The two definitions of the index difference
Given two metrics of positive scalar curvature on a closed spin manifold, there is a secondary index invariant in real -theory. There exist two definitions of this invariant, one of homotopical flavor, the other one defined by an index problem of Atiyah-Patodi-Singer type. We give a complete and detailed proof of the folklore result that both constructions yield the same answer. Moreover, we generalize this result to the case of two families of positive scalar curvature metrics, parametrized by a finite CW complex. In essence, we prove a generalization of the classical “spectral-flow-index theorem” to the case of families of real operators.
Key words and phrases:Fredholm model for K-theory, Bott periodicity, spectral flow, Dirac operator, positive scalar curvature
1991 Mathematics Subject Classification:19K56, 53C21, 53C27, 55N15, 58J30, 58J40
1.1. Two secondary index invariants for positive scalar curvature metrics
The space of metrics of positive scalar curvature on a closed spin manifold of dimension has attracted the attention of homotopically minded geometric topologists. One tool for its study comes from index theory. Given any Riemannian metric on , one can consider the Atiyah-Singer-Dirac operator , which is an odd, -linear and symmetric elliptic operator that acts on the space of sections of the real spinor bundle . Thus it yields a point in , the space of self-adjoint odd -linear Fredholm operators. By a classical result of Atiyah-Singer and Karoubi, represents the real -theory functor , and thus we get an index . If has positive scalar curvature, then : the well-known argument using the Schrödinger-Lichnerowicz formula proves that is invertible, which shows that at first glance, the index does not convey any information. There are, however, useful secondary index-theoretic invariants that one can attach not to single metric of positive scalar curvature, but to a pair of such. There are two constructions of such invariants. What they have in common is that one starts by considering the family of metrics, parametrized by . Of course, and this is the point, the metric typically does not have positive scalar curvature.
For the first construction, we consider the path in the space (in this introduction, we are glossing over the detail that and hence depends on ). This path begins and ends at an invertible operator, since has positive scalar curvature. As the space of invertible operators is contractible (Kuiper’s theorem), the path contains the homotopical information of a point in the loop space . This space represents the functor , and in this way we obtain the first version of the index difference . This viewpoint was introduced by Hitchin [Hit].
For the second construction, we extend the metric on to a metric on the cylinder (constant outside ) and consider the Dirac operator on of this metric. As the dimension of is , this is a -linear operator. It defines a Fredholm operator on , and we can consider it as a point in , giving another element (one could view this an index of an Atiyah-Patodi-Singer boundary value problem as in [APS1], but we use a different setup). This viewpoint on the index difference is due to Gromov and Lawson [GL].
An obvious question is whether both constructions yield the same result and this question is most cleanly formulated in the family case, to which both constructions can be generalized. There are well-defined homotopy classes of maps
Both map the diagonal to the (contractible) space of invertible operators. The Bott periodicity map is a weak equivalence
The two maps and are weakly homotopic. In other words, if is a finite CW complex and is a map, then and are homotopic. Moreover, if the subcomplex is mapped to the diagonal, the homotopy can be chosen to be through invertible operators on .
The restriction to finite CW pairs is out of convenience; it can probably be removed, but we opine that this is not worth the effort. Certainly, Theorem A does not come as a surprise at all and in fact it has the status of a folklore result. However, we are not aware of a published adequate exposition. As indicated, the secondary index is an important tool to detect homotopy classes in the space , compare [GL, Hit, HSS, CS]. The authors of these works have been careful to avoid any use of Theorem A. In [BERW], a stronger detection theorem for is proven, for manifolds of all dimensions . In that paper, Theorem A is used in an essential way to pass from -manifolds to -dimensional manifolds. Thus, filling this gap was motivated not by an encyclopedic striving for completeness, but by necessity.
1.2. Spectral-flow-index theorem
Let us explain the strategy for the proof of Theorem A, first under the assumption that and . In this case, is isomorphic to , detected by an ordinary Fredholm index. The space is homeomorphic to the space of self-adjoint Fredholm operators on a real Hilbert space. The fundamental group is isomorphic to , and it is detected by the “spectral flow” : if is a family of operators such that invertible, then is the number of eigenvalues of that cross , counted with multiplicities. This concept was introduced by Atiyah-Patodi-Singer [APS3, §7]. To such a family, one can associate the operator , acting on . This is Fredholm, and so has an index. The “spectral-flow index theorem” states that (for the moment, we ignore the sign). The basic idea why Theorem A is expected to be true is that Hitchin’s index difference is given by the spectral flow of the family , while Gromov-Lawson’s index difference is the index of the operator .
There are many approaches to spectral-flow-index theorems in the literature. Atiyah-Patodi-Singer gave a proof for a special case [APS3, Theorem 7.4]: has to be an elliptic operator on a closed manifold, and the crucial assumption is that . This is important, because their proof is by gluing the ends together in order to reduce to an index problem on , which can be solved by the usual Atiyah-Singer index theorem. However, this assumption is not satisfied in our case. One obvious first idea for such a reduction would be to use that and are homotopic through invertible operators, by Kuiper’s theorem. Composing the family with such a homotopy would result in a closed family , and the operator would have the same index. But the index problem for the new family still cannot be reduced to a problem on , for a very fundamental reason: for that to work the homotopy must be through pseudo-differential operators. Even though and are homotopic through elliptic differential operators and through invertible operators on the Hilbert space, we cannot fulfill both requirements at once! In fact, our proof will clearly show that this is the essential information captured by the spectral flow.
Before we describe our argument, let us discuss several other approaches that appeared in the literature. Bunke [Bu93] considers the case when is a family of differential operators with the same symbol. He reduces the problem to the closed case; but his answer is in terms of cohomology, and he does not treat the parametrized situation. Translating his argument to real -theory would, as far as we can see, not have resulted in a shorter proof of Theorem A. Robbin and Salamon [RoSa] worked in an abstract functional analytic setting, ignoring that the operators are pseudo-differential. For the case , , they gave a detailed proof in this abstract setting. We will use a special case of their result in our proof; but we failed with an attempt at a straightforward generalization of their argument to the family case. Another proof in the framework of -theory is due to Kaad and Lesch [KL], again the details are only for the complex case and ; our knowledge of Kasparov theory does not suffice to carry out the generalization to the case we need.
1.3. Overview of the paper
Let us now give a description of what we actually do. Chapter 2 surveys background material on Clifford algebras and -Theory. In section 2.1, we collect the conventions on Clifford algebras that we use (we use all Clifford algebras to make the linear algebra work better). Section 2.2 recalls the classical Fredholm model for real -Theory. We have to generalize the Fredholm model in such a way that a Hilbert bundle, together with a Fredholm family represents an element in -Theory. There are well-known difficulties with the structural group and the continuity condition on a Fredholm family. Therefore, we spend some pages explaining these conditions (section 2.3). Section 2.4 discusses the generalized Fredholm model; the proof that the construction gives the correct answer is deferred to the appendix LABEL:appendix. A side-purpose of chapter 2 is to close a gap in the literature. Classically, the family index theorem is only formulated for compact base spaces, and even a definition of the family index over a noncompact base does not seem to be discussed properly in the literature. In [BERW], we need to consider family indices over not even locally compact bases, and chapter 2, together with the appendix, was partially written with that goal in mind.
The goal of chapter 3 is to give the rigorous definition of the secondary index invariants and the formulation of the main result of this paper (Theorem 3.22). Section 3.1 collects some facts on elliptic regularity for manifolds with cylindrical ends; the key result is Proposition 3.7. In order to cover the other index-theoretic arguments that appear in [BERW], we prove more general versions than necessary for Theorem A. The overall structure of the proof of Theorem A forces us to leave the realm of Dirac operators; we have to work with pseudo-differential operators that have the leading symbol of a Dirac operator (we call them “pseudo-Dirac operators”). The general setting for our index theorem are families of -linear pseudo-Dirac operators on a closed manifolds. Given such a family, we get a new operator on the manifold , which is -linear (called suspension). While the family yields , the operator corresponds to . We organize the curves of Dirac operators on , parametrized by a space , in a suitable -group that we call . The two constructions (family index of and family index of ) give maps and our main result (Theorem 3.22) says that both maps are equal. In section 3.4, we deduce Theorem A from Theorem 3.22.
Chapter 4 contains the proof of Theorem 3.22 and we follow a common strategy for proving index theorems. The crucial analytical ingredient for the proof of Theorem 3.22 is Proposition LABEL:thm:invertiblepseudos which states that the space of curves of -linear pseudo-Dirac operators , invertible, is rich enough to realize . This is inspired by a theorem of Booß-Wojciechowski [BW] and would be false if we tried to use differential operators. In section LABEL:formal-structures, we use Proposition LABEL:thm:invertiblepseudos and formal properties of -theory to reduce everything to the special case and , which is the case that was dealt with by Robbin and Salamon (in fact, we use an explicit index computation instead).
I am grateful to Boris Botvinnik and Oscar Randal-Williams for the exciting collaboration (of which the present paper is an outsourced part); moreover I want to thank Boris and his wife Irina for the warm welcome in their home. Thomas Schick made several useful comments on this paper, but in particular I want to thank the anonymous referee for reading the paper in an extremely careful way.
2. Preliminaries on -Theory and Clifford algebras
2.1. Clifford algebra
Throughout the paper, we work over the real numbers; the proofs can easily be “complexified”. Without any further mentioning, we assume (or claim implicitly) that all Hilbert spaces are separable.
Let and be two finite-dimensional euclidean vector spaces and let be a Hilbert space. A -structure on is a pair , where is a self-adjoint involution of and is a linear map to the space of bounded operators on such that
for all , . A -Hilbert space is a Hilbert space, equipped with a -structure. The opposite -Hilbert space is and is shortly denoted by . A bounded linear operator of -Hilbert spaces will be called Clifford-linear if holds all . A Clifford-linear bounded operator is even if , and odd if .
If there is no risk of confusion, we write for . Of particular interest to us is the case , , both with the standard scalar product. In this case, we write instead of . A -structure on is given by orthogonal automorphisms of , satisfying the relations
There are various functors between the categories of -Hilbert spaces for different values of , the classical Morita equivalences. If is a -Hilbert space, we obtain a -Hilbert space , and
(here and are the standard basis vectors of ). If is a Clifford-linear endomorphism of , then is a Clifford-linear endomorphism of , and if is even or odd, then so is . Vice versa, if is a -Hilbert space with and being the actions of the basis vectors in the -summands, consider ; this inherits a -structure from the given one on . Clifford-linear endomorphisms of restricts to Clifford-linear endomorphisms of , and if is even or odd, then so is . Both procedures are mutually inverse. In a similar fashion, -Hilbert spaces and -Hilbert spaces are equivalent. Let be a -Hilbert space and put , so that . Then we obtain a -Hilbert space ; , for . Clifford-linear, even (or odd) endomorphisms are preserved under this procedure. By a similar recipe one transforms -Hilbert spaces back into -Hilbert spaces. Combining both types of equivalences, one gets equivalences between -, - and -Hilbert spaces. All these definitions and constructions generalize without effort to nontrivial Riemannian vector bundles , and Hilbert bundles .
The structure theory of -Hilbert spaces is well-known [LM, §I.5]. For brevity, we call finite-dimensional -Hilbert spaces just -modules. Each -Hilbert space decomposes into a Hilbert sum of (finite-dimensional) irreducible ones. If , then there is exactly one irreducible -module, up to isomorphism. If , there are exactly two irreducible -modules, up to isomorphism. These are mutually opposite and distinguished by their chirality: consider the operator . If , then is -linear, even and satisfies . In an irreducible -module, must act by (Schur’s lemma), and this sign is the chirality.
A -Hilbert space is called ample if it contains each irreducible -module with infinite multiplicity.
Note that two ample -Hilbert spaces are isomorphic. If is an ample -Hilbert space, , , one can extend the -structure on to an ample -structure. There are strong homotopy theoretic versions of these statements. Let be a real Hilbert space and let be the set of ample -structures; a point in is given by linear isometries and we topologize as a subspace of , equipped with the norm topology. If is understood, we write . The following result is well-hidden in [AS69] and [Karoubi-paper], and we make the proof explicit.
If is infinite dimensional, the space is contractible. Moreover, if , , then the forgetful map is a Serre fibration with contractible fibers.
Fix and let be obtained by restriction. Let be the group of isometries of , equipped with the norm topology. This group acts by conjugation on , and the action is transitive, by the structure theory of Clifford modules. Let be the stabilizer of . This is the group of all -linear and even isometries. By Kuiper’s theorem [Kui], . By Morita equivalences and ampleness, is homeomorphic to either the group of isometries of an infinite-dimensional Hilbert space over or or a product of such. Thus by [Kui] as well. The space is the homogeneous space and so is contractible. The forgetful map can be identified with the -bundle and so the proof is complete. To make this argument valid, it remains to be shown that has local sections (so that is a -principal bundle). For this, use that a -structure can be viewed as an orthogonal representation of the finite group generated by symbols , subject to the Clifford relations and invoke the following general lemma (take to be the sum of all irreducible -modules). ∎
Let be a finite group and let be a Hilbert space. Let be a unitary representation. Let be the space of representations such that (i.e., equipped with the -action induced by ) is -isomorphic to . Endow with the norm topology, as a subspace of . Then the map , , has local sections.
(This proof was suggested to us by the referee) We first construct the local section around the basepoint . We need to find a neighborhood of and a map , such that is -equivariant. For each , define the bounded operator
It is clear that and that . Moreover
so that is continuous. In particular, is invertible if is close to . Now let be the polar decomposition, with unitary. The operator is equivariant as a map , and so is equivariant as a map , as claimed.
Now let be arbitrary. Since is surjective, there is a unitary with . A section to near is given by . ∎
2.2. The Fredholm model for -Theory, version 1
For each Hilbert space , denotes the space of all bounded Fredholm operators , equipped with the operator norm topology. Let be a -Hilbert space. A -Fredholm operator on is a bounded, -linear, self-adjoint and odd Fredholm operator . If is ample and , let be the space of all -Fredholm operators on , equipped with the operator norm topology. If , then is the space of all such Fredholm operators with the property that the (self-adjoint) operator is neither essentially positive nor essentially negative. The subspace of invertible elements is denoted .
The relevance of the condition in the case is the following. One restricts to an invertible operator on . By a spectral deformation, is homotopic to an involution . The datum forms a -structure on , and the condition is that this structure is ample. The following is a classical result due to Atiyah-Singer and Karoubi.
[AS69, Karoubi-paper] If is an ample -Hilbert space, then is a representing space for the functor .
Using the Morita equivalences, it is easy to derive Theorem 2.7 from the version proven in [AS69].
If is ample, then is contractible.
This follows without pain from Lemma 2.4: inside , there is the subspace of involutions. By a spectral deformation argument, the inclusion is a homotopy equivalence (the details of this argument can be found in [AS69]). If , then defines an extension of the -structure to a -structure. Therefore, we can identify with the fiber of the restriction map , and so it is contractible by Lemma 2.4. ∎
By the symbol , we denote the space of continuous paths such that . This space indeed has the weak homotopy type of the homotopy-theoretic loop space of , provided that is ample. The proof is a standard exercise in elementary homotopy theory, using Lemma 2.8.
For many practical purposes, the description of -Theory given in Theorem 2.7 is not flexible enough. Before we can describe a more useful model, we need to say a few words about Fredholm families on Hilbert bundles.
2.3. A digression on Hilbert bundles
For us, a Hilbert bundle will always be a fiber bundle with structural group and fiber . Here, is a separable (possibly finite-dimensional) real Hilbert space and is its unitary group, with the compact-open topology. The fiber over will be denoted . The trivial Hilbert bundle with fiber over will be denoted by the symbol . A -structure on will be described by a tuple of isometric automorphisms of , satisfying the relations (2.2). A Fredholm family on will be given by a collection , a Fredholm operator on , and likewise a -Fredholm family will be a Fredholm family such that is a -Fredholm operator for each . One needs a continuity condition on , and the right formulation of it is a highly nontrivial insight by Dixmier and Douady [DD]. It is given in Definitions 2.9, 2.12 and 2.14 below (we were also strongly influenced by Atiyah-Segal [Atiseg] and Kasparov’s -Theory).
A homomorphism of Hilbert bundles is a fiber-preserving continuous map which is linear in each fiber.
A homomorphism is determined by a collection of bounded linear operators . If is locally compact or metrizable, then the function , is locally bounded, by the Banach-Steinhaus theorem (but in general not continuous).
Let be a separable Hilbert space, a compactly generated topological space and be a map. The map , is a homomorphism of Hilbert bundles if and only if is continuous when the target is equipped with the compact-open topology.
The collection of adjoint operators may or may not form a homomorphism, and if it does, we say that is adjointable. Note that a homomorphism which is pointwise self-adjoint is adjointable. The set of adjointable endomorphisms of is a unital -algebra . We say that is an isomorphism or invertible if it is a unit in , in other words: is an isomorphism if is an isomorphism for all and if the collection is a homomorphism. For self-adjoint homomorphisms , there is a functional calculus: if is self-adjoint and is a continuous function, then the collection is a homomorphism, see [DD, p. 245].
Let be a Hilbert bundle and let be a self-adjoint homomorphism. Assume that there is a continuous function such that for all . Then is invertible.
Invertibility of is a local (in ) property: if there is an open cover of such that is invertible for all , then is invertible. Under the assumptions of the Lemma, there is an open cover of and for each , there is such that for all . Let be an odd continuous function with for . Then is a homomorphism and for , we have . Therefore, is invertible for all and hence is invertible. ∎
The notion of a compact operator is more difficult to formulate and we follow [DD, §22] (see also Proposition 3 loc.cit.). Let be the space of continuous sections of , and for each , one defines the adjointable operator by (for ). The linear span of all these operators is a -ideal (if is adjointable, then and ). Clearly, the operator is of finite rank and hence an element of , the space of compact operators in .
An endomorphism of is compact if for each and each , there exists and a neighborhood of such that for all , one has . The space of all compact operators is denoted and is a two-sided -ideal in .
Note that if is compact, then is compact for all . It is not hard to characterize compact operators in trivial bundles.
The compact operators of the trivial Hilbert bundle are precisely the continuous maps (the target has the norm topology, in contrast to the situation of Example 2.10).
Let be a Hilbert bundle and . We say that is a Fredholm family if there exists such that . We say that is a parametrix to . If is equipped with a graded -structure, then a -Fredholm family is a Fredholm family such that each is self-adjoint, Clifford linear and odd.
We need criteria to check that a family is Fredholm. The first useful thing to know is that on reasonable spaces, being Fredholm is a local property (it is not a pointwise property).
Let be a Hilbert bundle over a paracompact space and . Assume that there is an open cover of such that is Fredholm for all . Then is Fredholm.
Let be a parametrix for . Let be a partition of unity subordinate to . Then is a parametrix to (since the property of being compact is local in ). ∎
Let be a Hilbert bundle over a space. Let be a collection of bounded operators . Assume that each admits a neighborhood of and a trivialization such that in this trivialization is given by a continuous map (the target has the norm topology). Then . If is compact (invertible) for each , then is compact (invertible). If is Fredholm for all , then is Fredholm, provided that is paracompact.
The statements on adjointability, compactness and invertibility are trivial consequences of what we have said so far. Next, recall that the quotient map has a continuous section (both spaces are equipped with the norm topology). This follows from a general theorem by Bartle and Graves which states that if is a surjective bounded operator between Banach spaces, then there is a continuous cross-section ([BartGrav], see also [Holm, p. 187] for the explicit statement). Using this section, one constructs a map such that is a parametrix for , for each . Therefore, if is Fredholm for all , then is locally Fredholm. By Lemma 2.15, it follows that is globally Fredholm. ∎
Let be a Hilbert bundle, paracompact and let be self-adjoint. The following statements are equivalent.
is a Fredholm family.
There is a continuous function such that in the sense that there exists with (pointwise).
If is Fredholm, then there is a parametrix , and we can assume that is self-adjoint. Then and are compact. For , we pick and a neighborhood of such that and hence for (the norm of a homomorphism is locally bounded). It follows that and therefore throughout . Put . This proves the desired inequality locally. Globally, one patches the constant functions on these neighborhoods together with a partition of unity.
For the reverse implication, pick and as in (2). Then is invertible (Lemma 2.11) and is compact, proving that is a right-parametrix of . Similarly, is a left-parametrix. ∎
2.4. The Fredholm model for -Theory, version 2
We denote the product of space pairs by .
Let be a space pair. A -cycle on is a pair , where is a -Hilbert bundle (with separable fibers) and a -Fredholm family on . Moreover, we require that is invertible. There are obvious notions of pullbacks, direct sum and isometric isomorphisms of -cycles. Two -cycles and are homotopic or concordant if there exists a -cycle on such that the restriction of to is isomorphic to . Homotopy is an equivalence relation. A -cycle is acyclic if is invertible. The set of homotopy classes of -cycles is an abelian monoid, and we define the abelian group as the quotient of that monoid by the submonoid of homotopy classes that contain acyclic representatives.
We implicitly said in the definition that is a group, and this is indeed true. For this and similar purposes, we use the following convenient criterion to prove that two -cycles are concordant. We say that a space pair is a paracompact pair if is paracompact and is closed. We denote the anticommutator of two operators by
Let and be two -cycles on the paracompact space pair . If the anticommutator is nonnegative, i.e. for all , then .
This is a generalization of a special case of a result in -theory due to Connes and Skandalis [Bla, Proposition 17.2.7]. One important feature of Lemma 2.19 is that the positivity condition can be checked pointwise, hence often the proof for general is only notationally more involved than that for .
Let be a paracompact pair. Then is a group, and the additive inverse to is or .
We claim that . The operator is an odd, -linear involution that anticommutes with and a straightforward application of Lemma 2.19 concludes the proof. The second formula is proven in the same way with . ∎
How do these new groups relate to -Theory?
Let be a compact pair. Then the obvious comparison map
Moreover, the functor is representable for paracompact pairs.
There exists a space pair and a natural map
which is bijective whenever is paracompact and compactly generated and closed. Moreover, the space is weakly contractible.
and a formal consequence of the above results is that it is a weak homotopy equivalence (because and are by definition metric spaces and therefore paracompact). Let us have a look at Bott periodicity in this framework. We write , and
Let and be a -cycle over . Put . Define a -cycle on by the formula
The operator is self-adjoint and odd and hence so is . Since anticommutes with each , the formula holds and is invertible if or if is invertible. Observe that commutes with and , but not with . When we define a -action by reindexing the generators, is -linear. This construction preserves the equivalence relation defining the -groups, and we obtain a group homomorphism, the Bott map
The Bott periodicity theorem states that (2.23) is an isomorphism for all paracompact and compactly generated pairs; this follows from the main result of [AS69], together with Theorems 2.21 and 2.22. Using the Morita equivalences, we obtain natural isomorphisms of functors
The formulas for the Morita equivalences that we gave make it clear that these maps are compatible with the Bott maps whenever this statement makes sense, i.e. if . Another fundamental structure is a -module structure on . To define it, we use the model for by complexes of vector bundles, see [ABS, Part II]. We can slightly reformulate this construction by saying:
Let be a compact pair. Consider the monoid of concordance classes of -cycles such that the underlying Hilbert bundle has finite rank and the same equivalence relations as in Definition 2.18. The quotient group is naturally isomorphic to .
The natural map is an isomorphism for compact pairs . For and , we define
It is easy to see that this definition yields a well-defined bilinear map
If , this defines the usual cross product on under the isomorphism from Theorems 2.21 and 2.22. This product is compatible with Morita equivalences and Bott periodicity, in the sense that these maps are linear over .
3. Analytical arguments
3.1. Preliminaries on elliptic theory
In this subsection, we recollect some purely analytical results from the literature. It was written for the convenience of the readers with less background in analysis (including the author), and we also want to give a reference for further analytical results needed in [BERW], which is why we formulate the analysis in greater generality.
Even though our main result only involve differential operators, the proof requires to use the bigger class of pseudo-differential operators. Let us first summarize the essential properties of pseudo-differential operators that we will use, following Atiyah and Singer [AS1, ASIV]. Let be a closed manifold and let be a smooth vector bundle. In [AS1, §5], a vector space of linear maps , called pseudo-differential operators of order (denoted in loc. cit.), is considered whose properties we briefly recall.
and is a filtered algebra. Pseudo-differential operators have (formal) adjoints.
Pseudo-differential operators have symbols. Let be the cotangent bundle without the zero section. The leading symbol of is a smooth section of which is positively homogeneous of degree