L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES
SARA AZZALI
Abstract. We define the secondary invariants L2-eta and -rho forms for families
of generalized Dirac operators on normal coverings of fibre bundles. On the cover-
ing family we assume transversally smooth spectral projections and Novikov–Shubin
invariants bigger than 3(dimB + 1) to treat the large time asymptotic for general
operators. In the case of a bundle of spin manifolds, we study the L2-rho class in
relation to the space R+(M/B) of positive scalar curvature vertical metrics.
1. Introduction
Secondary invariants of Dirac operators are a distinctive issue of the heat equation
approach to index theory. The eta invariant of a Dirac operator first appeared as the
boundary term in the Atiyah–Patodi–Singer index theorem [2]: this spectral invariant,
highly nonlocal and therefore unstable, became a major object of investigation, because
of its subtle relation to geometry. With the introduction of superconnnections in index
theory by Quillen and Bismut, it became possible to employ heat equation techniques
in higher geometric situations, where the primary invariant, the index, is no longer a
number, but a class in a K-theory group [44, 10, 35]. This led to so called local index
theorems, which are refinements of the cohomological index theorems at the level of
differential forms, and gave as new fundamental byproduct the eta forms, coming from
the transgression of the index class [11, 12, 13], which are the higher analogue of eta
invariants [41, 38, 34].
Rho invariants are differences (or, more generally, delocalized parts) of eta invariants,
so they naturally possess stability properties when computed for geometrically relevant
operators, mainly the spin Dirac operator and the signature operator [3, 31, 42]. Fur-
thermore, they can be employed to detect geometric structures: the Cheeger–Gromov
L2-rho invariant, for example, has major applications in distinguishing positive scalar
curvature metrics on spin manifolds [15, 43], and can show the existence of infinitely
many manifolds that are homotopy equivalent but not diffeomorphic to a fixed one [16].
As secondary invariants always accompany primary ones, it is very natural to ask what
are the L2-eta and L2-rho forms in the case of a families, and what are their properties.
We consider the easiest L2-setting one could think of, namely a normal covering of a fibre
bundle. This interesting model contains yet all the features and problems offered by the
presence of continuos spectrum. Since the fibres of the covering family are noncompact,
the large time asymptotic of the superconnection Chern character is in general not
Date: September 4, 2010.
http://arxiv.org/abs/0704.0909v2
2 SARA AZZALI
converging to a differential form representative of the index class, and the same problem
is reflected when trying to integrate on [1,∞) the transgression term involved in the
definition of the L2-eta form.
The major result in this sense is by Heitsch and Lazarov, who gave the first families index
theorem for foliations with Hausdorff graph [30]. They computed the large time limit
of the superconnection Chern character as Haefliger form, assuming smooth spectral
projections and Novikov–Shubin invariants bigger than 3 times the codimension of the
foliation. Their result implies an index theorem in Haefliger cohomology (not a local
one, because they do not deal with the transgression term), which in particular applies
to the easier L2-setting under consideration.
We use the techniques of Heitsch–Lazarov to investigate the integrability on [1,∞) of
the transgression term, in order to define the L2-eta form for families D of generalised
Dirac operators on normal coverings of fibre bundles. Our main result, Theorem 3.4,
implies that the L2-eta form η̂(2)(D) is well defined as a continuos differential form on
the base B if the spectral projections of the family D are smooth, and the families
Novikov–Shubin invariants {αK}K⊂B are greater than 3(dimB + 1).
We define then naturally the L2-rho form ρ̂(2)(D) as the difference between the L2-eta
form for the covering family and the eta form of the family of compact manifolds. When
the fibre is odd dimensional, the zero degree term of ρ̂(2)(D) is the Cheeger–Gromov
L2-rho invariant of the induced covering of the fibre. We prove that the L2-form is
(weakly) closed when the fibres are odd dimensional (Prop. 4.3).
The strong assumptions of Theorem 3.4 are required because we want to define η̂(2) for
a family of generalised Dirac operators. In the particular case of de Rham and signature
operators one can put weaker assumptions: this is showed by Gong–Rothenberg’s result
for the L2-Bismut–Lott index theorem (proved under positivity of the Novikov–Shubin
invariants) [24], and from results in [4], where we develop a new approach to large time
estimate exclusive to the families of de Rham and signature operators. On the contrary,
a family of signature operators twisted by a fibrewise flat bundle has to be treated as a
general Dirac operator [7].
Next we investigate the L2-rho form in relation to the space R+(M/B) of positive scalar
curvature vertical metrics for a fibre bundle of spin manifolds. For this purpose, the
Dirac families D/ involved are uniformly invertible by Lichnerowicz formula, so that the
definition of the L2-rho form does not require Theorem 3.4, but follows from classical
estimates. Here the L2-rho form is always closed, and we prove the first step in order
to use this invariant for the study of R+(M/B), namely that the class [ρ̂(2)(D/)] is the
same for metrics in the same concordance classes of R+(M/B) (Prop.5.1). The action
of a fibrewise diffeomorphism is also taken into account.
Along the lines of [42] we can expect that if Γ is torsion-free and satisfies the Baum–
Connes conjecture, then the L2-rho class of a family of odd signature operators is an
oriented Γ- fibrewise homotopy invariant, and that [ρ̂(2)(D̃/ĝ)] vanishes correspondingly
to a vertical metric ĝ of positive scalar curvature.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 3
Acknowledgements This work was part of my researches for the doctoral thesis. I
would like to thank Paolo Piazza for having suggested the subject, for many interesting
discussions and for the help and encouragement. I wish to express my gratitude to
Moulay-Tahar Benameur for many interesting discussions.
2. Geometric families in the L2-setting
We recall local index theory’s machine, here adapted to the following L2-setting for
families.
Definition 2.1. Let π̃ : M̃ → B be a smooth fibre bundle, with typical fibre Z̃ con-
nected, and let Γ be a discrete group acting fibrewise freely and properly discontinuosly
on M , such that the quotient M = M̃/Γ is a fibration π : M → B with compact fibre Z.
Let p : M̃ → M̃/Γ = M denote the covering map. This setting will be called a normal
covering of the fibre bundle π and will be denoted with the pair (p : M̃ →M,π : M → B).
Let π : M → B be endowed with the structure of a geometric family (π : M →
B, gM/B ,V, E), meaning by definition:
• gM/B is a given metric on the vertical tangent bundle T (M/B)
• V the choice of a smooth projection V : TM → T (M/B) (equivalently, the choice
of a horizontal complement THM = KerV)
• E → M is a Dirac bundle, i.e. an Hermitian vector bundle of vertical Clifford
modules, with unitary action c : Cl(T ∗(M/B), gM/B) → End(E), and Clifford
connection ∇E.
To a gemetric family it is associated a family D = (Db)b∈B of Dirac operators along
the fibres of π, Db = cb ◦ ∇Eb : C∞(Mb, Eb) → C∞(Mb, Eb), where Mb = π−1(b), and
Eb := E|Mb .
If we have a normal Γ-covering p : M̃ → M of the fibre bundle π, the pull back of the
geometric family via p gives a Γ-invariant geometric family which we denote (π̃ : M̃ →
B, p∗gM/B , Ṽ , Ẽ).
2.0.1. The Bismut superconnection. The structure of a geometric family gives a distin-
guished metric connection ∇M/B on T (M/B), defined as follows: fix any metric gB on
the base and endow TM with the metric g = π∗gB ⊕ gM/B ; let ∇g the Levi-Civita con-
nection on M with respect to g; the connection ∇M/B := V∇gV on the vertical tangent
does not depend on gB ([9, Prop. 10.2]).
When X ∈ C∞(B,TB), let XH denote the unique section of THM s.t. π∗XH = X. For
any ξ1, ξ2 ∈ C∞(B,TB) let T (ξ1, ξ2) := [ξH1 , ξH2 ]− [ξ1, ξ2]H and let δ ∈ C∞(M, (THM)∗)
measuring the change of the volume of the fibres LξH vol =: δ(ξH ) vol. Following the
notation of [9], in formulas in local expression we denote as e1, . . . , en a local orthonormal
base of the vertical tangent bundle; f1, . . . fm will be a base of TyB and dy
1, . . . , dym will
denote the dual base. The indices i, j, k.. will be used for vertical vectors, while α, β, . . .
will be for the horizontal ones. The 2-form c(T ) =
α<β(T (fα, fβ), ei)eidy
αdyβ has
4 SARA AZZALI
values vertical vectors. Using the vertical metric, c(T )(fα, fβ) can be seen as a cotangent
vertical vector, hence it acts on E via Clifford multiplication.
Let H → B be the infinite dimensional bundle with fibres Hb = C∞(Mb, Eb). Its space of
sections is given by C∞(B,H) = C∞(M,E). We denote Ω(B,H) := C∞(M,π∗(ΛT ∗B)⊗
E). Let ∇H be the connection on H → B defined by ∇HU ξ = ∇EUHξ +
δ(ξH) where ξ is
on the right hand side is regarded as a section of E. ∇H is compatible with the inner
product < s, s′ >b:=
hE(s, s′) vol b , with s, s
′ ∈ C∞(B,H), and hE the fixed metric
on E.
Even dimensional fibre. When dimF = 2l the bundle E is naturally Z2-graded by chi-
raliry, E = E+ ⊕ E−, and D is odd. Correspondingly, the infinite dimensional bundle
is also Z2-graded: H = H+ ⊕ H−. The Bismut superconnection adapted to D is the
superconnection B = ∇H +D − c(T )
on H.
The corresponding bundle for the covering family π̃ is denoted H̃ → B where the same
construction for the family M̃ → B gives the Bismut superconnection B̃ = ∇H̃ + D̃ −
c(T̃ )
, adapted to D̃. It is Γ-invariant by construction, being the pull-back via p of B.
Odd dimensional fibre. When dimZ = 2l − 1, the appropriate notion is the one of
Cl(1)-superconnection, as introduced by Quillen in [44, sec. 5]. Let Cl(1) the Clifford
algebra Cl(1) = C⊕Cσ, where σ2 = 1, and consider EndE⊗Cl(1), adding therefore the
extra Clifford variable σ. On End(Eb) ⊗ Cl(1) = Endσ(Eb ⊕ Eb) define the supertrace
tr σ(A + Bσ) := trB, extended then to tr σ : C∞(M,π∗Λ∗B ⊗ EndE) → Ω(B) as usual
by tr σ(ω ⊗ (a+ bσ)) = ω tr b, for ω ∈ C∞(B,ΛT ∗B), ∀a, b ∈ C∞(B,EndE).
The family D, as well as c(T ) are even degree elements of the algebra C∞(B,EndH ⊗
Cl(1)⊗̂ΛT ∗B). On the other hand, ∇H is odd. By definition, the Bismut Cl(1)-
superconnection adapted to the family D is the operator of odd total degree Bσ :=
Dσ + ∇̃u − c(T )
Notation. In the odd case we will distinguish between the Cl(1)-superconnection de-
fined above Bσ acting on Ω(B,H) ⊗̂Cl(1), and the differential operator B : Ω(B,H) →
Ω(B,H) given by B := D +∇H − c(T )
, which is not a superconnection but is needed in
the computations.
2.1. The heat operator for the covering family. In this section we briefly discuss
the construction of the heat operator e−B̃
, which can be easily performed combining
the usual construction for compact fibres families in [9, Appendix of Chapter 9], with
Donnelly’s construction for the case of a covering of a compact manifolds [20]. We
integrate notations of [9, Ch. 9-10] with the ones of our appendix A. We refer to the
latter for the definitions of the spaces of operators used the rest of this section.
Let C∞(B,DiffΓ(Ẽ)) the algebra of smooth maps D : B → DiffΓ(Ẽ) satisfying that
∀z ∈ B, Dz is a Γ-invariant differential operator on M̃z, with coefficients depending
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 5
smoothly on the variables of B. In the same way, let N = C∞(B,ΛT ∗B⊗Op−∞Γ (Ẽ)) =
Ω(B,Op−∞Γ (Ẽ)) the space of smooth maps A : B → ΛT ∗B ⊗ Op
Γ (Ẽ). N contains
families of Γ-invariant operators of order −∞ with coefficients differential forms, hence
N is filtered by Ni = C∞(B,
j≥i Λ
jT ∗B ⊗Op−∞Γ (Ẽ)). The curvature of B̃ is a family
2 ∈ Ω(B,Diff2Γ(Ẽ)) and can be written as B̃2 = D̃2 − C̃, with C̃ ∈ Ω≥1(B,Diff1Γ(Ẽ)).
2.1.1. Definition and construction. For each point z ∈ B the operator e−tB̃2z is by defi-
nition an the one whose Schwartz kernel p̃zt (x, y) ∈ Ẽx⊗ Ẽ∗y ⊗ΛT ∗zB is the fundamental
solution of the heat equation, i.e.
• p̃zt (x, y) is C1 in t, C2 in x, y;
p̃zt (x, y) + B̃
z,II p̃
t (x, y) = 0 where B̃z,II means it acts on the second variable;
• lim
p̃zt (x, y) = δ(x, y)
• ∀T > 0 ∀t ≤ T ∃ c(T ) :
∥∂it∂
ypt(x, y)
∥ ≤ ct−
−i−j−ke−
2(x,y)
2 , 0 ≤ i, j, k ≤ 1.
Its construction is as follows: pose
e−tB̃
z := e−tD̃
tk e−σ0tD̃
z C̃e−σ1tD̃
z . . . C̃e−σktD̃
︸ ︷︷ ︸
dσ1 . . . dσk (2.1)
Since ∀σ = (σ0, . . . , σk) there exists σi > 1k+1 , then each term Ik ∈ ΛT
zB ⊗Op−∞(Ẽz)
and so does e−tB̃
z . Let p̃zt (x, y) = [e
−B̃2t,z ](x, y) be the Schwartz kernel of the operator
(2.1). Using arguments of [9, theorems 9.50 and 9.51], one proves that p̃zt (x, y) is smooth
in z ∈ B so that one can conclude e−B̃ ∈ Ω(B,Op−∞Γ ).
The next property, proved in [20] and [21], is needed in the t→ 0 asymptotic. For t < T0
−tB̃2 ](x̃, ỹ)
∣ ≤ c1t−
2 e−c2
2(x̃,ỹ)
t (2.2)
2.2. Transgression formulæ, eta integrands. For t > 0 let δt : Ω(B,H) → Ω(B,H)
the operator which on Ωi(B,H) is multiplication by t− i2 . Then consider the rescaled
superconnection Bt = t
2 δtBδ
t = ∇H +
tD − c(T ) 1
2.2.1. Even dimensional fibre. From (A.1) we have
Str Γe
−B̃2t = −dStr Γ
which on a finite interval (t, T ) gives the transgression formula
Str Γ
− Str Γ
Str Γ
ds (2.3)
6 SARA AZZALI
2.2.2. Odd dimensional fibre. Here it is convenient to use that tr σΓe
−(B̃σt )2 = tr oddΓ e
−B̃2t ,
(from [44] and (A.1)), where trodd means we take the odd degree part of the resulting
form. Then taking the odd part of the formula
tr Γe
−B2t = −d tr Γ
Tr oddΓ
− Tr oddΓ
Tr evenΓ
ds (2.4)
Remarks and notation 2.2. Since we wish now to look at the limits as t → 0 and
t → ∞ in (2.3) and 2.4, let us make precise what the convergences on the spaces of
forms are, and for families of operators. On Ω(B) we consider the topology of conver-
gence on compact sets. We say a family of forms ωt
C0→ ωt0 as t → t0 if ∀K
supz∈K ‖ωt(z)− ωt0(z)‖ΛT ∗z B → 0. We say ωt
C1→ ωt0 if the convergence also hold for
first derivatives of ωt with respect to the base variables. We say ωt = O(tδ) as t→ ∞ if
∃ a constant C = C(K) : supz∈K ‖ωt(z) − ωt0(z)‖ΛT ∗z B ≤ Ct
δ. We say ωt
= O(tδ) if
also the first derivatives with respect to base directions are O(tδ).
For a family Tt ∈ UC∞(B,Op−∞(Ẽ)) we say Tt
Ck→ Tt0 as t → t0 if ∀K
⊆ B, ∀r, s ∈ Z
supz∈K ‖Tt(z)− Tt0(z)‖r,s → 0 together with derivatives up to order k with respect to
the base variables.
On the space of kernels UC∞(M̃ ×B M̃, Ẽ4Ẽ∗ ⊗ π∗ΛT ∗B), we say kt → kt0 if ∀ϕ ∈
C∞c (B) ‖(π∗ϕ(x))(kt(x, y) − kt0(x, y))‖k → 0.
We stress that from (A.3) the map Ω(B,Op−∞Γ (Ẽ)) → UC∞(M̃ ×B M̃ , Ẽ4Ẽ∗ ⊗
π∗ΛT ∗B), T 7→ [T ] is continuos.
2.3. The t→ 0 asymptotic.
Proposition 2.3.
Str Γ
Â(M/B) chE/S if dim Z̃ = even
tr oddΓ
Â(M/B) chE/S if dim Z̃ = odd
The result is proved exactly as in the classic case of compact fibres, together with the
following argument of [33, Lemma 4, pag. 4]:
Lemma 2.4. [33] ∃A > 0, c > 0 s.t.
−B2t ](π(x̃), π(x̃))− [e−B̃2t ](x̃, x̃)
∣ = O(t−ce−
For the proof of the lemma see [32], or also [5], [24]. With the same technique we deduce
Proposition 2.5. The differential forms StrΓ
and tr σΓ
dB̃σt
e−(B̃
integrable on [0, 1], uniformly on compact subsets.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 7
Proof. The proof is as in [9, Ch.10, pag. 340]. We reason for example in the even case.
Consider the rescaled superconnection B̃s as a one-parameter family of superconnections,
s ∈ R+, and construct the new family M̆ = M̃ ×R+ → B ×R+ =: B̆. On Ĕ = Ẽ ×R+
there is a naturally induced family of Dirac operators whose Bismut superconnection
is B̆ = B̃s + dR+ − n4sds, and its rescaling is B̆t = B̃st + dR+ −
ds. Its curvature is
t = B̃
st + t
∧ ds, so that
t = e−B̃
e−uB̃
e−(1−u)B
st ∧ ds = e−F̃st − ∂B̃st
e−B̃st ∧ ds.
Str Γ
= Str Γ(e
−B̃2st)− Str Γ
∂B̃st
e−B̃st
ds (2.5)
At t = 0 we have the asymptotic expansion Str Γ(e
−B̆t) ∼
j=0 t
2 (Φ j
− α j
without singular terms. Computing (2.5) in s = 1, since
∂B̃st
, one has
Str Γ
e−F̃t
, and therefore Str Γ
e−F̃t
j=0 t
−1α j
. Let’s
compute α0. From the local formula
Φ0 − α0ds = lim
Str Γ
e−F̆t
M̆/B̆
Â(M̆/B̆) (2.6)
since M̆(z,s) = M̃z×{s} and the differential forms are pulled back from those on M̃ → B,
then the right hand side of (2.6) does not contain ds so that α0 = 0. This implies that
Str Γ(
t ) ∼
−1α j
3. The L2-eta form
We prove in Theorem 3.4 the well definiteness of the L2-eta form η̂(2)(D̃) under opportune
regularity assumptions. We make use of the techniques of [30].
3.1. The family Novikov–Shubin invariants. The t → ∞ asymptotic of the heat
kernel is controlled by the behaviour of the spectrum near zero. Let P̃ = (P̃ z)z∈B
the family of projections onto ker D̃ and let P̃ǫ = χ(0,ǫ)(D̃) be the family of spectral
projections relative to the interval (0, ǫ); denote Q̃ǫ = 1− P̃ǫ − P̃ .
For any z ∈ B the operator D̃z is a Γ-invariant unbounded operator: let D̃2z =
λdEz(λ)
be the spectral decomposition of D̃2z , andN
z(λ) = trΓE
z(λ) its spectral density function
[27]. Denote bz = trΓ P̃
z. Then N z(ǫ) = bz + trΓ P̃
ǫ and from [22] the behaviour of
θz(t) = trΓ(exp(−tD̃z)) at ∞ is governed by
αz = sup{a : θz(t) = bz +O(t−a)} = sup{a : N z(ǫ) = bz +O(ǫa)} (3.1)
8 SARA AZZALI
where αz is called the Novikov–Shubin invariant of D̃z.
We shall later impose conditions on αz uniformly on compact subset of B, so we intro-
duce the following definition from [24]: let K ⊂ B be a compact, define αK := infz∈K αz.
We call {αK}K⊂B the family Novikov–Shubin invariants of the fibre bundle M̃ → B.
By results of Gromov and Shubin [27], when D̃2z is the Laplacian, αz is a Γ-homotopy
invariant of M̃z [27], in particular it does not depend on z. In that case αz is locally
constant on B. For a general Dirac type operator this is not true and we need to use
the αK ’s.
Definition 3.1. [30] We say the family D̃ has regular spectral projections if P̃ and P̃ǫ are
smooth with respect to z ∈ B, for ǫ small, and ∇H̃P̃ ,∇H̃P̃ǫ are in N and are bounded
independently of ǫ. We say that the family D̃ has regularity A, if ∀K
⊆ B it holds
αK ≥ A.
Remark 3.2. To have regular projections is a strong condition, difficult to be verified
in general. The family of signature operators verifies the smoothness of P̃ [24, Theorem
2.2] but the smoothness of P̃ǫ is not clear even in that case.
The large time limit of the superconnection-Chern character StrΓ e
−B̃2t is computed in
[30, Theorem 5]. Specializing to our L2-setting it says the following.
Theorem 3.3. [30] Let ∇̃0 = P̃∇H̃P̃ . If D̃ has regular projections and regularity
> 3 dimB,
Str Γ(e
−B̃2t ) = Str Γe
3.2. The L2-eta form. We now use the same techniques of [30] to analyse the trans-
gression term in (2.3) and define the secondary invariant L2 eta form. We prove
Theorem 3.4. If D̃ has regular spectral projections and regularity > 3(dimB+1), then
= O(t−δ−1), for δ > 0. The same holds for trevenΓ
We start with some remarks and lemmas. In particular we shall repeatedly use the
following.
Remark 3.5. Let T ∈ N . From lemma A.6, ∀z ∈ B its Schwartz kernel [Tz] satis-
fies that for sufficiently large l, ∃ czl such that ∀x, y ∈ M̃z | [Tz](x, y) | ≤ czl ‖Tz‖−l,l
Therefore an estimate of ‖Tz‖−l,l produces directly via an estimate of TrΓ Tz.
Notation. Since in this section we are dealing only with the family of operators on the
covering, to simplify the notations let’s call D̃ = D, removing all tildes. Pose
Bǫ := (P +Qǫ)B(P +Qǫ) + PǫBPǫ
Aǫ = B− Bǫ
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 9
and write the rescaled operators as
Bǫ,t = (P +Qǫ)(Bt −
tD)(P +Qǫ) +
tD + Pǫ(Bt −
tD)Pǫ (3.2)
Aǫ,t = (P +Qǫ)(Bt −
tD)Pǫ + Pǫ(Bt −
tD)(P +Qǫ)
Denote also Tǫ = QǫBQǫ and Tǫ,t = QǫBtQǫ as in [30].
We will need the following two lemmas from [30]. The first is the “diagonalization” of
ǫ with respect to the spectral splitting of H.
Lemma 3.6. [30, Prop.6] Let M be the space of all maps f : B → ΛTB⊗End H̃. There
exists a measurable section gǫ ∈ M, with gǫ ∈ 1 +N1 such that
∇20 0 0
0 T 2ǫ 0
0 0 (PǫBPǫ)
N3 0 0
0 N2 0
0 0 0
The diagonalization procedure acts on (P ⊕Qǫ)H, in fact gǫ has the form gǫ = ĝǫ ⊕ 1,
with ĝǫ acting on (P ⊕Qǫ)H. From this lemma we get B2ǫ,t = tδtB2ǫδ−1t =
= tδtg
∇20 0 0
0 T 2ǫ 0
0 0 (PǫBtPǫ)
N3 0 0
0 N2 0
0 0 0
 gǫδt =
= δtg
tδt(∇20 +N3)δ−1t 0 0
0 tδt(T
ǫ +N2)δ−1t 0
0 0 PǫBtPǫ
δtgǫδ
The next lemma gives an estimate of the terms which are modded out.
Lemma 3.7. [30, lemma 9] If A ∈ Nk is a residual term in the diagonalization lemma
or is a term in gǫ − 1 or g−1ǫ − 1, then, posing ǫ = t−
a , At := δtAδ
t verifies: ∀r, s
‖At‖r,s = O(t
a ) as t→ ∞.
The lemma implies that at place (1,1) in the diagonalized matrix above we get ∇20 +
O(t− 32+ 3a+1) = O(t− 12+ 3a ). To have −1
< 0 we take a > 6. The term at place (2,2)
gives T 2ǫ,t +O(t
a ). Then
ǫ,t = δtg
∇20 +O(t−γ) 0 0
0 T 2ǫ +O(t
a ) 0
0 0 (PǫBPǫ)
δtgǫδ
t , with γ > 0
Now since gǫ = ĝǫ ⊕ 1
ǫ,t =
∇20 +O(t−γ) 0
0 T 2ǫ,t +O(t
δtĝǫδ
0 PǫBPǫ
10 SARA AZZALI
Observe that since gǫ − 1, g−1ǫ − 1 ∈ N1, we have δtĝ−1ǫ δ−1t = Id+
O(t− 12+ 1a ).
Denote w := O(t− 12+ 1a ). Then
∇20 +O(t−γ) 0
0 T 2ǫ,t +O(t
δtĝǫδ
1 + w w
w 1 + w
∇20 +O(t−γ) 0
0 T 2ǫ,t +O(t
1 + w w
w 1 + w
Since e−∇
0+O(t−γ ) = e−∇
0 +O(t−γ), then leaving (P +Qǫ) out of the notation
ǫ,t =
1 + w w
w 1 + w
0 +O(t−γ) 0
0 e−T
1 + θ w
w 1 + w
+ e−(PǫBPǫ)
= e−(PǫBPǫ)
where
(1 + w)2e−∇
0 w(1 + w)e−∇
w(1 + w)e−∇
0 w2e−∇
O(t−1+ 2a ) O(t− 12+ 1a )
O(t− 12+ 1a ) O(t−1+ 2a )
(1 + w)2O(t−γ) w(1 + w)[O(t−γ) + e−T ]
w(1 + w)[O(t−γ) + e−T ] w2O(t−γ) + (1 + w)2e−T
Proof of theorem 3.4. To fix notation, say Z is even dimensional. In the odd case use
trevenΓ instead of Str Γ.
Let K ⊆ B be a compact, and denote as β = αK the Novikov–Shubin invariant on it.
Write Bt = Bǫ,t + Aǫ,t as in (3.2), and define Bt(z) = Bt,ǫ + zAt,ǫ, z ∈ [0, 1], so that by
Duhamel’s principle (for example [30, eq. (3.10)])
t − e−B2t,ǫ =
e−Bt(z)
dz = −
e−(s−1)B
t (z)
dB2t (z)
t (z)dsdz =: Fǫ,t
Write then
Str Γ(
t ) = Str Γ(
dBt,ǫ
︸ ︷︷ ︸
+Str Γ(
Fǫ,t)
︸ ︷︷ ︸
(3.3)
For the family
we shall use that
D + c(T )
D+O(t−
2 ), as in
Remark 2.2.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 11
3.2.1. The term I.
t,ǫ =
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
+O(t−
e−(PǫBPǫ)
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
0 0 0
0 0 0
0 0 0
O(t−1+ 2a ) O(t− 12+ 1a ) 0
O(t− 12+ 1a ) O(t−1+ 2a ) 0
0 0 0
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
(1 + w)2O(t−γ) w(1 +w)2(O(t−γ) + e−T ) 0
w(1 + w)2(O(t−γ) + e−T ) w2O(t−γ) + (1 + w)2e−T 0
0 0 0
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
e−(PǫBPǫ)
2PǫDPǫe
−(PǫBPǫ)2 +
0 0 0
2QǫDQǫO(t−
a ) QǫDQǫO(t−
a ) 0
0 0 0
0 0 0
2QǫDQǫw(1 + w)(O(t−γ) + e−T ) t−
2QǫDQǫ(w
2O(t−γ) + (1 + w)2e−T ) 0
0 0 0
The choice of a > 6 implies 2
. Moreover only diagonal blocks contribute1 to the
StrΓ, therefore we only have to guarantee the integrability of StrΓ(t
2PǫDPǫe
−PǫB2tPǫ),
because from [30, Prop.11] Str Γ e
−T = O(t−δ), ∀δ > 0.
We reason as follows: Str Γ(t
2PǫDPǫe
−PǫB2tPǫ) = t−
2 tr Γ(UPǫ), where U =
τPǫDPǫe
−PǫB2tPǫ , and τ is the chirality grading.
Next we evaluate trΓ(UPǫ) = trΓ(UP
ǫ ) = trΓ(PǫUPǫ). To do this, since our trace has
values differential forms, let ω1, . . . , ωJ a base of ΛT
zB, for z fixed on K. U is a family
of operators and Uz acts on C∞(M̃z, Ẽz)⊗ ΛT ∗zB. Write Uz =
j Uj ⊗ ωj.
tr Γ(PǫUPǫ) =
tr Γ(PǫUjPǫ)⊗ ωj =
tr(χFPǫUjPǫχF )⊗ ωj.
Now tr(χFPǫUjPǫχF ) =
i < χFPǫUjPǫχFδvi , δvi >=
i < UjPǫχFδvi , PǫχFδvi >,
where {δvi} is a base of L2(M̃z |F , Ẽz |F). Therefore
| < UjPǫχFδvi , PǫχFδvi > | ≤ ‖UjPǫχFδvi‖ · ‖PǫχFδvi‖ ≤
≤ ‖Uj‖ ‖PǫχFδvi‖
2 ≤ ‖Uz‖ ‖PǫχFδvi‖
1In fact if Pi are orthogonal projections s.t.
Pi = 1, then for a fibrewise operator A we have
StrA = tr ηA = tr(
PiηAPi) + tr(
PiηAPj) = tr(
PiηAPi).
12 SARA AZZALI
i ‖PǫχFδvi‖ =
i < PǫχFδvi , PǫχFδvi >=
i < χFPǫχFδvi , δvi >= tr Γ(Pǫ) =
O(ǫβ) where β = αK . Hence
tr Γ(PǫUPǫ) ≤ ‖U‖O(ǫβ) = ‖U‖O(t−
a ) , with ǫ = t−
Claim ([30, Lemma 13]):
∥ is bounded independently of t, for t large. This follows
because (PǫBPǫ)
2 = PǫD
2Pǫ − C̄t, with C̄t is a fibrewise differential operator of order
at most one with uniformly bounded coefficients. Therefore
2 C̄t
l,l−1
is bounded
independently of t, for t large. Now writing the Volterra series for e−t(PǫD
2+C̄t ,
we have U = τPǫ
e−tσ0PǫD
2PǫC̄te
−tσ1PǫD2Pǫ . . . C̄te
−tσkPǫD2Pǫdσ, then estimating
each addend as
−tσ0PǫD2PǫC̄te
−tσ1PǫD2Pǫ
∥τPǫDe
−tσ0PǫD2Pǫ
l,l+1
l+1,l
−tσ1PǫD2Pǫ
l,l+1
·· · ··
l+1,l
−tσkPǫD2Pǫ
l,l+1
we get the Claim.
Thus t−
2 trΓ(UPǫ) ≤ c ‖U‖ t−
2 , and Str Γ(
t,ǫ) ≤ ct
2 . We require then
< −1 to have integrability hence we need finally a < 2β
. Because a was also
required to be a > 6 (see lines after Lemma 3.7), the hypothesis
β > 3(q + 1) (3.4)
is a sufficient condition to have the first term in (3.3) equal O(t−1−δ), with δ > 0.
3.2.2. The term II. Now let’s consider the second term in (3.3). As in [30, pag.197-
198], write Bt =
tD + B1 +
B2, and locally B1 = d + Φ. We have
dB2t (z)
Bt(z)Aǫ,t + Aǫ,tBt(z) =
tDA1 + A2
tD + A3, where Ai = Ci,1PǫCi,2, and Ci,j ∈ M1
are sums of words in Φ, d(Φ), t−
2B[2], t
2 d(B[2]). This implies that Ci,j are differential
operators with coefficients uniformly bounded in t.
Str Γ
= tr Γτ(t
2D − t−
2B[2])
e−(s−1)B
t (z)(
tDC1,1PǫC1,2+
+ C2,1PǫC2,2
tD + C3,1PǫC3,2)e
−sB2t (z)dsdz =
= tr Γ
C1,2e
−sB2t (z)τ
e−(s−1)B
t (z)
tDC1,1Pǫ +
+ C2,2
tDe−sB
t (z)τ
e−(s−1)B
t (z)C2,1Pǫ+
+C3,2e
−sB2t (z)τ
e−(s−1)B
t (z)C3,1Pǫ
dsdz = tr Γ(PǫWPǫ)
with W the term in square brackets.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 13
With a similar argument as in the Claim above and as in [30, p. 199], we have that
2 e−sB
t (z)τe−(s−1)B
t (z)
∥ is bounded independently of t as t→ ∞ so that the condition
(3.4) on the Novikov–Shubin exponent guaranties that the term II. is O(t−1−δ) as t → ∞
as well. �
Theorem 3.4 and Proposition 2.5 taken together imply
Corollary 3.8. If D̃ has regular spectral projections and regularity > 3(dimB + 1)
η̂(2)(D̃) =
dt if dim Z̃ = even
trevenΓ
dt if dim Z̃ = odd
is well defined as a continuos differential form on B.
Remark 3.9. Theorem 3.4 gives η̂(2) as a continuos form on B. Therefore η̂(2) fits
into a weak L2-local index theorem (see [24, 4]). To get a strong local index theorem
one should prove estimates for Str Γ(
t ) in C1-norm, assuming more regularity
on αK .
Remark 3.10. If Z odd dimensional, ρ̂(2) is an even degree differential form, whose
zero degree term is a continuos function on B with values the Cheeger–Gromov L2-eta
invariant of the fibre, η̂
(b) = η(2)(Db, M̃b →Mb).
3.3. Case of uniform invertibility. Suppose the two families D and D̃ are both
uniformly invertible, i.e.
∃µ > 0 such that ∀b ∈ B
spec(Db) ∩ (−µ, µ) = ∅
spec(D̃b) ∩ (−µ, µ) = ∅
(3.5)
In this case the t → ∞ asymptotic is easy and in particular StrΓ(
t ) = O(t−δ),
∀δ > 0 [5]. With the same estimates (see [30, p. 194]) one can look at ∂
StrΓ(
and obtain that StrΓ(
= O(t−δ), ∀δ > 0.
4. The L2 rho form
Definition 4.1. Let (π : M → B, gM/B ,V, E) be a geometric family, p : M̃ → M a
normal covering of it. Assume that kerD forms a vector bundle, and that the family D̃
has regular projections with family Novikov–Shubin invariants αK > 3(dimB + 1). We
define the L2-rho form to be the difference
ρ̂(2)(M,M̃,D) := η̂(2)(D̃)− η̂(D) ∈ C0(B,ΛT ∗B).
14 SARA AZZALI
Remark 4.2. When the fibres are odd dimensional, ρ̂(2) is an even degree differential
form, whose zero degree term is a continuos function on B with values the Cheeger–
Gromov L2-rho invariant of the fibre, ρ̂
(b) = ρ(2)(Db, M̃b →Mb).
We say a continuos k-form ϕ on B has weak exterior derivative ψ (a (k+1)-form) if, for
each smooth chain c : ∆k+1 → B, it holds
ϕ, and we write dϕ = ψ.
Proposition 4.3. If π : M → B has odd dimensional fibres, ρ̂(2)(D) is weakly closed.
Proof. From (2.4),
odde−B̃
odde−B̃
dt. Tak-
ing the limits t → 0, T → ∞ we get
Â(M/B) ch(E/S) =
η̂(2)(D̃)
because limT→∞ tr
odde−B
T = tr(e−∇
0)odd = 0 because tr(e−∇
0) is a form of even degree.
The same happens for the family D̃ where
Â(M/B) ch(E/S) = dη̂(D̃) (strongly).
ρ̂(2)(D) = 0, which gives the result. �
Corollary 4.4. Under uniform invertibility hypothesis (3.5) the form ρ̂(2)(D) is always
(strongly) closed.
Proof. The argument is standard: from transgression formulæ (2.3) (2.4), asymptotic
behaviour, and Remark 3.9, we have dη̂(D) =
Â(M/B) ch(E/S) = dη̂(2)(D̃). �
5. ρ̂(2) and positive scalar curvature for spin vertical bundle
Let π : M → B be a smooth fibre bundle with compact base B. If ĝ denotes a metric
on the vertical tangent bundle T (M/B), and b ∈ B, denote with ĝb the metric induced
on the fibre Mb, and write ĝ = (ĝb)b∈B . Define
R+(M/B) := {ĝ metric on T (M/B) | scal ĝb > 0 ∀b ∈ B}
to be the space of positive scalar curvature vertical metrics (= PSC).
Assume that T (M/B) is spin and let ĝ ∈ R+(M/B) 6= ∅. By Lichnerowicz formula
the family of Dirac operators D/ĝ is uniformly invertible. Let p : M̃ → M be a normal
Γ-covering of π, with M̃ → B having connected fibres, and denote with r : M → BΓ the
map classifying it. The same holds for D̃/ĝ, so that we are in the situation of (3.3).
On the space R+(M/B) we can define natural relations, following [43]. We say ĝ0,
ĝ1 ∈ R+(M/B) are path-connected if there exists a continuos path ĝt ∈ R+(M/B)
between them.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 15
We say ĝ0 and ĝ1 are concordant if on the bundle of the cylinders Π: M × I → B,
Π(m, t) = π(m), there exists a vertical metric Ĝ such that: ∀b ∈ B Ĝb is of product-
type near the boundary, scal(Ĝb) > 0, and onM×{i} → B it coincides with ĝi, i = 0, 1.
Proposition 5.1. Let π : M → B be a smooth fibre bundle with T (M/B) spin and
B compact. Let p : M̃ → M be a normal Γ-covering of the fibre bundle, such that
M̃ → B has connected fibres. Then the rho class [ρ̂(2)(D/)] ∈ H∗dR(B) is constant on the
concordance classes of R+(M/B).
Proof. Let ĝ0 and ĝ1 be concordant, and Ĝ the PSC vertical metric on the family of
cylinders. The family of Dirac operators D/
M×I/B,Ĝ has as boundary the two families
D/0 = (Dz , ĝ0,z)z∈B and D/1 = (Dz , ĝ1,z)z∈B , both invertible. Then the Bismut–Cheeger
theorem in [11] can be applied
M×I/B
Â(M × I/B)− 1
η̂(D/ĝ0) +
η̂(D/ĝ1) in H∗dR(B)
where Ch(IndDM×I,h) = 0 ∈ H∗dR(B).
On the family of coverings we reason as before and apply the index theorem in [36,
Theorem 4] to get
M×I/B
Â(M × I/B)− 1
η̂(2)(D̃/ĝ0) +
η̂(2)(D̃/ĝ1) in H
dR(B)
Subtracting we get [ρ̂(2)(D/g0)] = [ρ̂(2)(D/g1)] ∈ H∗dR(B). �
5.1. ρ̂(2) and the action of a fibre bundle diffeomorphism on R+(M/B). Let
(p, π) be as in Definition 2.1 and assume further that p is the universal covering of M .
If one wants to use [ρ̂(2)(D/)] for the study of R+(M/B) it is important to check how this
invariant changes when ĝ ∈ R+(M/B) is acted on by a fibre bundle diffeomorphism f
preserving the spin structure.
Proposition 5.2. Let f : M →M be a fibre bundle diffeomorphism preserving the spin
structure. Then [ρ̂(2)(D/ĝ)] = [ρ̂(2)(D/f∗ ĝ)]
Proof. We follow the proof [43, Prop. 2.10] for the Cheeger–Gromov rho invariant. Let
ĝ be a vertical metric and denote S = PSpin(M/B) a fixed spin structure, i.e. a 2-fold
covering2 of PSOĝ(T (M/B)) →M .
The eta form downstairs of D/ depends in fact on ĝ, on the spin structure, and on the
horizontal connection THM , so we write here explicitly η̂(D/ĝ) = η̂(D/ĝ,S , THM).
First of all η̂(D/ĝ,S , THM) = η̂(D/f∗ ĝ,f∗S , f∗THM), because f induces a unitary equiva-
lence between the superconnections constructed with the two geometric structures.
2or, equivalently, a 2-fold covering of PGL+(T (M/B)) which is not trivial along the fibres of
PGL+(T (M/B)) → M , [43, p. 8].
16 SARA AZZALI
Because f spin structure preserving, it induces an isomorphism βGL+ between the orig-
inal spin structure S and the pulled back one df∗S. Then βGL+ gives a unitary equiv-
alence between the operator obtained via the pulled back structures, and the Dirac
operator for f∗ĝ and the chosen fixed spin structure, so that η̂(D/f∗ĝ,f∗S , f∗THM) =
η̂(D/f∗ĝ,S , f∗THM). Taken together
η̂(D/ĝ,S , THM) = η̂(D/f∗ĝ,S , f∗THM)
Let p : M̃ →M be the universal covering. Now we look at η̂(2)(D̃/) = η̂(2)(D̃/ĝ,S , THM,p),
where on M̃ the metric, spin structure and connection are the lift via p as by defini-
tion. Again, if we construct the L2 eta form for the entirely pulled back structure, we
get η̂(2)(D̃/ĝ,S , THM,p) = η̂(2)(D̃/f∗ĝ,f∗S , f∗THM,f∗p). Proceeding as above on the spin
structure, η̂(2)(D̃/f∗ĝ,f∗S , f∗THM,f∗p) = η̂(2)(D̃/f∗ĝ,S , f∗THM,f∗p). Since M̃ is the uni-
versal covering we have a covering isomorphism between f∗M̃ and M̃ , which becomes
an isometry when M̃ is endowed of the lift of the pulled back metric f∗ĝ, therefore
η̂(2)(D̃/f∗ ĝ,S , f
∗THM,f∗p) = η̂(2)(D̃/f∗ ĝ,S , f
∗THM,p)
It remains to observe how η̂ and η̂(2) depends on the connection T
HM . We remove for
the moment the hat ˆ to simplify the notation. Let TH0 M,T
1 M two connections, say
given by ω0, ω1 ∈ Ω1(M,T (M/B)) and pose ωt = (1− t)ω0 + tω1. Construct the family
M̆ =M × [0, 1] π̆→ B× [0, 1] =: B̆ as in the proof of Prop. 2.5. On this fibre bundle put
the connection one form ω̆ + dt. Since d̆η̆ = dη̆(·, t)− ∂
η(t)dt we have
η0 − η1 =
d̆η̆ −
M̆/B̆
Â(M × I/B × I)− d
which is the sum of a local contribution plus an exact form. Writing the same for η(2)
we get that for the L2-rho form ρ̂(2)(D/, TH0 M) = ρ̂(2)(D/, TH1 M) ∈ Ω(B)/dΩ(B) and
therefore we get the result. �
5.2. Conjectures. Along the lines of [31, 42] we can state the following conjectures.
Conjecture 5.1. If Γ is torsion-free and satisfies the Baum-Connes conjecture for the
maximal C∗-algebra, then [ρ̂(2)(D/ĝ)] vanishes if ĝ ∈ R+(M/B).
Definition 5.3. Let π : M → B and θ : N → B be two smooth fibre bundles of compact
manifolds over the same base B. A continuos map h : N → M is called a fibrewise
homotopy equivalence if π ◦ h = θ, and there exists g : N →M such that θ ◦ g = π and
such that h ◦ g, g ◦ h are homotopic to the identity by homotopies that take each fibre
into itself.
We work in the following with smooth fibrewise homotopy equivalences.
Definition 5.4. Let Γ be a discrete group and (π : M → M,p : M̃ → M), (θ : N →
B, q : Ñ → N) be two normal Γ-coverings of the fibre bundles π and θ. Denote as
r : M → BΓ, s : N → BΓ the two classifying maps. We say (π, p) and (θ, q) are Γ-
fibrewise homotopy equivalent if there exists a fibrewise homotopy equivalence h : N →
M such that s ◦ h is homotopic to r.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 17
Let Dsign denote the family of signature operators.
Conjecture 5.2. Assume Γ is a torsion-free group that satisfies the Baum-Connes
conjecture for the maximal C∗-algebra. Let h be a orientation preserving Γ-fibrewise
homotopy equivalence between (π, p) and (θ, q) and suppose D̃sign
and D̃sign
have smooth
spectral projections and Novikov–Shubin invariants > 3(dimB + 1).
Then [ρ̂(2)(D̃signM/B)] = [ρ̂(2)(D̃
)] ∈ H∗dR(B).
Appendix A. Analysis on normal coverings
We summarize the analytic tools we use to investigate L2 spectral invariants, namely
NΓ-Hilbert spaces and Sobolev spaces on manifolds of bounded geometry, following the
nice exposition in [46].
A.1. NΓ-Hilbert spaces and von Neumann dimension. Let Γ be a discrete count-
able group and l2(Γ) the Hilbert space of complex valued, square integrable functions
on Γ. Denote with δγ ∈ CΓ the function with value 1 on γ, and zero elsewhere. The
convolution law on CΓ is δγ ∗ δβ = δγβ. Let L be the action of Γ on l2(Γ) by left con-
volution L : Γ → U(l2(Γ)), Lγ(f) = (δγ ∗ f)(x) = f(γ−1x). Right convolution action is
denoted by R.
Definition A.1. The group von Neumann algebra NΓ is defined to be the weak closure
NΓ := L(CΓ)weak in B(l2(Γ)). By the double commutant theorem NΓ = R(CΓ)′, so
that NΓ is the algebra of operators commuting with the right action of Γ. An important
feature of the group von Neumann algebra is its standard trace trΓ : NΓ −→ C defined
as trΓA =< Aδe, δe >l2(Γ). In particular for A =
aγLγ ∈ NΓ, then trΓ(A) = ae.
Definition A.2. A free NΓ-Hilbert space is a Hilbert space of the form W ⊗ l2(Γ),
where W is a Hilbert space and Γ acts on l2(Γ) on the right.
A NΓ-Hilbert space H is a Hilbert space with a unitary right-action of Γ such
that there exists a Γ-equivariant immersion H → V ⊗ l2(Γ) in some free NΓ-
Hilbert space. For H1,H2 NΓ-Hilbert spaces, define BΓ(H1H2) : = {T : H1 →
H2 bounded and Γ-equivariant}.
Let H = V ⊗ l2(Γ) be a free NΓ-Hilbert space. Then BΓ(V ⊗ l2(Γ)) ≃ B(V) ⊗ NΓ.
There exist a trace on the positive elements of this von Neumann algebra, with values
in [0,∞]: let (ψj)j∈N is a orthonormal base of V; if f ∈ B(H)+, its trace is given by
trΓ(f) =
j∈N < f(ψj ⊗ δe), ψj ⊗ δe >. A Γ-trace can be defined also on any NΓ-
Hilbert-space H using the immersion j : H →֒ V ⊗ l2Γ and proveing that the trace does
not depend on the choice of j (see [17] or [39, pag. 17]).
Definition A.3. Let H be a NΓ-Hilbert space. Its von Neumann dimension is defined
as dim Γ(H) = trΓ(id : H → H) ∈ [0,+∞).
Definition A.4. Let H1 and H2 be NΓ-Hilbert spaces. Define
18 SARA AZZALI
• BfΓ(H1,H2) := {A ∈ BΓ(H1,H2)′|dim Γ
<∞} are the Γ-finite rank oper-
ators
• B∞Γ (H1,H2) := B
Γ(H1,H2)
, are the Γ-compact operators
• B2Γ(H) := {A ∈ BΓ(H)s.t. trΓ (AA∗) <∞}, are the Γ-Hilbert-Schmidt operators
• B1Γ(H) := B2Γ(H)B2Γ(H)∗ the Γ-trace class operators.
Their main properties are:
1) Bf (H),B∞(H),B2(H),B1(H) are ideals and Bf ⊂ B1 ⊂ B2 ⊂ B∞;
2) A ∈ Bi(H) if and only if |A| ∈ Bi(H) for i = 1, 2, f,∞.
A.2. Covering spaces, bounded geometry techniques. Let p : Z̃ → Z a normal
Γ-covering of a compact Riemannian manifold Z. Let I ⊂ Z̃ be a fundamental domain
for the (right) action of Γ on Z̃ (I is an open subset s.t. I · γ ∩ I and Z̃ \
I · γ have
zero measure ∀γ 6= e).
Let E → Z a Hermitian vector bundle, and Ẽ = p∗E the pull-back. The sections
C∞c (Z̃, Ẽ) form a CΓ-right module for the action (ξ · f)(m̃) =
(R∗gξ)(m̃)f(g
−1) where
(R∗gξ)(m̃) := ξ(m̃g). Its Hilbert space completion L
2(Z̃, Ẽ) is a Γ-free Hilbert space
in the sense of definition A.2, in fact the map ψ : L2(Z̃, Ẽ) −→ L2(I, Ẽ|I) ⊗ l2(Γ),
|I ⊗ δγ is an isomorphism.
The Γ-trace class operators are characterized as follows: let A ∈ BΓ(L2(Z̃, Ẽ))
A ∈ B1Γ(L2(Z̃, Ẽ)) if and only if χI |A|χI ∈ B1(L2(I, E|I))
If A ∈ B1Γ(L2(Z̃, Ẽ)) then trΓ(A) = tr(χIAχI). If A ∈ B1Γ(L2(Z̃, Ẽ)) has Schwartz
kernel [A] continuos, then
trA =
([A](x, x)) dx =
π∗ tr Ẽx ([A](x, x)) dx . (A.1)
The covering of a compact manifold and the pulled back bundle Ẽ above are the most
simple examples of manifolds of bounded geometry3.
The analysis on manifolds of bounded geometry was developped in [45]. We specialize
here to the case of a normal covering Z̃.
The Sobolev spaces of sections are defined, for k ≥ 0, as the completion Hk(Z̃, Ẽ) :=
C∞c (Z̃, Ẽ)
where ‖f‖k :=
L2(Z̃,Ẽ⊗jT ∗Z̃); for k < 0 H
k(Z̃, Ẽ) is defined
as the dual of H−k(Z̃, Ẽ).
3Let (N, g) be a Riemannian manifold. N is of bounded geometry if
(1) it has positive injectivity radius i(N, g);
(2) the curvature RN and all its covariant derivatives are bounded.
A hermitian vector bundle E → N is of bounded geometry if the curvature RE and all its covariant
derivatives are bounded. This can be characterized in normal coordinates with conditions on g, coordi-
nate transformations and ∇ (see for example in [45] and [46]).
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 19
The spaces of uniform Ck sections are defined as follows: UCk(M̃) = {f : M̃ → C | f ∈
Ck and ‖f‖k ≤ c(k) ∀k}, where ‖f‖k = supm̃∈M̃,Xi{|∇X1 . . .∇Xkf(m̃)|}, and analo-
gously for sections UCk(M̃, Ẽ). UC∞(M̃, Ẽ) is the Fréchet space :=
k UCk(M̃ , Ẽ).
The following Sobolev embedding property holds [45]: if dim M̃ = n, then for j > n
there is a continuos inclusion Hj(M̃ , Ẽ) →֒ UCk(M̃ , Ẽ).
The algebra UDiff(M̃, Ẽ) of uniform differential operators is the algebra generated by
operators in UC∞(M̃ ,End Ẽ) and derivatives {∇ẼX}X∈UC∞(M̃ ,TM̃) with respect to uni-
form vector fields. P ∈ UDiff(M̃ , Ẽ) extends to a continuos operator Hj(M̃, Ẽ) →
Hj−k(M̃ , Ẽ) ∀j ∈ Z.
P ∈ UDiff(M̃ , Ẽ) is called uniformly elliptic if its principal symbol σpr ∈
UC∞(T ∗M̃, π∗ End Ẽ) is invertible out of an ǫ-neighborhood of 0 ∈ T ∗M̃ , with inverse
section which can be uniformly estimated.
For a uniformly elliptic operator T the G̊arding inequality holds:
Hs+k(M̃,Ẽ)
≤ c(s, k) (‖ϕ‖Hs + ‖Tϕ‖Hs) ∀s ∈ R (A.2)
If T is a continuos operator T : C∞c (N,E) → (C∞c (M̃ , Ẽ))′ we will denote its Schwartz
kernel with [T ] ∈ C∞(M̃ × M̃, Ẽ4Ẽ∗).
Definition A.5. We say that T : C∞c (N,E) → (C∞c (M̃ , Ẽ))′ has order k ∈ Z if ∀s ∈
Z it admits a bounded extension Hs(M̃, Ẽ) → Hs−k(M̃, Ẽ). Hence it is closable as
unbounded operator on L2(M̃ , Ẽ).
The space of order k operators is denoted Opk(M̃, Ẽ), and comes with the seminorms
on B(Hs(M̃ , Ẽ),Hs−k(M̃ , Ẽ)). The space Op−∞(M̃, Ẽ) =
k(M̃, Ẽ) is a Fréchet
space.
Finally, an operator T ∈ Opk(M̃ , Ẽ) is called elliptic if it satisfies G̊arding inequality.
We will denote as OpkΓ(M̃, Ẽ) the subspace of Γ-invariant operators in Op
k(M̃, Ẽ).
Consider the Fréchet space of continuos rapidly decreasing functions
RB(R) = {f : R → C : f continuos, and
∣(1 + x
2 f(x)
∣ <∞ ∀k}
Let T ∈ Opk(M̃ , Ẽ) , k ≥ 1 an elliptic, formally self-adjoint operator. Denote again by T
its closure, with domain Dom T = Hk(M̃ , Ẽ). From G̊arding inequality (A.2) the map
RC(R) −→ B(Hj(M̃ , Ẽ),H l(M̃ , Ẽ)), f 7→ f(T ) is continuos ∀j, l ∈ Z, so that
RC(R) −→ Op−∞(Z̃, Ẽ) , f 7→ f(T )
is continuos. One can prove that the Schwartz kernel of such operator is smooth: by
(A.2) and Sobolev embedding, for L = [n
+ 1] the map
Op−2L−l(Z̃, Ẽ) −→ UC l(Z̃ × M̃,E4E∗) , T 7→ [T ] (A.3)
is continuos ∀l ∈ N; then in particular for f ∈ RC(R), the kernel [f(T )] ∈ UC∞(Z̃ ×
Z̃, Ẽ4Ẽ∗) and the map RB(R) −→ UC∞(Z̃ × Z̃, Ẽ4Ẽ∗) , f 7→ [f(T )] is continuos.
20 SARA AZZALI
Lemma A.6. Since elements in Op−∞Γ (M̃, Ẽ) are Γ-trace class, then for T ∈
Op kΓ(M̃, Ẽ) elliptic and selfadjoint, the map RC(R) → B1Γ(M̃ , Ẽ) , f 7→ f(T ) is contin-
uos. As a consequence ∀m ∃l such that
| trΓ f(T )|m ≤ C ‖f(T )‖−l,l . (A.4)
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bonn.de/people/strohmai/globan/boris/
Mathematisches Institut, Georg-August Universität Göttingen
E-mail address: azzali@uni-math.gwdg.de
	1. Introduction
	2. Geometric families in the L2-setting
	2.1. The heat operator for the covering family
	2.2. Transgression formulæ, eta integrands
	2.3. The t0 asymptotic
	3. The L2-eta form
	3.1. The family Novikov–Shubin invariants
	3.2. The L2-eta form
	3.3. Case of uniform invertibility
	4. The L2 rho form
	5. (2) and positive scalar curvature for spin vertical bundle
	5.1. (2) and the action of a fibre bundle diffeomorphism on R+(M/B)
	5.2. Conjectures
	Appendix A. Analysis on normal coverings
	A.1. N-Hilbert spaces and von Neumann dimension
	A.2. Covering spaces, bounded geometry techniques
	References

  We define the secondary invariants L^2- eta and -rho forms for families of
generalized Dirac operators on normal coverings of fibre bundles. On the
covering family we assume transversally smooth spectral projections, and
Novikov--Shubin invariants bigger than 3(dim B+1) to treat the large time
asymptotic for general operators. In the particular case of a bundle of spin
manifolds, we study the L^2- rho class in relation to the space of positive
scalar curvature vertical metrics.

Introduction
Secondary invariants of Dirac operators are a distinctive issue of the heat equation
approach to index theory. The eta invariant of a Dirac operator first appeared as the
boundary term in the Atiyah–Patodi–Singer index theorem [2]: this spectral invariant,
highly nonlocal and therefore unstable, became a major object of investigation, because
of its subtle relation to geometry. With the introduction of superconnnections in index
theory by Quillen and Bismut, it became possible to employ heat equation techniques
in higher geometric situations, where the primary invariant, the index, is no longer a
number, but a class in a K-theory group [44, 10, 35]. This led to so called local index
theorems, which are refinements of the cohomological index theorems at the level of
differential forms, and gave as new fundamental byproduct the eta forms, coming from
the transgression of the index class [11, 12, 13], which are the higher analogue of eta
invariants [41, 38, 34].
Rho invariants are differences (or, more generally, delocalized parts) of eta invariants,
so they naturally possess stability properties when computed for geometrically relevant
operators, mainly the spin Dirac operator and the signature operator [3, 31, 42]. Fur-
thermore, they can be employed to detect geometric structures: the Cheeger–Gromov
L2-rho invariant, for example, has major applications in distinguishing positive scalar
curvature metrics on spin manifolds [15, 43], and can show the existence of infinitely
many manifolds that are homotopy equivalent but not diffeomorphic to a fixed one [16].
As secondary invariants always accompany primary ones, it is very natural to ask what
are the L2-eta and L2-rho forms in the case of a families, and what are their properties.
We consider the easiest L2-setting one could think of, namely a normal covering of a fibre
bundle. This interesting model contains yet all the features and problems offered by the
presence of continuos spectrum. Since the fibres of the covering family are noncompact,
the large time asymptotic of the superconnection Chern character is in general not
Date: September 4, 2010.
http://arxiv.org/abs/0704.0909v2
2 SARA AZZALI
converging to a differential form representative of the index class, and the same problem
is reflected when trying to integrate on [1,∞) the transgression term involved in the
definition of the L2-eta form.
The major result in this sense is by Heitsch and Lazarov, who gave the first families index
theorem for foliations with Hausdorff graph [30]. They computed the large time limit
of the superconnection Chern character as Haefliger form, assuming smooth spectral
projections and Novikov–Shubin invariants bigger than 3 times the codimension of the
foliation. Their result implies an index theorem in Haefliger cohomology (not a local
one, because they do not deal with the transgression term), which in particular applies
to the easier L2-setting under consideration.
We use the techniques of Heitsch–Lazarov to investigate the integrability on [1,∞) of
the transgression term, in order to define the L2-eta form for families D of generalised
Dirac operators on normal coverings of fibre bundles. Our main result, Theorem 3.4,
implies that the L2-eta form η̂(2)(D) is well defined as a continuos differential form on
the base B if the spectral projections of the family D are smooth, and the families
Novikov–Shubin invariants {αK}K⊂B are greater than 3(dimB + 1).
We define then naturally the L2-rho form ρ̂(2)(D) as the difference between the L2-eta
form for the covering family and the eta form of the family of compact manifolds. When
the fibre is odd dimensional, the zero degree term of ρ̂(2)(D) is the Cheeger–Gromov
L2-rho invariant of the induced covering of the fibre. We prove that the L2-form is
(weakly) closed when the fibres are odd dimensional (Prop. 4.3).
The strong assumptions of Theorem 3.4 are required because we want to define η̂(2) for
a family of generalised Dirac operators. In the particular case of de Rham and signature
operators one can put weaker assumptions: this is showed by Gong–Rothenberg’s result
for the L2-Bismut–Lott index theorem (proved under positivity of the Novikov–Shubin
invariants) [24], and from results in [4], where we develop a new approach to large time
estimate exclusive to the families of de Rham and signature operators. On the contrary,
a family of signature operators twisted by a fibrewise flat bundle has to be treated as a
general Dirac operator [7].
Next we investigate the L2-rho form in relation to the space R+(M/B) of positive scalar
curvature vertical metrics for a fibre bundle of spin manifolds. For this purpose, the
Dirac families D/ involved are uniformly invertible by Lichnerowicz formula, so that the
definition of the L2-rho form does not require Theorem 3.4, but follows from classical
estimates. Here the L2-rho form is always closed, and we prove the first step in order
to use this invariant for the study of R+(M/B), namely that the class [ρ̂(2)(D/)] is the
same for metrics in the same concordance classes of R+(M/B) (Prop.5.1). The action
of a fibrewise diffeomorphism is also taken into account.
Along the lines of [42] we can expect that if Γ is torsion-free and satisfies the Baum–
Connes conjecture, then the L2-rho class of a family of odd signature operators is an
oriented Γ- fibrewise homotopy invariant, and that [ρ̂(2)(D̃/ĝ)] vanishes correspondingly
to a vertical metric ĝ of positive scalar curvature.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 3
Acknowledgements This work was part of my researches for the doctoral thesis. I
would like to thank Paolo Piazza for having suggested the subject, for many interesting
discussions and for the help and encouragement. I wish to express my gratitude to
Moulay-Tahar Benameur for many interesting discussions.
2. Geometric families in the L2-setting
We recall local index theory’s machine, here adapted to the following L2-setting for
families.
Definition 2.1. Let π̃ : M̃ → B be a smooth fibre bundle, with typical fibre Z̃ con-
nected, and let Γ be a discrete group acting fibrewise freely and properly discontinuosly
on M , such that the quotient M = M̃/Γ is a fibration π : M → B with compact fibre Z.
Let p : M̃ → M̃/Γ = M denote the covering map. This setting will be called a normal
covering of the fibre bundle π and will be denoted with the pair (p : M̃ →M,π : M → B).
Let π : M → B be endowed with the structure of a geometric family (π : M →
B, gM/B ,V, E), meaning by definition:
• gM/B is a given metric on the vertical tangent bundle T (M/B)
• V the choice of a smooth projection V : TM → T (M/B) (equivalently, the choice
of a horizontal complement THM = KerV)
• E → M is a Dirac bundle, i.e. an Hermitian vector bundle of vertical Clifford
modules, with unitary action c : Cl(T ∗(M/B), gM/B) → End(E), and Clifford
connection ∇E.
To a gemetric family it is associated a family D = (Db)b∈B of Dirac operators along
the fibres of π, Db = cb ◦ ∇Eb : C∞(Mb, Eb) → C∞(Mb, Eb), where Mb = π−1(b), and
Eb := E|Mb .
If we have a normal Γ-covering p : M̃ → M of the fibre bundle π, the pull back of the
geometric family via p gives a Γ-invariant geometric family which we denote (π̃ : M̃ →
B, p∗gM/B , Ṽ , Ẽ).
2.0.1. The Bismut superconnection. The structure of a geometric family gives a distin-
guished metric connection ∇M/B on T (M/B), defined as follows: fix any metric gB on
the base and endow TM with the metric g = π∗gB ⊕ gM/B ; let ∇g the Levi-Civita con-
nection on M with respect to g; the connection ∇M/B := V∇gV on the vertical tangent
does not depend on gB ([9, Prop. 10.2]).
When X ∈ C∞(B,TB), let XH denote the unique section of THM s.t. π∗XH = X. For
any ξ1, ξ2 ∈ C∞(B,TB) let T (ξ1, ξ2) := [ξH1 , ξH2 ]− [ξ1, ξ2]H and let δ ∈ C∞(M, (THM)∗)
measuring the change of the volume of the fibres LξH vol =: δ(ξH ) vol. Following the
notation of [9], in formulas in local expression we denote as e1, . . . , en a local orthonormal
base of the vertical tangent bundle; f1, . . . fm will be a base of TyB and dy
1, . . . , dym will
denote the dual base. The indices i, j, k.. will be used for vertical vectors, while α, β, . . .
will be for the horizontal ones. The 2-form c(T ) =
α<β(T (fα, fβ), ei)eidy
αdyβ has
4 SARA AZZALI
values vertical vectors. Using the vertical metric, c(T )(fα, fβ) can be seen as a cotangent
vertical vector, hence it acts on E via Clifford multiplication.
Let H → B be the infinite dimensional bundle with fibres Hb = C∞(Mb, Eb). Its space of
sections is given by C∞(B,H) = C∞(M,E). We denote Ω(B,H) := C∞(M,π∗(ΛT ∗B)⊗
E). Let ∇H be the connection on H → B defined by ∇HU ξ = ∇EUHξ +
δ(ξH) where ξ is
on the right hand side is regarded as a section of E. ∇H is compatible with the inner
product < s, s′ >b:=
hE(s, s′) vol b , with s, s
′ ∈ C∞(B,H), and hE the fixed metric
on E.
Even dimensional fibre. When dimF = 2l the bundle E is naturally Z2-graded by chi-
raliry, E = E+ ⊕ E−, and D is odd. Correspondingly, the infinite dimensional bundle
is also Z2-graded: H = H+ ⊕ H−. The Bismut superconnection adapted to D is the
superconnection B = ∇H +D − c(T )
on H.
The corresponding bundle for the covering family π̃ is denoted H̃ → B where the same
construction for the family M̃ → B gives the Bismut superconnection B̃ = ∇H̃ + D̃ −
c(T̃ )
, adapted to D̃. It is Γ-invariant by construction, being the pull-back via p of B.
Odd dimensional fibre. When dimZ = 2l − 1, the appropriate notion is the one of
Cl(1)-superconnection, as introduced by Quillen in [44, sec. 5]. Let Cl(1) the Clifford
algebra Cl(1) = C⊕Cσ, where σ2 = 1, and consider EndE⊗Cl(1), adding therefore the
extra Clifford variable σ. On End(Eb) ⊗ Cl(1) = Endσ(Eb ⊕ Eb) define the supertrace
tr σ(A + Bσ) := trB, extended then to tr σ : C∞(M,π∗Λ∗B ⊗ EndE) → Ω(B) as usual
by tr σ(ω ⊗ (a+ bσ)) = ω tr b, for ω ∈ C∞(B,ΛT ∗B), ∀a, b ∈ C∞(B,EndE).
The family D, as well as c(T ) are even degree elements of the algebra C∞(B,EndH ⊗
Cl(1)⊗̂ΛT ∗B). On the other hand, ∇H is odd. By definition, the Bismut Cl(1)-
superconnection adapted to the family D is the operator of odd total degree Bσ :=
Dσ + ∇̃u − c(T )
Notation. In the odd case we will distinguish between the Cl(1)-superconnection de-
fined above Bσ acting on Ω(B,H) ⊗̂Cl(1), and the differential operator B : Ω(B,H) →
Ω(B,H) given by B := D +∇H − c(T )
, which is not a superconnection but is needed in
the computations.
2.1. The heat operator for the covering family. In this section we briefly discuss
the construction of the heat operator e−B̃
, which can be easily performed combining
the usual construction for compact fibres families in [9, Appendix of Chapter 9], with
Donnelly’s construction for the case of a covering of a compact manifolds [20]. We
integrate notations of [9, Ch. 9-10] with the ones of our appendix A. We refer to the
latter for the definitions of the spaces of operators used the rest of this section.
Let C∞(B,DiffΓ(Ẽ)) the algebra of smooth maps D : B → DiffΓ(Ẽ) satisfying that
∀z ∈ B, Dz is a Γ-invariant differential operator on M̃z, with coefficients depending
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 5
smoothly on the variables of B. In the same way, let N = C∞(B,ΛT ∗B⊗Op−∞Γ (Ẽ)) =
Ω(B,Op−∞Γ (Ẽ)) the space of smooth maps A : B → ΛT ∗B ⊗ Op
Γ (Ẽ). N contains
families of Γ-invariant operators of order −∞ with coefficients differential forms, hence
N is filtered by Ni = C∞(B,
j≥i Λ
jT ∗B ⊗Op−∞Γ (Ẽ)). The curvature of B̃ is a family
2 ∈ Ω(B,Diff2Γ(Ẽ)) and can be written as B̃2 = D̃2 − C̃, with C̃ ∈ Ω≥1(B,Diff1Γ(Ẽ)).
2.1.1. Definition and construction. For each point z ∈ B the operator e−tB̃2z is by defi-
nition an the one whose Schwartz kernel p̃zt (x, y) ∈ Ẽx⊗ Ẽ∗y ⊗ΛT ∗zB is the fundamental
solution of the heat equation, i.e.
• p̃zt (x, y) is C1 in t, C2 in x, y;
p̃zt (x, y) + B̃
z,II p̃
t (x, y) = 0 where B̃z,II means it acts on the second variable;
• lim
p̃zt (x, y) = δ(x, y)
• ∀T > 0 ∀t ≤ T ∃ c(T ) :
∥∂it∂
ypt(x, y)
∥ ≤ ct−
−i−j−ke−
2(x,y)
2 , 0 ≤ i, j, k ≤ 1.
Its construction is as follows: pose
e−tB̃
z := e−tD̃
tk e−σ0tD̃
z C̃e−σ1tD̃
z . . . C̃e−σktD̃
︸ ︷︷ ︸
dσ1 . . . dσk (2.1)
Since ∀σ = (σ0, . . . , σk) there exists σi > 1k+1 , then each term Ik ∈ ΛT
zB ⊗Op−∞(Ẽz)
and so does e−tB̃
z . Let p̃zt (x, y) = [e
−B̃2t,z ](x, y) be the Schwartz kernel of the operator
(2.1). Using arguments of [9, theorems 9.50 and 9.51], one proves that p̃zt (x, y) is smooth
in z ∈ B so that one can conclude e−B̃ ∈ Ω(B,Op−∞Γ ).
The next property, proved in [20] and [21], is needed in the t→ 0 asymptotic. For t < T0
−tB̃2 ](x̃, ỹ)
∣ ≤ c1t−
2 e−c2
2(x̃,ỹ)
t (2.2)
2.2. Transgression formulæ, eta integrands. For t > 0 let δt : Ω(B,H) → Ω(B,H)
the operator which on Ωi(B,H) is multiplication by t− i2 . Then consider the rescaled
superconnection Bt = t
2 δtBδ
t = ∇H +
tD − c(T ) 1
2.2.1. Even dimensional fibre. From (A.1) we have
Str Γe
−B̃2t = −dStr Γ
which on a finite interval (t, T ) gives the transgression formula
Str Γ
− Str Γ
Str Γ
ds (2.3)
6 SARA AZZALI
2.2.2. Odd dimensional fibre. Here it is convenient to use that tr σΓe
−(B̃σt )2 = tr oddΓ e
−B̃2t ,
(from [44] and (A.1)), where trodd means we take the odd degree part of the resulting
form. Then taking the odd part of the formula
tr Γe
−B2t = −d tr Γ
Tr oddΓ
− Tr oddΓ
Tr evenΓ
ds (2.4)
Remarks and notation 2.2. Since we wish now to look at the limits as t → 0 and
t → ∞ in (2.3) and 2.4, let us make precise what the convergences on the spaces of
forms are, and for families of operators. On Ω(B) we consider the topology of conver-
gence on compact sets. We say a family of forms ωt
C0→ ωt0 as t → t0 if ∀K
supz∈K ‖ωt(z)− ωt0(z)‖ΛT ∗z B → 0. We say ωt
C1→ ωt0 if the convergence also hold for
first derivatives of ωt with respect to the base variables. We say ωt = O(tδ) as t→ ∞ if
∃ a constant C = C(K) : supz∈K ‖ωt(z) − ωt0(z)‖ΛT ∗z B ≤ Ct
δ. We say ωt
= O(tδ) if
also the first derivatives with respect to base directions are O(tδ).
For a family Tt ∈ UC∞(B,Op−∞(Ẽ)) we say Tt
Ck→ Tt0 as t → t0 if ∀K
⊆ B, ∀r, s ∈ Z
supz∈K ‖Tt(z)− Tt0(z)‖r,s → 0 together with derivatives up to order k with respect to
the base variables.
On the space of kernels UC∞(M̃ ×B M̃, Ẽ4Ẽ∗ ⊗ π∗ΛT ∗B), we say kt → kt0 if ∀ϕ ∈
C∞c (B) ‖(π∗ϕ(x))(kt(x, y) − kt0(x, y))‖k → 0.
We stress that from (A.3) the map Ω(B,Op−∞Γ (Ẽ)) → UC∞(M̃ ×B M̃ , Ẽ4Ẽ∗ ⊗
π∗ΛT ∗B), T 7→ [T ] is continuos.
2.3. The t→ 0 asymptotic.
Proposition 2.3.
Str Γ
Â(M/B) chE/S if dim Z̃ = even
tr oddΓ
Â(M/B) chE/S if dim Z̃ = odd
The result is proved exactly as in the classic case of compact fibres, together with the
following argument of [33, Lemma 4, pag. 4]:
Lemma 2.4. [33] ∃A > 0, c > 0 s.t.
−B2t ](π(x̃), π(x̃))− [e−B̃2t ](x̃, x̃)
∣ = O(t−ce−
For the proof of the lemma see [32], or also [5], [24]. With the same technique we deduce
Proposition 2.5. The differential forms StrΓ
and tr σΓ
dB̃σt
e−(B̃
integrable on [0, 1], uniformly on compact subsets.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 7
Proof. The proof is as in [9, Ch.10, pag. 340]. We reason for example in the even case.
Consider the rescaled superconnection B̃s as a one-parameter family of superconnections,
s ∈ R+, and construct the new family M̆ = M̃ ×R+ → B ×R+ =: B̆. On Ĕ = Ẽ ×R+
there is a naturally induced family of Dirac operators whose Bismut superconnection
is B̆ = B̃s + dR+ − n4sds, and its rescaling is B̆t = B̃st + dR+ −
ds. Its curvature is
t = B̃
st + t
∧ ds, so that
t = e−B̃
e−uB̃
e−(1−u)B
st ∧ ds = e−F̃st − ∂B̃st
e−B̃st ∧ ds.
Str Γ
= Str Γ(e
−B̃2st)− Str Γ
∂B̃st
e−B̃st
ds (2.5)
At t = 0 we have the asymptotic expansion Str Γ(e
−B̆t) ∼
j=0 t
2 (Φ j
− α j
without singular terms. Computing (2.5) in s = 1, since
∂B̃st
, one has
Str Γ
e−F̃t
, and therefore Str Γ
e−F̃t
j=0 t
−1α j
. Let’s
compute α0. From the local formula
Φ0 − α0ds = lim
Str Γ
e−F̆t
M̆/B̆
Â(M̆/B̆) (2.6)
since M̆(z,s) = M̃z×{s} and the differential forms are pulled back from those on M̃ → B,
then the right hand side of (2.6) does not contain ds so that α0 = 0. This implies that
Str Γ(
t ) ∼
−1α j
3. The L2-eta form
We prove in Theorem 3.4 the well definiteness of the L2-eta form η̂(2)(D̃) under opportune
regularity assumptions. We make use of the techniques of [30].
3.1. The family Novikov–Shubin invariants. The t → ∞ asymptotic of the heat
kernel is controlled by the behaviour of the spectrum near zero. Let P̃ = (P̃ z)z∈B
the family of projections onto ker D̃ and let P̃ǫ = χ(0,ǫ)(D̃) be the family of spectral
projections relative to the interval (0, ǫ); denote Q̃ǫ = 1− P̃ǫ − P̃ .
For any z ∈ B the operator D̃z is a Γ-invariant unbounded operator: let D̃2z =
λdEz(λ)
be the spectral decomposition of D̃2z , andN
z(λ) = trΓE
z(λ) its spectral density function
[27]. Denote bz = trΓ P̃
z. Then N z(ǫ) = bz + trΓ P̃
ǫ and from [22] the behaviour of
θz(t) = trΓ(exp(−tD̃z)) at ∞ is governed by
αz = sup{a : θz(t) = bz +O(t−a)} = sup{a : N z(ǫ) = bz +O(ǫa)} (3.1)
8 SARA AZZALI
where αz is called the Novikov–Shubin invariant of D̃z.
We shall later impose conditions on αz uniformly on compact subset of B, so we intro-
duce the following definition from [24]: let K ⊂ B be a compact, define αK := infz∈K αz.
We call {αK}K⊂B the family Novikov–Shubin invariants of the fibre bundle M̃ → B.
By results of Gromov and Shubin [27], when D̃2z is the Laplacian, αz is a Γ-homotopy
invariant of M̃z [27], in particular it does not depend on z. In that case αz is locally
constant on B. For a general Dirac type operator this is not true and we need to use
the αK ’s.
Definition 3.1. [30] We say the family D̃ has regular spectral projections if P̃ and P̃ǫ are
smooth with respect to z ∈ B, for ǫ small, and ∇H̃P̃ ,∇H̃P̃ǫ are in N and are bounded
independently of ǫ. We say that the family D̃ has regularity A, if ∀K
⊆ B it holds
αK ≥ A.
Remark 3.2. To have regular projections is a strong condition, difficult to be verified
in general. The family of signature operators verifies the smoothness of P̃ [24, Theorem
2.2] but the smoothness of P̃ǫ is not clear even in that case.
The large time limit of the superconnection-Chern character StrΓ e
−B̃2t is computed in
[30, Theorem 5]. Specializing to our L2-setting it says the following.
Theorem 3.3. [30] Let ∇̃0 = P̃∇H̃P̃ . If D̃ has regular projections and regularity
> 3 dimB,
Str Γ(e
−B̃2t ) = Str Γe
3.2. The L2-eta form. We now use the same techniques of [30] to analyse the trans-
gression term in (2.3) and define the secondary invariant L2 eta form. We prove
Theorem 3.4. If D̃ has regular spectral projections and regularity > 3(dimB+1), then
= O(t−δ−1), for δ > 0. The same holds for trevenΓ
We start with some remarks and lemmas. In particular we shall repeatedly use the
following.
Remark 3.5. Let T ∈ N . From lemma A.6, ∀z ∈ B its Schwartz kernel [Tz] satis-
fies that for sufficiently large l, ∃ czl such that ∀x, y ∈ M̃z | [Tz](x, y) | ≤ czl ‖Tz‖−l,l
Therefore an estimate of ‖Tz‖−l,l produces directly via an estimate of TrΓ Tz.
Notation. Since in this section we are dealing only with the family of operators on the
covering, to simplify the notations let’s call D̃ = D, removing all tildes. Pose
Bǫ := (P +Qǫ)B(P +Qǫ) + PǫBPǫ
Aǫ = B− Bǫ
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 9
and write the rescaled operators as
Bǫ,t = (P +Qǫ)(Bt −
tD)(P +Qǫ) +
tD + Pǫ(Bt −
tD)Pǫ (3.2)
Aǫ,t = (P +Qǫ)(Bt −
tD)Pǫ + Pǫ(Bt −
tD)(P +Qǫ)
Denote also Tǫ = QǫBQǫ and Tǫ,t = QǫBtQǫ as in [30].
We will need the following two lemmas from [30]. The first is the “diagonalization” of
ǫ with respect to the spectral splitting of H.
Lemma 3.6. [30, Prop.6] Let M be the space of all maps f : B → ΛTB⊗End H̃. There
exists a measurable section gǫ ∈ M, with gǫ ∈ 1 +N1 such that
∇20 0 0
0 T 2ǫ 0
0 0 (PǫBPǫ)
N3 0 0
0 N2 0
0 0 0
The diagonalization procedure acts on (P ⊕Qǫ)H, in fact gǫ has the form gǫ = ĝǫ ⊕ 1,
with ĝǫ acting on (P ⊕Qǫ)H. From this lemma we get B2ǫ,t = tδtB2ǫδ−1t =
= tδtg
∇20 0 0
0 T 2ǫ 0
0 0 (PǫBtPǫ)
N3 0 0
0 N2 0
0 0 0
 gǫδt =
= δtg
tδt(∇20 +N3)δ−1t 0 0
0 tδt(T
ǫ +N2)δ−1t 0
0 0 PǫBtPǫ
δtgǫδ
The next lemma gives an estimate of the terms which are modded out.
Lemma 3.7. [30, lemma 9] If A ∈ Nk is a residual term in the diagonalization lemma
or is a term in gǫ − 1 or g−1ǫ − 1, then, posing ǫ = t−
a , At := δtAδ
t verifies: ∀r, s
‖At‖r,s = O(t
a ) as t→ ∞.
The lemma implies that at place (1,1) in the diagonalized matrix above we get ∇20 +
O(t− 32+ 3a+1) = O(t− 12+ 3a ). To have −1
< 0 we take a > 6. The term at place (2,2)
gives T 2ǫ,t +O(t
a ). Then
ǫ,t = δtg
∇20 +O(t−γ) 0 0
0 T 2ǫ +O(t
a ) 0
0 0 (PǫBPǫ)
δtgǫδ
t , with γ > 0
Now since gǫ = ĝǫ ⊕ 1
ǫ,t =
∇20 +O(t−γ) 0
0 T 2ǫ,t +O(t
δtĝǫδ
0 PǫBPǫ
10 SARA AZZALI
Observe that since gǫ − 1, g−1ǫ − 1 ∈ N1, we have δtĝ−1ǫ δ−1t = Id+
O(t− 12+ 1a ).
Denote w := O(t− 12+ 1a ). Then
∇20 +O(t−γ) 0
0 T 2ǫ,t +O(t
δtĝǫδ
1 + w w
w 1 + w
∇20 +O(t−γ) 0
0 T 2ǫ,t +O(t
1 + w w
w 1 + w
Since e−∇
0+O(t−γ ) = e−∇
0 +O(t−γ), then leaving (P +Qǫ) out of the notation
ǫ,t =
1 + w w
w 1 + w
0 +O(t−γ) 0
0 e−T
1 + θ w
w 1 + w
+ e−(PǫBPǫ)
= e−(PǫBPǫ)
where
(1 + w)2e−∇
0 w(1 + w)e−∇
w(1 + w)e−∇
0 w2e−∇
O(t−1+ 2a ) O(t− 12+ 1a )
O(t− 12+ 1a ) O(t−1+ 2a )
(1 + w)2O(t−γ) w(1 + w)[O(t−γ) + e−T ]
w(1 + w)[O(t−γ) + e−T ] w2O(t−γ) + (1 + w)2e−T
Proof of theorem 3.4. To fix notation, say Z is even dimensional. In the odd case use
trevenΓ instead of Str Γ.
Let K ⊆ B be a compact, and denote as β = αK the Novikov–Shubin invariant on it.
Write Bt = Bǫ,t + Aǫ,t as in (3.2), and define Bt(z) = Bt,ǫ + zAt,ǫ, z ∈ [0, 1], so that by
Duhamel’s principle (for example [30, eq. (3.10)])
t − e−B2t,ǫ =
e−Bt(z)
dz = −
e−(s−1)B
t (z)
dB2t (z)
t (z)dsdz =: Fǫ,t
Write then
Str Γ(
t ) = Str Γ(
dBt,ǫ
︸ ︷︷ ︸
+Str Γ(
Fǫ,t)
︸ ︷︷ ︸
(3.3)
For the family
we shall use that
D + c(T )
D+O(t−
2 ), as in
Remark 2.2.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 11
3.2.1. The term I.
t,ǫ =
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
+O(t−
e−(PǫBPǫ)
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
0 0 0
0 0 0
0 0 0
O(t−1+ 2a ) O(t− 12+ 1a ) 0
O(t− 12+ 1a ) O(t−1+ 2a ) 0
0 0 0
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
(1 + w)2O(t−γ) w(1 +w)2(O(t−γ) + e−T ) 0
w(1 + w)2(O(t−γ) + e−T ) w2O(t−γ) + (1 + w)2e−T 0
0 0 0
0 0 0
2QǫDQǫ 0
0 0 t−
2PǫDPǫ
e−(PǫBPǫ)
2PǫDPǫe
−(PǫBPǫ)2 +
0 0 0
2QǫDQǫO(t−
a ) QǫDQǫO(t−
a ) 0
0 0 0
0 0 0
2QǫDQǫw(1 + w)(O(t−γ) + e−T ) t−
2QǫDQǫ(w
2O(t−γ) + (1 + w)2e−T ) 0
0 0 0
The choice of a > 6 implies 2
. Moreover only diagonal blocks contribute1 to the
StrΓ, therefore we only have to guarantee the integrability of StrΓ(t
2PǫDPǫe
−PǫB2tPǫ),
because from [30, Prop.11] Str Γ e
−T = O(t−δ), ∀δ > 0.
We reason as follows: Str Γ(t
2PǫDPǫe
−PǫB2tPǫ) = t−
2 tr Γ(UPǫ), where U =
τPǫDPǫe
−PǫB2tPǫ , and τ is the chirality grading.
Next we evaluate trΓ(UPǫ) = trΓ(UP
ǫ ) = trΓ(PǫUPǫ). To do this, since our trace has
values differential forms, let ω1, . . . , ωJ a base of ΛT
zB, for z fixed on K. U is a family
of operators and Uz acts on C∞(M̃z, Ẽz)⊗ ΛT ∗zB. Write Uz =
j Uj ⊗ ωj.
tr Γ(PǫUPǫ) =
tr Γ(PǫUjPǫ)⊗ ωj =
tr(χFPǫUjPǫχF )⊗ ωj.
Now tr(χFPǫUjPǫχF ) =
i < χFPǫUjPǫχFδvi , δvi >=
i < UjPǫχFδvi , PǫχFδvi >,
where {δvi} is a base of L2(M̃z |F , Ẽz |F). Therefore
| < UjPǫχFδvi , PǫχFδvi > | ≤ ‖UjPǫχFδvi‖ · ‖PǫχFδvi‖ ≤
≤ ‖Uj‖ ‖PǫχFδvi‖
2 ≤ ‖Uz‖ ‖PǫχFδvi‖
1In fact if Pi are orthogonal projections s.t.
Pi = 1, then for a fibrewise operator A we have
StrA = tr ηA = tr(
PiηAPi) + tr(
PiηAPj) = tr(
PiηAPi).
12 SARA AZZALI
i ‖PǫχFδvi‖ =
i < PǫχFδvi , PǫχFδvi >=
i < χFPǫχFδvi , δvi >= tr Γ(Pǫ) =
O(ǫβ) where β = αK . Hence
tr Γ(PǫUPǫ) ≤ ‖U‖O(ǫβ) = ‖U‖O(t−
a ) , with ǫ = t−
Claim ([30, Lemma 13]):
∥ is bounded independently of t, for t large. This follows
because (PǫBPǫ)
2 = PǫD
2Pǫ − C̄t, with C̄t is a fibrewise differential operator of order
at most one with uniformly bounded coefficients. Therefore
2 C̄t
l,l−1
is bounded
independently of t, for t large. Now writing the Volterra series for e−t(PǫD
2+C̄t ,
we have U = τPǫ
e−tσ0PǫD
2PǫC̄te
−tσ1PǫD2Pǫ . . . C̄te
−tσkPǫD2Pǫdσ, then estimating
each addend as
−tσ0PǫD2PǫC̄te
−tσ1PǫD2Pǫ
∥τPǫDe
−tσ0PǫD2Pǫ
l,l+1
l+1,l
−tσ1PǫD2Pǫ
l,l+1
·· · ··
l+1,l
−tσkPǫD2Pǫ
l,l+1
we get the Claim.
Thus t−
2 trΓ(UPǫ) ≤ c ‖U‖ t−
2 , and Str Γ(
t,ǫ) ≤ ct
2 . We require then
< −1 to have integrability hence we need finally a < 2β
. Because a was also
required to be a > 6 (see lines after Lemma 3.7), the hypothesis
β > 3(q + 1) (3.4)
is a sufficient condition to have the first term in (3.3) equal O(t−1−δ), with δ > 0.
3.2.2. The term II. Now let’s consider the second term in (3.3). As in [30, pag.197-
198], write Bt =
tD + B1 +
B2, and locally B1 = d + Φ. We have
dB2t (z)
Bt(z)Aǫ,t + Aǫ,tBt(z) =
tDA1 + A2
tD + A3, where Ai = Ci,1PǫCi,2, and Ci,j ∈ M1
are sums of words in Φ, d(Φ), t−
2B[2], t
2 d(B[2]). This implies that Ci,j are differential
operators with coefficients uniformly bounded in t.
Str Γ
= tr Γτ(t
2D − t−
2B[2])
e−(s−1)B
t (z)(
tDC1,1PǫC1,2+
+ C2,1PǫC2,2
tD + C3,1PǫC3,2)e
−sB2t (z)dsdz =
= tr Γ
C1,2e
−sB2t (z)τ
e−(s−1)B
t (z)
tDC1,1Pǫ +
+ C2,2
tDe−sB
t (z)τ
e−(s−1)B
t (z)C2,1Pǫ+
+C3,2e
−sB2t (z)τ
e−(s−1)B
t (z)C3,1Pǫ
dsdz = tr Γ(PǫWPǫ)
with W the term in square brackets.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 13
With a similar argument as in the Claim above and as in [30, p. 199], we have that
2 e−sB
t (z)τe−(s−1)B
t (z)
∥ is bounded independently of t as t→ ∞ so that the condition
(3.4) on the Novikov–Shubin exponent guaranties that the term II. is O(t−1−δ) as t → ∞
as well. �
Theorem 3.4 and Proposition 2.5 taken together imply
Corollary 3.8. If D̃ has regular spectral projections and regularity > 3(dimB + 1)
η̂(2)(D̃) =
dt if dim Z̃ = even
trevenΓ
dt if dim Z̃ = odd
is well defined as a continuos differential form on B.
Remark 3.9. Theorem 3.4 gives η̂(2) as a continuos form on B. Therefore η̂(2) fits
into a weak L2-local index theorem (see [24, 4]). To get a strong local index theorem
one should prove estimates for Str Γ(
t ) in C1-norm, assuming more regularity
on αK .
Remark 3.10. If Z odd dimensional, ρ̂(2) is an even degree differential form, whose
zero degree term is a continuos function on B with values the Cheeger–Gromov L2-eta
invariant of the fibre, η̂
(b) = η(2)(Db, M̃b →Mb).
3.3. Case of uniform invertibility. Suppose the two families D and D̃ are both
uniformly invertible, i.e.
∃µ > 0 such that ∀b ∈ B
spec(Db) ∩ (−µ, µ) = ∅
spec(D̃b) ∩ (−µ, µ) = ∅
(3.5)
In this case the t → ∞ asymptotic is easy and in particular StrΓ(
t ) = O(t−δ),
∀δ > 0 [5]. With the same estimates (see [30, p. 194]) one can look at ∂
StrΓ(
and obtain that StrΓ(
= O(t−δ), ∀δ > 0.
4. The L2 rho form
Definition 4.1. Let (π : M → B, gM/B ,V, E) be a geometric family, p : M̃ → M a
normal covering of it. Assume that kerD forms a vector bundle, and that the family D̃
has regular projections with family Novikov–Shubin invariants αK > 3(dimB + 1). We
define the L2-rho form to be the difference
ρ̂(2)(M,M̃,D) := η̂(2)(D̃)− η̂(D) ∈ C0(B,ΛT ∗B).
14 SARA AZZALI
Remark 4.2. When the fibres are odd dimensional, ρ̂(2) is an even degree differential
form, whose zero degree term is a continuos function on B with values the Cheeger–
Gromov L2-rho invariant of the fibre, ρ̂
(b) = ρ(2)(Db, M̃b →Mb).
We say a continuos k-form ϕ on B has weak exterior derivative ψ (a (k+1)-form) if, for
each smooth chain c : ∆k+1 → B, it holds
ϕ, and we write dϕ = ψ.
Proposition 4.3. If π : M → B has odd dimensional fibres, ρ̂(2)(D) is weakly closed.
Proof. From (2.4),
odde−B̃
odde−B̃
dt. Tak-
ing the limits t → 0, T → ∞ we get
Â(M/B) ch(E/S) =
η̂(2)(D̃)
because limT→∞ tr
odde−B
T = tr(e−∇
0)odd = 0 because tr(e−∇
0) is a form of even degree.
The same happens for the family D̃ where
Â(M/B) ch(E/S) = dη̂(D̃) (strongly).
ρ̂(2)(D) = 0, which gives the result. �
Corollary 4.4. Under uniform invertibility hypothesis (3.5) the form ρ̂(2)(D) is always
(strongly) closed.
Proof. The argument is standard: from transgression formulæ (2.3) (2.4), asymptotic
behaviour, and Remark 3.9, we have dη̂(D) =
Â(M/B) ch(E/S) = dη̂(2)(D̃). �
5. ρ̂(2) and positive scalar curvature for spin vertical bundle
Let π : M → B be a smooth fibre bundle with compact base B. If ĝ denotes a metric
on the vertical tangent bundle T (M/B), and b ∈ B, denote with ĝb the metric induced
on the fibre Mb, and write ĝ = (ĝb)b∈B . Define
R+(M/B) := {ĝ metric on T (M/B) | scal ĝb > 0 ∀b ∈ B}
to be the space of positive scalar curvature vertical metrics (= PSC).
Assume that T (M/B) is spin and let ĝ ∈ R+(M/B) 6= ∅. By Lichnerowicz formula
the family of Dirac operators D/ĝ is uniformly invertible. Let p : M̃ → M be a normal
Γ-covering of π, with M̃ → B having connected fibres, and denote with r : M → BΓ the
map classifying it. The same holds for D̃/ĝ, so that we are in the situation of (3.3).
On the space R+(M/B) we can define natural relations, following [43]. We say ĝ0,
ĝ1 ∈ R+(M/B) are path-connected if there exists a continuos path ĝt ∈ R+(M/B)
between them.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 15
We say ĝ0 and ĝ1 are concordant if on the bundle of the cylinders Π: M × I → B,
Π(m, t) = π(m), there exists a vertical metric Ĝ such that: ∀b ∈ B Ĝb is of product-
type near the boundary, scal(Ĝb) > 0, and onM×{i} → B it coincides with ĝi, i = 0, 1.
Proposition 5.1. Let π : M → B be a smooth fibre bundle with T (M/B) spin and
B compact. Let p : M̃ → M be a normal Γ-covering of the fibre bundle, such that
M̃ → B has connected fibres. Then the rho class [ρ̂(2)(D/)] ∈ H∗dR(B) is constant on the
concordance classes of R+(M/B).
Proof. Let ĝ0 and ĝ1 be concordant, and Ĝ the PSC vertical metric on the family of
cylinders. The family of Dirac operators D/
M×I/B,Ĝ has as boundary the two families
D/0 = (Dz , ĝ0,z)z∈B and D/1 = (Dz , ĝ1,z)z∈B , both invertible. Then the Bismut–Cheeger
theorem in [11] can be applied
M×I/B
Â(M × I/B)− 1
η̂(D/ĝ0) +
η̂(D/ĝ1) in H∗dR(B)
where Ch(IndDM×I,h) = 0 ∈ H∗dR(B).
On the family of coverings we reason as before and apply the index theorem in [36,
Theorem 4] to get
M×I/B
Â(M × I/B)− 1
η̂(2)(D̃/ĝ0) +
η̂(2)(D̃/ĝ1) in H
dR(B)
Subtracting we get [ρ̂(2)(D/g0)] = [ρ̂(2)(D/g1)] ∈ H∗dR(B). �
5.1. ρ̂(2) and the action of a fibre bundle diffeomorphism on R+(M/B). Let
(p, π) be as in Definition 2.1 and assume further that p is the universal covering of M .
If one wants to use [ρ̂(2)(D/)] for the study of R+(M/B) it is important to check how this
invariant changes when ĝ ∈ R+(M/B) is acted on by a fibre bundle diffeomorphism f
preserving the spin structure.
Proposition 5.2. Let f : M →M be a fibre bundle diffeomorphism preserving the spin
structure. Then [ρ̂(2)(D/ĝ)] = [ρ̂(2)(D/f∗ ĝ)]
Proof. We follow the proof [43, Prop. 2.10] for the Cheeger–Gromov rho invariant. Let
ĝ be a vertical metric and denote S = PSpin(M/B) a fixed spin structure, i.e. a 2-fold
covering2 of PSOĝ(T (M/B)) →M .
The eta form downstairs of D/ depends in fact on ĝ, on the spin structure, and on the
horizontal connection THM , so we write here explicitly η̂(D/ĝ) = η̂(D/ĝ,S , THM).
First of all η̂(D/ĝ,S , THM) = η̂(D/f∗ ĝ,f∗S , f∗THM), because f induces a unitary equiva-
lence between the superconnections constructed with the two geometric structures.
2or, equivalently, a 2-fold covering of PGL+(T (M/B)) which is not trivial along the fibres of
PGL+(T (M/B)) → M , [43, p. 8].
16 SARA AZZALI
Because f spin structure preserving, it induces an isomorphism βGL+ between the orig-
inal spin structure S and the pulled back one df∗S. Then βGL+ gives a unitary equiv-
alence between the operator obtained via the pulled back structures, and the Dirac
operator for f∗ĝ and the chosen fixed spin structure, so that η̂(D/f∗ĝ,f∗S , f∗THM) =
η̂(D/f∗ĝ,S , f∗THM). Taken together
η̂(D/ĝ,S , THM) = η̂(D/f∗ĝ,S , f∗THM)
Let p : M̃ →M be the universal covering. Now we look at η̂(2)(D̃/) = η̂(2)(D̃/ĝ,S , THM,p),
where on M̃ the metric, spin structure and connection are the lift via p as by defini-
tion. Again, if we construct the L2 eta form for the entirely pulled back structure, we
get η̂(2)(D̃/ĝ,S , THM,p) = η̂(2)(D̃/f∗ĝ,f∗S , f∗THM,f∗p). Proceeding as above on the spin
structure, η̂(2)(D̃/f∗ĝ,f∗S , f∗THM,f∗p) = η̂(2)(D̃/f∗ĝ,S , f∗THM,f∗p). Since M̃ is the uni-
versal covering we have a covering isomorphism between f∗M̃ and M̃ , which becomes
an isometry when M̃ is endowed of the lift of the pulled back metric f∗ĝ, therefore
η̂(2)(D̃/f∗ ĝ,S , f
∗THM,f∗p) = η̂(2)(D̃/f∗ ĝ,S , f
∗THM,p)
It remains to observe how η̂ and η̂(2) depends on the connection T
HM . We remove for
the moment the hat ˆ to simplify the notation. Let TH0 M,T
1 M two connections, say
given by ω0, ω1 ∈ Ω1(M,T (M/B)) and pose ωt = (1− t)ω0 + tω1. Construct the family
M̆ =M × [0, 1] π̆→ B× [0, 1] =: B̆ as in the proof of Prop. 2.5. On this fibre bundle put
the connection one form ω̆ + dt. Since d̆η̆ = dη̆(·, t)− ∂
η(t)dt we have
η0 − η1 =
d̆η̆ −
M̆/B̆
Â(M × I/B × I)− d
which is the sum of a local contribution plus an exact form. Writing the same for η(2)
we get that for the L2-rho form ρ̂(2)(D/, TH0 M) = ρ̂(2)(D/, TH1 M) ∈ Ω(B)/dΩ(B) and
therefore we get the result. �
5.2. Conjectures. Along the lines of [31, 42] we can state the following conjectures.
Conjecture 5.1. If Γ is torsion-free and satisfies the Baum-Connes conjecture for the
maximal C∗-algebra, then [ρ̂(2)(D/ĝ)] vanishes if ĝ ∈ R+(M/B).
Definition 5.3. Let π : M → B and θ : N → B be two smooth fibre bundles of compact
manifolds over the same base B. A continuos map h : N → M is called a fibrewise
homotopy equivalence if π ◦ h = θ, and there exists g : N →M such that θ ◦ g = π and
such that h ◦ g, g ◦ h are homotopic to the identity by homotopies that take each fibre
into itself.
We work in the following with smooth fibrewise homotopy equivalences.
Definition 5.4. Let Γ be a discrete group and (π : M → M,p : M̃ → M), (θ : N →
B, q : Ñ → N) be two normal Γ-coverings of the fibre bundles π and θ. Denote as
r : M → BΓ, s : N → BΓ the two classifying maps. We say (π, p) and (θ, q) are Γ-
fibrewise homotopy equivalent if there exists a fibrewise homotopy equivalence h : N →
M such that s ◦ h is homotopic to r.
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 17
Let Dsign denote the family of signature operators.
Conjecture 5.2. Assume Γ is a torsion-free group that satisfies the Baum-Connes
conjecture for the maximal C∗-algebra. Let h be a orientation preserving Γ-fibrewise
homotopy equivalence between (π, p) and (θ, q) and suppose D̃sign
and D̃sign
have smooth
spectral projections and Novikov–Shubin invariants > 3(dimB + 1).
Then [ρ̂(2)(D̃signM/B)] = [ρ̂(2)(D̃
)] ∈ H∗dR(B).
Appendix A. Analysis on normal coverings
We summarize the analytic tools we use to investigate L2 spectral invariants, namely
NΓ-Hilbert spaces and Sobolev spaces on manifolds of bounded geometry, following the
nice exposition in [46].
A.1. NΓ-Hilbert spaces and von Neumann dimension. Let Γ be a discrete count-
able group and l2(Γ) the Hilbert space of complex valued, square integrable functions
on Γ. Denote with δγ ∈ CΓ the function with value 1 on γ, and zero elsewhere. The
convolution law on CΓ is δγ ∗ δβ = δγβ. Let L be the action of Γ on l2(Γ) by left con-
volution L : Γ → U(l2(Γ)), Lγ(f) = (δγ ∗ f)(x) = f(γ−1x). Right convolution action is
denoted by R.
Definition A.1. The group von Neumann algebra NΓ is defined to be the weak closure
NΓ := L(CΓ)weak in B(l2(Γ)). By the double commutant theorem NΓ = R(CΓ)′, so
that NΓ is the algebra of operators commuting with the right action of Γ. An important
feature of the group von Neumann algebra is its standard trace trΓ : NΓ −→ C defined
as trΓA =< Aδe, δe >l2(Γ). In particular for A =
aγLγ ∈ NΓ, then trΓ(A) = ae.
Definition A.2. A free NΓ-Hilbert space is a Hilbert space of the form W ⊗ l2(Γ),
where W is a Hilbert space and Γ acts on l2(Γ) on the right.
A NΓ-Hilbert space H is a Hilbert space with a unitary right-action of Γ such
that there exists a Γ-equivariant immersion H → V ⊗ l2(Γ) in some free NΓ-
Hilbert space. For H1,H2 NΓ-Hilbert spaces, define BΓ(H1H2) : = {T : H1 →
H2 bounded and Γ-equivariant}.
Let H = V ⊗ l2(Γ) be a free NΓ-Hilbert space. Then BΓ(V ⊗ l2(Γ)) ≃ B(V) ⊗ NΓ.
There exist a trace on the positive elements of this von Neumann algebra, with values
in [0,∞]: let (ψj)j∈N is a orthonormal base of V; if f ∈ B(H)+, its trace is given by
trΓ(f) =
j∈N < f(ψj ⊗ δe), ψj ⊗ δe >. A Γ-trace can be defined also on any NΓ-
Hilbert-space H using the immersion j : H →֒ V ⊗ l2Γ and proveing that the trace does
not depend on the choice of j (see [17] or [39, pag. 17]).
Definition A.3. Let H be a NΓ-Hilbert space. Its von Neumann dimension is defined
as dim Γ(H) = trΓ(id : H → H) ∈ [0,+∞).
Definition A.4. Let H1 and H2 be NΓ-Hilbert spaces. Define
18 SARA AZZALI
• BfΓ(H1,H2) := {A ∈ BΓ(H1,H2)′|dim Γ
<∞} are the Γ-finite rank oper-
ators
• B∞Γ (H1,H2) := B
Γ(H1,H2)
, are the Γ-compact operators
• B2Γ(H) := {A ∈ BΓ(H)s.t. trΓ (AA∗) <∞}, are the Γ-Hilbert-Schmidt operators
• B1Γ(H) := B2Γ(H)B2Γ(H)∗ the Γ-trace class operators.
Their main properties are:
1) Bf (H),B∞(H),B2(H),B1(H) are ideals and Bf ⊂ B1 ⊂ B2 ⊂ B∞;
2) A ∈ Bi(H) if and only if |A| ∈ Bi(H) for i = 1, 2, f,∞.
A.2. Covering spaces, bounded geometry techniques. Let p : Z̃ → Z a normal
Γ-covering of a compact Riemannian manifold Z. Let I ⊂ Z̃ be a fundamental domain
for the (right) action of Γ on Z̃ (I is an open subset s.t. I · γ ∩ I and Z̃ \
I · γ have
zero measure ∀γ 6= e).
Let E → Z a Hermitian vector bundle, and Ẽ = p∗E the pull-back. The sections
C∞c (Z̃, Ẽ) form a CΓ-right module for the action (ξ · f)(m̃) =
(R∗gξ)(m̃)f(g
−1) where
(R∗gξ)(m̃) := ξ(m̃g). Its Hilbert space completion L
2(Z̃, Ẽ) is a Γ-free Hilbert space
in the sense of definition A.2, in fact the map ψ : L2(Z̃, Ẽ) −→ L2(I, Ẽ|I) ⊗ l2(Γ),
|I ⊗ δγ is an isomorphism.
The Γ-trace class operators are characterized as follows: let A ∈ BΓ(L2(Z̃, Ẽ))
A ∈ B1Γ(L2(Z̃, Ẽ)) if and only if χI |A|χI ∈ B1(L2(I, E|I))
If A ∈ B1Γ(L2(Z̃, Ẽ)) then trΓ(A) = tr(χIAχI). If A ∈ B1Γ(L2(Z̃, Ẽ)) has Schwartz
kernel [A] continuos, then
trA =
([A](x, x)) dx =
π∗ tr Ẽx ([A](x, x)) dx . (A.1)
The covering of a compact manifold and the pulled back bundle Ẽ above are the most
simple examples of manifolds of bounded geometry3.
The analysis on manifolds of bounded geometry was developped in [45]. We specialize
here to the case of a normal covering Z̃.
The Sobolev spaces of sections are defined, for k ≥ 0, as the completion Hk(Z̃, Ẽ) :=
C∞c (Z̃, Ẽ)
where ‖f‖k :=
L2(Z̃,Ẽ⊗jT ∗Z̃); for k < 0 H
k(Z̃, Ẽ) is defined
as the dual of H−k(Z̃, Ẽ).
3Let (N, g) be a Riemannian manifold. N is of bounded geometry if
(1) it has positive injectivity radius i(N, g);
(2) the curvature RN and all its covariant derivatives are bounded.
A hermitian vector bundle E → N is of bounded geometry if the curvature RE and all its covariant
derivatives are bounded. This can be characterized in normal coordinates with conditions on g, coordi-
nate transformations and ∇ (see for example in [45] and [46]).
L2-RHO FORM FOR NORMAL COVERINGS OF FIBRE BUNDLES 19
The spaces of uniform Ck sections are defined as follows: UCk(M̃) = {f : M̃ → C | f ∈
Ck and ‖f‖k ≤ c(k) ∀k}, where ‖f‖k = supm̃∈M̃,Xi{|∇X1 . . .∇Xkf(m̃)|}, and analo-
gously for sections UCk(M̃, Ẽ). UC∞(M̃, Ẽ) is the Fréchet space :=
k UCk(M̃ , Ẽ).
The following Sobolev embedding property holds [45]: if dim M̃ = n, then for j > n
there is a continuos inclusion Hj(M̃ , Ẽ) →֒ UCk(M̃ , Ẽ).
The algebra UDiff(M̃, Ẽ) of uniform differential operators is the algebra generated by
operators in UC∞(M̃ ,End Ẽ) and derivatives {∇ẼX}X∈UC∞(M̃ ,TM̃) with respect to uni-
form vector fields. P ∈ UDiff(M̃ , Ẽ) extends to a continuos operator Hj(M̃, Ẽ) →
Hj−k(M̃ , Ẽ) ∀j ∈ Z.
P ∈ UDiff(M̃ , Ẽ) is called uniformly elliptic if its principal symbol σpr ∈
UC∞(T ∗M̃, π∗ End Ẽ) is invertible out of an ǫ-neighborhood of 0 ∈ T ∗M̃ , with inverse
section which can be uniformly estimated.
For a uniformly elliptic operator T the G̊arding inequality holds:
Hs+k(M̃,Ẽ)
≤ c(s, k) (‖ϕ‖Hs + ‖Tϕ‖Hs) ∀s ∈ R (A.2)
If T is a continuos operator T : C∞c (N,E) → (C∞c (M̃ , Ẽ))′ we will denote its Schwartz
kernel with [T ] ∈ C∞(M̃ × M̃, Ẽ4Ẽ∗).
Definition A.5. We say that T : C∞c (N,E) → (C∞c (M̃ , Ẽ))′ has order k ∈ Z if ∀s ∈
Z it admits a bounded extension Hs(M̃, Ẽ) → Hs−k(M̃, Ẽ). Hence it is closable as
unbounded operator on L2(M̃ , Ẽ).
The space of order k operators is denoted Opk(M̃, Ẽ), and comes with the seminorms
on B(Hs(M̃ , Ẽ),Hs−k(M̃ , Ẽ)). The space Op−∞(M̃, Ẽ) =
k(M̃, Ẽ) is a Fréchet
space.
Finally, an operator T ∈ Opk(M̃ , Ẽ) is called elliptic if it satisfies G̊arding inequality.
We will denote as OpkΓ(M̃, Ẽ) the subspace of Γ-invariant operators in Op
k(M̃, Ẽ).
Consider the Fréchet space of continuos rapidly decreasing functions
RB(R) = {f : R → C : f continuos, and
∣(1 + x
2 f(x)
∣ <∞ ∀k}
Let T ∈ Opk(M̃ , Ẽ) , k ≥ 1 an elliptic, formally self-adjoint operator. Denote again by T
its closure, with domain Dom T = Hk(M̃ , Ẽ). From G̊arding inequality (A.2) the map
RC(R) −→ B(Hj(M̃ , Ẽ),H l(M̃ , Ẽ)), f 7→ f(T ) is continuos ∀j, l ∈ Z, so that
RC(R) −→ Op−∞(Z̃, Ẽ) , f 7→ f(T )
is continuos. One can prove that the Schwartz kernel of such operator is smooth: by
(A.2) and Sobolev embedding, for L = [n
+ 1] the map
Op−2L−l(Z̃, Ẽ) −→ UC l(Z̃ × M̃,E4E∗) , T 7→ [T ] (A.3)
is continuos ∀l ∈ N; then in particular for f ∈ RC(R), the kernel [f(T )] ∈ UC∞(Z̃ ×
Z̃, Ẽ4Ẽ∗) and the map RB(R) −→ UC∞(Z̃ × Z̃, Ẽ4Ẽ∗) , f 7→ [f(T )] is continuos.
20 SARA AZZALI
Lemma A.6. Since elements in Op−∞Γ (M̃, Ẽ) are Γ-trace class, then for T ∈
Op kΓ(M̃, Ẽ) elliptic and selfadjoint, the map RC(R) → B1Γ(M̃ , Ẽ) , f 7→ f(T ) is contin-
uos. As a consequence ∀m ∃l such that
| trΓ f(T )|m ≤ C ‖f(T )‖−l,l . (A.4)
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bonn.de/people/strohmai/globan/boris/
Mathematisches Institut, Georg-August Universität Göttingen
E-mail address: azzali@uni-math.gwdg.de
	1. Introduction
	2. Geometric families in the L2-setting
	2.1. The heat operator for the covering family
	2.2. Transgression formulæ, eta integrands
	2.3. The t0 asymptotic
	3. The L2-eta form
	3.1. The family Novikov–Shubin invariants
	3.2. The L2-eta form
	3.3. Case of uniform invertibility
	4. The L2 rho form
	5. (2) and positive scalar curvature for spin vertical bundle
	5.1. (2) and the action of a fibre bundle diffeomorphism on R+(M/B)
	5.2. Conjectures
	Appendix A. Analysis on normal coverings
	A.1. N-Hilbert spaces and von Neumann dimension
	A.2. Covering spaces, bounded geometry techniques
	References
