Index and secondary index theory for flat bundles with duality

Ulrich Bunke^{∗}and Xiaonan Ma^{†}

June 29, 2001

Abstract

We discuss some aspects of index and secondary index theory for flat bundles with duality. This theory was first developed by Lott [9]. Our main purpose in the present paper is provide a modification with better functorial properties.

Contents

1 The functor L 1 1.1 Definition of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Computation of L(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 The natural transformation to K-theory . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Secondary L-groups. The ℝ/ℤ-variant . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Index and secondary index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Geometry and extended secondary L-groups 5 2.1 Definition of L̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Functorial properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Eta invariants and index maps 8 3.1 The η-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 The secondary index map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 The index map for L^{ex} and L̄^{ex} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

∗Göttingen, bunke@uni-math.gwdg.de †Berlin, ma@mathematik.hu-berlin.de

1 The functor L

1.1 Definition of L

We define a contravariant functor L from the category T op of topological spaces and continuous maps to ℤ_{2}-graded rings. Let X be a topological space and ϵ ∈ ℤ_{2}. If F is a locally constant sheaf of finite-dimensional ℝ-modules over X, then an ϵ-symmetric duality structure on F is an isomorphism of sheafs q : F → F^{∗} satisfying q^{∗} = (−1)^{ϵ}q, where F^{∗} := Hom(F, \underline{ℝ}). We first define a semigroup L̂_{ϵ}(X) and then obtain L_{ϵ}(X) by introducing a relation. A generator of the semigroup L̂_{ϵ}(X) is a pair (F, q) consisting of a locally constant sheaf of finite-dimensional ℝ-modules and an ϵ-symmetric duality structure q. The operation in L̂_{ϵ}(X) is given by direct sum

(F, q) + (F′, q′) := (F ⊕ F′, q ⊕ q′) .

The group L_{ϵ}(X) is obtained from L̂_{ϵ}(X) by introducing the relation lagrangian reduction. We require (F, q) ∼ 0 if there exists a locally constant lagrangian subsheaf i : L ↪ F, i.e. the composition L →^{i} F →^{q} F^{∗} →^{i^{∗}} L^{∗} vanishes and L = L^{⊥} := ker(i^{∗} ∘ q). The class of (F, q) in L_{ϵ}(X) will be denoted by [F, q]. The ring structure L_{ϵ}(X) ⊗ L_{ϵ′}(X) → L_{ϵ+ϵ′}(X) is induced by the tensor product:

[F, q][F′, q′] := \begin{matrix}\sqrt{(−1)^{ϵ}} & \sqrt{(−1)^{ϵ′}} \\ √ & \overline{(−1)^{ϵ+ϵ′}} \end{matrix} [F ⊗ F′, q ⊗ q′] .

The sign-convention is made such that later we have a natural transformation of rings from L to complex K-theory K^{0}. If f : Y → X is a morphism in T op, then f^{∗} : L(X) → L(Y) is defined by f^{∗}[F, q] = [f^{∗}F, f^{∗}q]. The map f^{∗} only depends on the homotopy class of f. \underline{Remark:} A version L(X)^{Lott} of this ring was first introduced by Lott [9]. His definition differs from ours since our relation ”lagrangian reduction” is repaced by ”hyperbolic reduction” in the definition of Lott. Here a generator (F, q) is called hyperbolic if there is an lagrangian subsheaf L ⊂ F which admits a complement inside F. In particular, L(X) is a quotient of L(X)^{Lott}.

1.2 Computation of L(X)

2 1 THE FUNCTOR L

Assume that X is path connected and fix a base point x. Let G := π_{1}(X, x) be the fundamental group. We consider the semigroup L̂_{ϵ}(G) of all tuples (F, q, ρ), where F is a finite-dimensional real vector space, q : F → F^{∗} is an ϵ-symmetric duality structure, and ρ : G → Aut(F, q) is a representation of G. The semigroup operation is given by direct sum. We then define the group L_{ϵ}(G) by introducing the relation (F, q, ρ) ∼ 0 if there exists an invariant lagrangian subspace L ⊂ F. Taking the holonomy representation on the fibre over x we obtain a bijection L(X) ≅ L(G). Let Ĝ := G×ℤ_{2}. If (F, q, ρ) ∈ L̂(G), then we form the representation (F̂, ρ̂) of Ĝ by F̂ := F ⊕F^{∗}, ρ̂((g, 0)) := ρ(g) ⊕ ρ(g^{−1})^{∗},

ρ̂(1, 1) := ( \begin{matrix}0 & q^{−1} \\ q & 0 \end{matrix} ) .

If (F, q, ρ) is an irreducible representation of G, then (F̂, ρ̂) is an irreducible representation of Ĝ. An irreducible representation (V, σ) of Ĝ is called real, complex, or quaternionic, if End_{G}(V) ≅ ℝ, ≅ ℂ or ≅ ℍ, respectively. For each irreducible representation (F, ρ) of G we define a ℤ_{2}-graded group A(F, ρ) as follows. If (F, ρ) admits an invariant ϵ-symmetric form q, then we define A_{ϵ}(F, ρ) := ℤ, iff (F̂, ρ̂) is real or quaternionic, A_{ϵ}(F, ρ) := ℤ_{2}, iff (F̂, ρ̂) is complex, and we set A_{ϵ}(F, ρ) := 0, if (F, ρ) does not admit an invariant ϵ-symmetric form. If A(F, ρ) ≠ 0, then we fix an invariant ϵ-symmetric form q_{F,ρ}. Let Rep(G) denote the set of isomorphism classes of irreducible representations of G.

Theorem 1.1 There is a natural isomorphism of ℤ_{2}-graded groups

L(G) = ⨁_{(F,ρ)∈Rep(G)} A(F, ρ)

fixed by the condition that (F, q, q_{F,q}) ↦ 1_{(F,q)}.

1.3 The natural transformation to K-theory

By T op_{para} we denote the full subcategory of T op of paracompact topological spaces. Let K^{0}(X) be the complex K-theory functor. We construct a natural transformation b : L → K^{0} of functors from T op_{para} to rings. A locally constant sheaf of finite-dimensional ℝ-modules on X gives rise to a locally trivial real vector bundle bundle(F) in a natural way. The correspondence bundle is functorial and compatible with direct sum, tensor product, and duality. Thus applying the bundle construction to (F, q) we obtain a pair (F, Q) consisting of a finite-dimensional real vector bundle and an isomorphism Q : F → F^{∗}. Let (F, Q) be a real vector bundle with an isomorphism Q : F → F^{∗} such that Q^{∗} = (−1)^{ϵ}Q for ϵ ∈ ℤ_{2}. Following the language introduced by Lott [9] we call an isomorphism J : F → F a metric structure, if

1. J^{∗} ∘ Q defines a scalar product on F, 1.4 Secondary L-groups. The ℝ/ℤ-variant 3

2. J^{2} = (−1)^{ϵ}id_{F},

3. J^{∗} ∘ Q = (−1)^{ϵ}Q ∘ J.

Since we assume that X is paracompact it admits partitions of unity. This implies that metric structures exist, and that the space of all metric structures is contractible. Given (F, Q) as above we choose a metric structure J. Let F_{ℂ} be the complexification of F. Then \frac{1}{\sqrt{(−1)^{ϵ}}} J is a ℤ_{2}-grading of F_{ℂ}, and the pair (F_{ℂ}, \frac{1}{\sqrt{(−1)^{ϵ}}} J) represents an element of K^{0}(X) which does not depend on the choice of J. The transformation b : L → K^{0} is obtained by composing the latter construction with bundle.

1.4 Secondary L-groups. The ℝ/ℤ-variant

We first recall the definition of the 2-periodic cohomology theory K_{ℝ/ℤ} introduced by [7], [8]. Let BU be the classifying space of complex K-theory. The Chern character (with real coefficients) is induced by a map ch : BU → ∏_{n=1}^{∞} K(ℝ, 2n). The homotopy fibre of this map classifies K_{ℝ/ℤ}. In particular, for any paracompact space X there is a natural exact sequence of K^{0}(X)-modules

→ K^{−1}(X) →^{ch} H^{odd}(X, ℝ) → K_{ℝ/ℤ}^{−1}(X) →^{β} K^{0}(X) →^{ch} H^{ev}(X, ℝ) → ,

where K^{0}(X) acts on cohomology via the Chern character. We now define the functor X ↦ L̄^{ℝ/ℤ}(X) from paracompact topological spaces to ℤ_{2}-graded groups by the following pull-back diagram:

\begin{matrix}L̄^{ℝ/ℤ}(X) & → & L(X) \\ ↓ & & ↓ b \\ K_{ℝ/ℤ}^{−1}(X) & →^{β} & K^{0}(X) \end{matrix} .

The grading is induced from that of L(X). On morphisms the functor L^{ℝ/ℤ} only depends on homotopy classes. Note that K^{0}(X) and K_{ℝ/ℤ}^{−1}(X) are L(X)-modules via b. This induces a graded L(X)-module structure on L̄^{ℝ/ℤ}(X). We have the following natural exact sequence of L(X)-modules

K^{−1}(X) →^{ch} H^{odd}(X, ℝ) → L̄^{ℝ/ℤ}(X) → L(X) →^{ch∘b} H^{ev}(X, ℝ) .

1.5 Index and secondary index

Let X → B be a smooth locally trivial fibre bundle over a compact base B with compact evendimensional fibres such that the vertical bundle T X/B is orientable. In this case we have the following maps:

• π_{∗}^{sign} : H^{∗}(X, ℝ) → H^{∗}(B, ℝ) defined by π_{∗}(ω) = ∫_{X/B} ω ∩ L(T X/B), where ∫_{X/B} is integration over the fibre and L(T X/B) denotes the Hirzebruch L-class of the vertical bundle.

• π_{!}^{sign} : K^{0}(X) → K^{0}(B) defined by π^{sign}([E]) = index(D_{E}^{sign}), where D_{E}^{sign} is the fibrewise signature operator twisted by E and index(D_{E}^{sign}) ∈ K^{0}(B) denotes the class of the index bundle.

• π_{∗}^{L} : L(X) → L(B) defined by π_{∗}^{L}[F, q] = [H^{•}Rπ_{∗}F, π_{∗}(q)], where H^{•}Rπ_{∗}F = ⊕_{i=0}^{∞}R^{i}π_{∗}(F) is the direct sum of higher derived direct images of F. The orientation of T X/B provides an isomorphism p : R^{dim(T X/B)}π_{∗}\underline{ℝ} →^{∼} \underline{ℝ}. It induces isomorphisms R^{i}π_{∗}F^{∗} → (R^{dim(T X/B)−i}π_{∗}F)^{∗}, and π_{∗}(q) is the composition of the sum of these maps with the sum of R^{i}π_{∗}(q) : R^{i}π_{∗}F → R^{i}π_{∗}F^{∗}.

It is an excercise in spectral sequences to show that π_{∗}^{L} is well-defined. \underline{Remark :} In [9] Lott defines π_{∗}^{L,Lott} : L^{Lott}(X) → L^{Lott}(B) which induces π_{∗}^{L} by passing to quotients. By the index theorem for families and fibrewise Hodge theory the following diagram commutes

\begin{aligned}L(X) & → & K^{0}(X) & → & H^{ev}(X, ℝ) \\ π_{∗}^{L} ↓ & & π_{!}^{sign} ↓ & & π_{∗}^{sign} ↓ \\ L(B) & → & K^{0}(B) & → & H^{ev}(B, ℝ) \end{aligned} .

In order to define an index map for K_{ℝ/ℤ} we need the further assumption that π is K-oriented. Thus assume that T X/B has a Spin_{c}-structure. Then there are maps π_{!}^{Spin_{c}} : K^{0}(X) → K^{0}(B) and π_{!}^{Spin_{c},ℝ/ℤ} : K_{ℝ/ℤ}^{−1}(X) → K_{ℝ/ℤ}^{−1}(B) (compare e.g. [8]), such that the following diagram commutes:

\begin{aligned}H^{odd}(X, ℝ) & → & K_{ℝ/ℤ}^{−1}(X) & → & K^{0}(X) & → & H^{ev}(X, ℝ) \\ π_{∗}^{Spin_{c}} ↓ & & π_{!}^{Spin_{c},ℝ/ℤ} ↓ & & π_{!}^{Spin_{c}} ↓ & & π_{∗}^{Spin_{c}} ↓ \\ H^{odd}(B, ℝ) & → & K_{ℝ/ℤ}^{−1}(B) & → & K^{0}(B) & → & H^{ev}(B, ℝ) \end{aligned} ,

where π_{∗}^{Spin_{c}}(ω) = ∫_{X/B} Â(T X/B) ∩ e^{c_{1}/2} ∩ ω and c_{1} is the first Chern class determined by the Spin_{c}-structure. There is an unique element E_{sign} ∈ K_{0}(X) such that π_{!}^{sign}(x) = π_{!}^{Spin_{c}}(E_{sign} • x). Note that ch(E_{sign}) ∩ Â(T X/B) ∩ e^{c_{1}/2} = L(T X/B).

• π^{sign,ℝ/ℤ} : K_{ℝ/ℤ}^{−1}(X) → K_{ℝ/ℤ}^{−1}(B) is defined by π^{sign,ℝ/ℤ}(x) := π^{Spin_{c},ℝ/ℤ}(E_{sign} • x) so that the following diagram commutes

\begin{aligned}H^{odd}(X, ℝ) & → & K_{ℝ/ℤ}^{−1}(X) & → & K^{0}(X) & → & H^{ev}(X, ℝ) \\ π_{∗}^{sign} ↓ & & π_{!}^{sign,ℝ/ℤ} ↓ & & π_{!}^{sign} ↓ & & π_{∗}^{sign} ↓ \\ H^{odd}(B, ℝ) & → & K_{ℝ/ℤ}^{−1}(B) & → & K^{0}(B) & → & H^{ev}(B, ℝ) \end{aligned} .

• π_{∗}^{L̄,ℝ/ℤ} : L̄^{ℝ/ℤ}(X) → L̄^{ℝ/ℤ}(B) is the map induced by π_{!}^{sign,ℝ/ℤ} and π_{∗}^{L}. The following diagramm commutes:

\begin{aligned}H^{odd}(X, ℝ) & → & L̄^{ℝ/ℤ}(X) & → & L(X) & → & H^{ev}(X, ℝ) \\ π_{∗}^{sign} ↓ & & π_{∗}^{L̄,ℝ/ℤ} ↓ & & π_{∗}^{L} ↓ & & π_{∗}^{sign} ↓ \\ H^{odd}(B, ℝ) & → & L̄^{ℝ/ℤ}(B) & → & L(B) & → & H^{ev}(B, ℝ) \end{aligned} .

All index maps are natural with respect to pull-back of fibre bundles. Let π_{1} : X → X_{1} and π_{2} : X_{1} → B be locally trivial smooth fibre bundles with closed evendimensional fibres and compact base. Further assume that the vertical bundles T X_{1}/X_{2} and T X_{2}/B carry Spin_{c}-structures (and are therefore oriented). Then the composition π = π_{2} ∘ π_{1} : X → B is a locally trivial fibre bundle with closed even-dimensional fibres, and the vertical bundle T X/B carries an induced Spin_{c}-structure. In this situation the index maps on complex K-theory, K_{ℝ/ℤ}-theory, and in cohomology behave functorially with respect to the iterated fibre bundle. It is again an excercise in spectral sequences to show that π_{∗}^{L} is also functorial. As a consequence, π_{∗}^{L̄,ℝ/ℤ} is functorial, i.e. π_{∗}^{L̄,ℝ/ℤ} = (π_{2}^{L̄,ℝ/ℤ})_{∗} ∘ (π_{1}^{L̄,ℝ/ℤ})_{∗}. \underline{Remark :} The motivation of introducing the quotient L(X) of Lott’s group L^{Lott}(X) is to implement this functoriality for π_{∗}^{L}, which is not true for Lott’s definition.

2 Geometry and extended secondary L-groups

2.1 Definition of L̄

The functor X ↦ L̄^{ℝ/ℤ}(X) from T op_{para} to ℤ_{2}-graded L(X)-modules was defined by a purely homotopy-theoretic construction as an extension of the functor

X ↦ ker (ch ∘ b : L(X) → H^{ev}(X, ℝ))

by X ↦ H^{odd}(X, ℝ)/ch(K^{−1}(X)). Let T op_{smooth} denote the full subcategory of topological spaces T op_{para} which are homotopy equivalent to smooth manifolds. In the present section we use a differential geometric construction in order to define on T op_{smooth} a functor X ↦ L̄(X) to graded L(X)-modules which extends X ↦ ker(ch ∘ b) by X ↦ H^{odd}(X, ℝ). First we define L̄(M) for a smooth manifold M. Again we first define a semigroup ˆ L̄(M). A generator of ˆ L̄_{ϵ}(M) is a tuple (F, q, J, ρ), where (F, q) is a generator of L̂_{ϵ}(M), J is a metric structure on F := bundle(F), and ρ ∈ Ω^{4∗−(−1)^{ϵ}}(M)/im(d) satisfies dρ = p(∇^{F} , J). Here (Ω^{∗}(M), d) is the real deRham complex of M and ∇^{F} is the canonical flat connection on F (such that ker(∇^{F}) ≅ F). The metric structure J induces a ℤ_{2}-grading of F_{ℂ}, and if ∇^{F_{ℂ},J} denotes the even part of the extension of ∇^{F} to 6 2 GEOMETRY AND EXTENDED SECONDARY L-GROUPS

F_{ℂ}, then p(∇^{F} , J) := ch(∇^{F_{ℂ},J}) = tr_{s} exp(−R^{∇^{F_{ℂ},J}} /2πi) is the characteristic form representing ch ∘ b([F, q]). The semigroup operation is given by

(F, q, J, ρ) + (F′, q′, J′, ρ′) := (F ⊕ F′, q ⊕ q′, J ⊕ J′, ρ + ρ′) .

We form L̄_{ϵ}(M) by introducing the relation lagrangian reduction. Let L ⊂ F be a locally constant lagrangian subsheaf and L := bundle(L). Then we have a decomposition F = L⊕J(L). Let ∇^{⊕} be the part of ∇^{F_{ℂ},J} which preserves this decomposition, and let c̃h(∇^{⊕}, ∇^{F_{ℂ},J}) be the transgression Chern form such that dc̃h(∇^{⊕}, ∇^{F_{ℂ}}) = ch(∇^{⊕})−ch(∇^{F_{ℂ}}). Note that ch(∇^{⊕}) = 0. In L̄_{ϵ}(M) we require

(F, q, J, ρ) ∼ (0, 0, 0, ρ + c̃h(∇^{⊕}, ∇^{F_{ℂ},J})) .

By [F, q, J, ρ] we denote the class in L̄ represented by (F, q, J, ρ). \underline{Remark :} In [9] similar functors L̄^{Lott} were defined replacing the relation ”lagrangian reduction” by ”hyperbolic reduction” and ”change of metric structure”. Lott’s relation is smaller than ours, and we have a natural surjective map L̄^{Lott}(M) → L̄(M). The graded module structure of L̄(X) over L(X) is defined by

[F, q, J, ρ] • [E, p] := \begin{matrix}\sqrt{(−1)^{ϵ}} & \sqrt{(−1)^{ϵ′}} \\ √ & \overline{(−1)^{ϵ+ϵ′}} \end{matrix} [F ⊗ E, q ⊗ p, J ⊗ J^{E}, ρ ∧ ch(∇^{E_{ℂ},J^{E}})] ,

where J^{E} is any metric structure on bundle(E).

2.2 Functorial properties

If f : M → N is a smooth map of manifolds, then we obtain an induced map f^{∗} : L̄(N) → L̄(M), which is given on generators by pull-back of structures. f^{∗} only depends on the smooth homotopy class of f and is compatible with the L(M)-module structures. We define natural maps H^{odd}(M, ℝ) → L̄(M) and L̄(M) → L(M) by [ρ] ↦ [0, 0, 0, ρ] and [F, q, J, ρ] ↦ [F, q]. Then we have the following exact sequence of L(M)-modules

H^{odd}(M, ℝ) → L̄(M) → L(M) → H^{ev}(M, ℝ)

(see [9], Prop. 21, for a similar argument).

Proposition 2.1 The map H^{odd}(M, ℝ) → L̄(M) is injective.

Indeed, let ω ∈ Ω^{odd}(M) be a closed form. If (0, 0, 0, ω) ∼ 0 in L̄_{ϵ}(M), there there exists (F, q, J, ρ) together with two lagrangian subsheaves L_{0}, L_{1} such that

[ω] = [c̃h(∇^{⊕_{0}}, ∇^{F_{ℂ}}) − c̃h(∇^{⊕_{1}}, ∇^{F_{ℂ}})] = [c̃h(∇^{⊕_{0}}, ∇^{⊕_{1}})]

in H^{odd}(M), where ∇^{⊕_{i}}, i = 0, 1 are defined using the decompositions F = L_{i} ⊕ J(L_{i}). The right-hand side belongs to ch(K^{−1}(X)) and is therefore rational. On the other hand it depends continuously on J. Therefore it is independent of J. First reducing to the case that L_{0}∩L_{1} = {0} and then choosing J such that JL_{i} = L_{1−i}, i = 0, 1, we see that [ω] = 0. We now construct a natural morphism of L(M)-modules γ̄ : L̄(M) → K_{ℝ/Z}^{−1}(M). Here we use the definition of K_{ℝ/Z}^{−1}(M) in terms of generators and relations given in [8], Def. 5 + 6. A generator of K_{ℝ/Z}^{−1}(M) is a tuple (E, h^{E}, ∇^{E}, ρ), where E is a ℤ_{2}-graded complex vector bundle of vertual dimension zero, h^{E} is a hermitean metric and ∇^{E} is a metric connection, both being compatible with the grading, and ρ ∈ Ω^{odd}(M)/im(d) satisfies dρ = ch(∇^{E}). The relations of K_{ℝ/Z}^{−1}(M) are generated by

1. isomorphism (E, h^{E}, ∇^{E}, ρ) ∼ (E′, h^{E′}, ∇^{E′}, ρ) if there exists an isomorphism from E to E′ which is compatible with metrics and connections.

2. direct sum (E, h^{E}, ∇^{E}, ρ) + (E′, h^{E′}, ∇^{E′}, ρ′) = (E ⊕ E′, h^{E⊕E′}, ∇^{E⊕E′}, ρ + ρ′)

3. change of connections (E, h^{E}, ∇, ρ) ∼ (E, h^{E}, ∇′, ρ′) if ρ′ = ρ + c̃h(∇′, ∇).

4. trivial elements If (E, h^{E}, ∇^{E}) is a Z_{2}-graded hermitean vector bundle with connection, then (E ⊕ E^{op}, h^{E⊕E}, ∇^{E⊕E}, 0) ∼ 0, where E^{op} denotes E with the opposite grading.

Let [E, h^{E}, ∇^{E}, ρ] denote the class of (E, h^{E}, ∇^{E}, ρ) in K_{ℝ/Z}^{−1}(M). We define L̄(M) → K_{ℝ/Z}^{−1}(M) by γ̄[F, q, J, ρ] = [F_{ℂ}, h^{F_{ℂ}}, ∇^{F_{ℂ},J}, ρ], where h^{F_{ℂ}} is the hermitean extension of the metric J^{∗} ∘ Q on F. The following diagram commutes

\begin{aligned}L̄(M) & → & L(M) \\ ↓ & & ↓ \\ K_{ℝ/Z}^{−1}(M) & → & K^{0}(M) \end{aligned} .

We therefore obtain a natural map L̄(M) → L̄^{ℝ/ℤ}(M) which is in fact surjective. We now extend the functor L̄ to T op_{smooth} by setting

L̄(X) := lim_{f:M→X^{⟵}} L̄(M) ,

where the limit is taken over the category of manifolds over X. This extension has all functorial properties discussed above.

3 Eta invariants and index maps

3.1 The η-invariant

Using the η-invariant of the twisted signature operator for a closed odd-dimensional oriented manifold M in [9] Lott constructed a group homomorphism η^{Lott} : L̄^{Lott}(M) → ℝ. Its reduction modulo ℤ factors over the homomorphism η^{ℝ/ℤ} : K_{ℝ/ℤ}^{−1}(M) → ℝ/ℤ constructed in [8], which is given by the pairing with the K-homology class induced by the odd signature operator. Unfortunately, the homomorphism η^{Lott} does not factor over the quotient L̄^{Lott}(M) → L̄(M). In order to fix this for a given closed odd-dimensional oriented manifold M we define an extension

0 → ℤ → L̄_{ϵ}^{ex}(M) → L_{ϵ}(M) → 0 ,

such that

η̄: L̄_{ϵ}^{ex}(M) → L̄_{ϵ}(M) → K_{ℝ/ℤ}^{−1}(M) →^{η^{ℝ/ℤ}} ℝ/ℤ

lifts to η : L̄_{ϵ}^{ex}(M) → ℝ. Note that L̄^{ex}(M) is not a functor on M. Here n := dim(M), ϵ_{n} := [\frac{n(n−1)}{2}] ∈ ℤ_{2}, and we assume that ϵ = 1 − ϵ_{n}. Let (F, q) be a generator of L_{ϵ}(M) and L be a lagrangian locally constant subsheaf of F. We define an integer τ(F, q, L) by the following construction. We consider the complex of sheaves

K^{•} : L →^{i} F →^{i^{∗}∘q} L^{∗} .

Let (E_{r}^{••}, d_{r}), r ≥ 1 be the associated hyper cohomology spectral sequence. The duality q and the pairing between L and L^{∗} induce a duality q_{K} : K →^{∼} K^{∗}[−2] (the argument ”[−2]” indicates that q_{K} is a map of degree −2). The orientation of M and q induce a duality q_{E_{r}} : E_{r} →^{∼} E_{r}^{∗}[−2, − dim(M)] (i.e. of bidegree (−2, − dim(M))). Let N_{r} denote the total grading on E_{r}. Then we define the symmetric form Q_{r} on E_{r} by Q_{r}(v, w) = q_{E_{r}}((−1)^{\frac{Nr(Nr−1)}{2}} v)(d_{r}w). The integer τ(F, q, L) is now given by

τ(F, q, L) := 2(sign(Q_{1}) + sign(Q_{2})) .

We first define an extension L_{ϵ}^{ex}(M) of L_{ϵ}(M). A generator of the semigroup group L̂_{ϵ}^{ex}(M) is a triple (F, q, z) consisting of a locally constant sheaf of finite-dimensional ℝ-modules, an ϵ-symmetric duality structure q, and an integer z. The semigroup operation is given by

(F, q, z) + (F′, q′, z′) := (F ⊕ F′, q ⊕ q′, z + z′) .

Then we form L_{ϵ}^{ex}(M) by introducing the relation lagrangian reduction

(F, q, z) ∼ (0, 0, z + τ(F, q, L))

if F admits a locally constant lagrangian subsheaf L. There is an exact sequence

0 → ℤ → L_{ϵ}^{ex}(M) → L_{ϵ}(M) → 0 ,

where the maps are the obvious inclusion and projection. We now define L̄_{ϵ}^{ex}(M) by the following pull-back diagram:

\begin{aligned}L̄_{ϵ}^{ex}(M) & → & L_{ϵ}^{ex}(M) \\ ↓ & & ↓ \\ L̄_{ϵ}(M) & → & L_{ϵ}(M) \end{aligned} .

Then we have exact sequences

0 → ℤ → L̄_{ϵ}^{ex}(M) → L̄_{ϵ}(M) → 0

and

H^{4∗−(−1)^{ϵ}}(M, ℝ) → L̄_{ϵ}^{ex}(M) → L_{ϵ}^{ex}(M) → H^{4∗+1−(−1)^{ϵ}}(M, ℝ)

in a natural way. An element of L̄_{ϵ}^{ex}(M) can be represented by a tuple (F, q, J^{F} , ρ, z). We obtain a class [F, q, J^{F} , ρ] ∈ L̄_{ϵ}^{Lott}(M). If we set

η(F, q, J^{F} , ρ, z) := η^{Lott}([F, q, J^{F} , ρ]) − z ,

then we have the following result.

Theorem 3.1 The map (F, q, J^{F} , ρ, z) ↦ η(F, q, J^{F} , ρ, z) induces a well-defined homomorphism η : L̄_{ϵ}^{ex}(M) → ℝ which lifts η̄.

We must show that η is well-defined with respect to lagrangian reduction. The idea is to consider the η-invariant of the signature operator on M twisted with the complex K = bundle(K) and to rescale its differential. The corresponding adiabatic limits can be understood by the methods developed in [3] and [6] without investing essentially new ideas.

3.2 The secondary index map

Let M → B be a smooth locally trivial fibre bundle with even-dimensional closed fibres over a compact base B such that the vertical bundle T M/B is oriented. We set n := dim(T M/B). In [9] Lott constructed a secondary index map π_{∗}^{L̄,Lott} : L̄_{ϵ}^{Lott}(M) → L_{ϵ+ϵ_{n}}^{Lott}(B) which fits into the commutative diagram

\begin{aligned}H^{4∗−(−1)^{ϵ}}(M, ℝ) & → & L̄_{ϵ}^{Lott}(M) & → & L̄_{ϵ}^{ℝ/ℤ}(M) \\ π_{∗}^{sign} ↓ & & π_{∗}^{L̄,Lott} ↓ & & π_{∗}^{L̄,ℝ/ℤ} ↓ \\ H^{4∗−(−1)^{ϵ+ϵn}} (M, ℝ) & → & L̄_{ϵ+ϵ_{n}}^{Lott}(B) & → & L̄_{ϵ+ϵ_{n}}^{ℝ/ℤ}(B) \end{aligned} .

Theorem 3.2 By passing to quotients the map π_{∗}^{L̄,Lott} induces a well-defined secondary index map

πL̄ ; ∗ : L̄_{ϵ}(M) → L̄_{ϵ+ϵ_{n}}(B) .

The idea of the proof consists in investigating the adiabatic limits of the η-form of the fibrewise signature operator twisted with the complex K under rescaling the differential. The arguments are similar to the case of analytic torsion forms. It suffices to adapt the methods developed in [4], [1], [2], [10], [11], [12], [13]. The following diagram commutes

\begin{aligned}H^{4∗−(−1)^{ϵ}}(M, ℝ) & → & L̄_{ϵ}(M) & → & L_{ϵ}(M) \\ π_{∗}^{sign} ↓ & & πL̄ ; ∗ ↓ & & π_{∗}^{L} ↓ \\ H^{4∗−(−1)^{ϵ+ϵn}} (M, ℝ) & → & L̄_{ϵ+ϵ_{n}}(B) & → & L_{ϵ+ϵ_{n}}(B) \end{aligned} .

The secondary index map is natural with respect to pull-back of fibre bundles. Furthermore, if π_{1} : X → X_{1} and π_{2} : X_{2} → B is an iterated bundle with oriented vertical bundles, and π = π_{2} ∘ π_{1}, then

Theorem 3.3 πL̄ ; ∗ = (πL̄ ; 2)_{∗} ∘ (πL̄ ; 1)_{∗}

This follows from the functoriality of the η -form studied in [11] in a similar way as the functoriality of the secondary index in [5] was deduced from the functoriality of the higher analytic torsion form [10].

3.3 The index map for L^{ex} and L̄^{ex}

Let π : M → B as above and assume that B is oriented, closed, of dimension m. Furthermore, assume that ϵ = 1 − ϵ_{n} − ϵ_{m}. Let (F, q) be a generator of L_{ϵ}(M). We define an integer τ(F, q, M →^{π} B) by the following construction. Let (E^{••}, d_{r}) be the Leray-Serre spectral sequence converging to sheaf cohomology H^{∗}(M, F) with E_{2}^{p,q} = H^{p}(B, R^{q}π_{∗}F). The orientations of T M/B and B and the duality q induce dualities q_{E_{r}} : E_{r} → E_{r}^{∗}[−m, −n] which give the Poincaré duality on the limit. We define the symmetric form Q_{r} on E_{r} by Q_{r}(v, w) = q_{E_{r}}((−1)^{\frac{Nr(Nr−1)}{2}} v)(d_{r}w), where N_{r} denotes the total degree. Then we set

τ(F, q, M →^{π} B) := 2 ∑_{r≥2} sign(Q_{r}) .

Theorem 3.4 1. The prescription

π_{∗}^{L^{ex}}[F, q, z] := [H^{•}Rπ_{∗}F, π_{∗}(q), z − τ(F, q, M →^{π} B)] .

defines an extended index map π_{∗}^{L^{ex}} : L_{ϵ}^{ex}(M) → L_{ϵ+ϵ_{n}}^{ex}(B).

REFERENCES 11

2. The extented index map is functorial with respect to iterated fibre bundles. The following diagram

L̄_{ϵ}(M) → L_{ϵ}(M) ← L_{ϵ}^{ex}(M)
πL̄ ; ∗ ↓ π_{∗}^{L} ↓ π_{∗}^{L^{ex}} ↓
L̄_{ϵ+ϵ_{n}}(B) → L_{ϵ+ϵ_{n}}(B) ← L_{ϵ+ϵ_{n}}^{ex}(B)

commutes and induces an extended secondary index map πL̄^{ex} ; ∗ : L̄^{ex}(M)_{ϵ} → L̄_{ϵ+ϵ_{n}}^{ex}(B) which is functorial with respect to iterated fibre bundles. We have the following compatibility of this extended secondary index map with the η-homomorphism. Theorem 3.5

L̄_{ϵ}^{ex}(M) →^{η} ℝ
πL̄^{ex} ; ∗ ↓  ∥
L̄_{ϵ+ϵ_{n}}^{ex}(B) →^{η} ℝ

This theorem essentially follows from Dai’s adabatic limit formula for the η-invariant [6].

References

[1] A. Berthomieu and J.M. Bismut. Quillen metrics and higher analytic torsion forms. J. Reine Angew. Math., 457(1994), 85–184.

[2] J. M. Bismut. Families of immersions, and higher analytic torsion. Asterisque, 244, 1997.

[3] J. M. Bismut and J. Cheeger. η-invariants and their adiabatic limits. J. AMS, 2(1989), 33–70.

[4] J.M. Bismut and G. Lebeau. Complex immersions and Quillen metrics. Publ. Math. IHES, 74, 1991.

[5] U. Bunke. On the functoriality of Lott’s secondary analytic index. Preprint, 2000.

[6] X. Dai. Adiabatic limits, non-multiplicativity of signature and Leray spectral sequence. J. A.M.S., 4(1991), 265–321.

[7] M. Karoubi. Homologie cyclique et K-théorie. Asterisque, 149, 1987.

[8] J. Lott. R/Z-index theory. Comm. Anal. Geom., 2(1994), 279–311.

[9] J. Lott. Secondary analytic indices. Preprint, 1999.

12 REFERENCES

[10] X. Ma. Functoriality of real analytic torsion form. Preprint, 1999.

[11] X. Ma and U. Bunke. Flat duality bundle and the eta form. Preprint, 2000.

[12] X. Ma. Formes de torsion analytique et familles de submersions I. Bull. Soc. Math. France, 127(1999), 541–621.

[13] X. Ma. Formes de torsion analytique et familles de submersions II. Asian J. of Math., 4(2000), 633-668.