(N) = 0.

Proof: Consider B_{t} = D + tzF for small t > 0. B_{t} is invertible at infinity. Thus, by Theorem 2.28 and the deformation invariance α(N) = {M_{1}} = {M_{t}}. Then B_{t}^{2} = Δ + τ/4 − zt grad(F) + t^{2}|F |^{2}. If t is small enough, then τ/4 − zt grad(F) ≥ 0 and B_{t} is invertible, but then {M_{t}} = 0. □

This sort of argument was used in Lesch [22] in order to prove

Corollary 2.33 ([22]) If α(N) ≠ 0, then there is no complete Riemannian metric on R × N that has non-negative scalar curvature which is positive on some section {a} × N ⊂ R × N.

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