K-Theory 24: 341–359, 2001.
c
2001 Kluwer Academic Publishers. Printed in the Netherlands.
341
On Higher Eta-Invariants and Metrics of
Positive Scalar Curvature
ERIC LEICHTNAM
1,
and PAOLO PIAZZA
2,
1
Institut de Jussieu-Chevaleret, Plateau E, 7
me
´etage (Alg`ebres d’op´erateurs), 175 rue de
Chevaleret, 75013 Paris, France. e-mail: leicht@math.jussieu.fr
2
Universit`a di Roma ‘La Sapienza’, Dipartimento di Matematica ‘Guido Castelnuovo’,
P.le A. Moro 2, I-00185 Roma, Italy. e-mail: piazza@mat.uniroma1.it
(Received: June 2001)
Abstract. Let N be a closed connected spin manifold admitting one metric of positive scalar
curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order
to distinguish metrics of positive scalar curvature on N . In particular, we give sufficient conditions,
involving π
1
(N) and dim N,forN to admit an infinite number of metrics of positive scalar curvature
that are nonbordant.
Mathematics Subject Classifications (2000): 55N22, 19L41.
Key words: bordism groups, positive scalar curvature metrics, Galois coverings, higher eta-inva-
riants, higher ρ-invariants, b-pseudodifferential calculus, higher APS index formula.
0. Introduction
Let N
n
be a closed connected spin manifold of dimension n. In this Introduction we
shall always assume that N admits at least one metric of positive scalar curvature
g. One may ask how many such metrics there are on N. As a small perturbation
of g produces a different metric which is still of positive scalar curvature (≡ PSC),
it is clear that in order to give a rigorous meaning to the above question one first
needs to introduce a way for distinguishing two metrics of PSC g
0
, g
1
.Thereare
three increasingly stronger conditions for distinguishing two metrics of PSC g
0
and g
1
.
The first one is to say that g
0
and g
1
are not path-connected in R
+
(N), the space
of PSC metrics on N. Thus, in this case, we are interested in π
0
(R
+
(N)), the set
of arcwise connected components of R
+
(N).
The second way for distinguishing two PSC metrics employs the notion of con-
cordance: g
0
and g
1
are concordant if there exists a metric of PSC on N × [0, 1]
extending g
0
on N ×{0}, g
1
on N ×{1} and of product-type near the boundary.
Notice that two metrics which are path connected are concordant; it is not known
Research partially supported by a CNR-CNRS cooperation project.
Research partially supported by a CNR-CNRS cooperation project and by MURST.
342 ERIC LEICHTNAM AND PAOLO PIAZZA
whether the converse is true. In this case one is interested in the set of concord-
ance classes of PSC metrics on N , denoted by π
0
(R
+
(N)).
The third and more subtle way for distinguishing two PSC metrics g
0
, g
1
on
our spin manifold N employs the notion of bordism. Intuitively we allow ourself
the freedom of taking any spin manifold W with boundary ∂W = N
−N and
a PSC metric G on W which is of product-type near the boundary and restricts
to g
0
g
1
on ∂W. See Section 1 for the precise definition. Clearly, two metrics
that are concordant are also bordant. (On the other hand there are example of
nonconcordant metrics that are bordant, see [LM, p. 329].) Summarizing, as far
as the problem of distinguishing metrics of positive scalar curvature is concerned,
we have: nonbordant ⇒ nonconcordant ⇒ nonpathconnected. In this paper we
shall be mainly concerned with nonbordant metrics.
Interesting results by Gromov and Lawson [GL], Lawson and Michelson [LM],
Kreck and Stolz [KS], Botvinnik and Gilkey [Bo-Gi] and others show that, under
some assumptions on N,ifN admits one metric of PSC then N admits an infinite
number of ‘distinct’ metrics of PSC. It is important to remark that all these papers
use index theory for Dirac operators in an essential way; the main point is to define
invariants that can distinguish two metrics of PSC. In the case of [GL, LM, Bo-Gi]
and [KS] these invariants involve, one way or another, the Atiyah–Patodi–Singer
index theory on manifolds with boundary. Of particular importance for us is the
paper by Botvinnik and Gilkey [Bo-Gi]. The fundamental idea used in [Bo-Gi] is
that the twisted eta-invariant is a bordism invariant and can, therefore, be used in
order to distinguish non-bordant metrics; using the twisted eta-invariants together
with some deep facts about PSC metrics, Botvinnik and Gilkey are able to give
sufficient conditions on π
1
(N) and dimN ensuring the existence of an infinite
number of nonbordant metrics.
Our main goal in this paper is to give an extension of the Botvinnik–Gilkey’s
result by replacing the twisted eta-invariant used in [Bo-Gi] by Lott’s higher eta-
invariant [L2, 3] and its delocalized part, which is, by definition, the higher ρ-
invariant. That the higher ρ-invariant could be used in connection with PSC
problems was already observed in [L2], where it was proved that it is constant on
the connected components of R
+
(N). In this paper we elaborate on these ideas,
relying heavily on the arguments given in [Bo-Gi]; the higher Atiyah–Patodi–
Singer index theory developed in [L2] and [LP1, 3] will play an essential role
throughout.
Although we shall give results that might be amenable to further appli-
cations, we prefer to state here the geometric outcome of our arguments. Let F
be a finite group and let m be a odd positive integer. Botvinnik and Gilkey have
introduced a nonnegative integer r
m
(F ), see [Bo-Gi]. We are interested in finite
groups F and positive integers m such that r
m
(F ) > 0. In this direction we recall
that if m ≡ 3mod4orif|F | is odd, then, automatically, r
m
(F ) > 0. If m ≡ 1mod4
then r
m
(F ) > 0iffF contains an element, which is not conjugate to its
inverse.
HIGHER ETA-INVARIANTS AND METRICS OF POSITIVE SCALAR CURVATURE 343
THEOREM 0.1. Let n ∈ N, n 5.LetF be a finite group such that r
m
(F ) > 0
for m odd, 5 m n.Let,
, be finitely generated groups. On these groups
we make the following assumptions: (i) =
>F ; (ii) is virtually nilpotent
or Gromov hyperbolic; (iii) there exists i ∈ N such that H
n−m−4i
(, Q) = 0 .
Let now N be a closed connected spin manifold of dimension n with fundamental
group × admitting one metric of positive scalar curvature. Then, N admits an
infinite number of nonbordant metrics of PSC.
1. Spin Bordism Groups
In this Section we recall the definition of the bordism groups that will be used
in the sequel. Metrics on manifolds with boundary will be understood to be
of product-type near the boundary; we make this assumption throughout the
paper.
Let be a finitely generated group. Let B be the associated classifying space
and let E →B be the universal -covering. We denote by
spin
n
(B) the bord-
ism group consisting of triples (N,s,f)with N a spin n-dimensional closed com-
pact manifold, s a spin structure on N and f a continuous map N →B.Welet
spin,+
n
(B) be the subgroup of
spin
n
(B) consisting of triples (N,s,f) with N
admitting a metric of positive scalar curvature.
Finally, let Pos
spin
n
(B) be the bordism group consisting of quadruples (N, s,
f, g) with g a metric of PSC on N. Two quadruples (N,s,f,g)and (N
,s
,f
,g
)
are bordant in Pos
spin
n
(B) if there exists a spin (n + 1)-dimensional manifold
with boundary W with a continuous map F : W → B, a spin structure S and a
positive scalar curvature metric H on W such that (with obvious notation) ∂W =
N
(−N
), F |
∂W
= f
f
, S
∂W
= s
s
, H |
∂W
= g
g
.
DEFINITION 1.1. Let N be a closed spin manifold of dimension n admitting one
metric of PSC. Let f : N → Bπ
1
(N) be the classifying map for the universal
cover of N. We shall say that two metrics of PSC g
0
and g
1
are bordant if for any
spin structure s we have: [N,s,f,g
0
] = [N,s,f,g
1
]inPos
spin
n
(Bπ
1
(N)) .
We now pass to the relative bordism groups. These were first introduced by
Hajduk [Ha]. They were successfully employed by Stolz in problems of classific-
ation of PSC metrics. See [St1] (p. 630) [St2]. Following [St1], we denote them
by R
spin
n
(B). Thus, we only consider the spin case; this corresponds to the choice
of supergroup γ = (, 0, × Z
2
) in [St2]. We recall that R
spin
n
(B) is the bord-
ism group of quadruples (N,s,f,h),whereN is a spin manifold of dimension n
possibly with boundary, s is a spin structure, f : N →B is a continuous map, h
is a PSC metric on ∂N. Two such quadruples (N,s,f,h), (N
,s
,f
,h
) are said
to be bordant if: (i) there is a bordism (V,S,F,H) between (∂N, s|
∂N
,f|
∂N
,h)
and (∂N
,s
|
∂N
,f
|
∂N
,h
) viewed as elements in Pos
spin
n−1
(B); (ii) the closed
(glued) spin manifold N ∪
∂N
V ∪
∂N
N
is the boundary of some spin manifold W .
344 ERIC LEICHTNAM AND PAOLO PIAZZA
Moreover, there is a spin structure S
W
on W and a continuous map E
W
: W → B
restricting to s (resp. s
, S)andf (resp. f
, resp. F )onN (resp. N
, resp. V ).
We also recall that there exists a long exact sequence
···→
spin
n+1
(B) → R
spin
n+1
(B)
∂
−→Pos
spin
n
(B) →
spin
n
(B) →··· .
(1.1)
2. A Product-Formula for the Higher Eta-Invariant
Let N
n
be a spin manifold with a fixed spin structure s and let f : N → B be
a continuous map. We assume that N admits a metric g of PSC. We can consider
the Dirac operator D/ associated to s and g.Wealsohavethe-covering f
∗
E
with the -invariant Riemannian metric g induced by g. We denote by
D/ the lifted
Dirac operator and by D the Dirac operator on N twisted by the flat bundle
V
f
= f
∗
E ×
B
∞
, (2.1)
here B
∞
denotes the Connes–Moscovici algebra [CM]. As g has PSC, the Dirac
operator D is invertible in the B
∞
-Mishchenko–Fomenko calculus. Thus, accord-
ing to [L2, §4.4], [L3] the higher eta-invariant η of D is well defined (see also
Section 2 of [LP1] and [LP3, Appendix]); it is an element in the vector space
of noncommutative differential forms modulo graded commutators:
∗
(B
∞
) =
∗
(B
∞
)/[
∗
(B
∞
),
∗
(B
∞
)]. By definition η =
+∞
0
η(t) dt with
η(t) =
2
√
π
STR
Cl(1)
YD exp −(tYD +∇⊗Id
C
2 )
2
, if n is odd,
2
√
π
STR D exp −(tD +∇)
2
, if n is even,
where Y =
01
10
and ∇ is the B
∞
-connection defined by Lott in [L1] p. 436.
Recall that
∗
(B
∞
) is an algebra; whereas
∗
(B
∞
) is only a vector space [Ka,
L1]. We shall denote the higher eta-invariant associated to this Dirac operator by
η(N, s, f, g); as in the case of the usual eta-invariant, η(N, s, f, g) depends in a
nontrivial way on the choice of g, f ,ands. This follows, for example, by the
variational formulae proved in [L1]; in particular η(N, s, f, g) does not descend to
Pos
spin
n
(B).
Let (N,s,f,g) be a quadruple as above and let (M,σ,φ) be a triple as in the
definition of
spin
n
(B).Leth be a smooth metric on M; then for >0small
enough, (1/h ×g) is a metric of PSC on M ×N. Let us also consider the product
spin structure and the product classifying map φ × f into B( × ). We consider
the following subalgebra of the reduced C
∗
-algebra C
∗
r
(×): A
×
= B
∞
⊗B
∞
.
The classifying map φ ×f defines a ×-covering
M × N of M × N.We
then consider the flat A
×
-vector bundle over M ×N defined as the fiber product
HIGHER ETA-INVARIANTS AND METRICS OF POSITIVE SCALAR CURVATURE 345
V
φ
⊗f
= (
M × N) ×
(×)
A
×
. These data define a A
×
-linear Dirac operator
D
M×N
acting on C
∞
(M ×N;V
φ
⊗f
⊗S
M×N
). We shall also consider the connection
∇
M×N
introduced by Lott (see [L1, p. 456]) acting on
C
∞
(M × N ;
∗
(A
×
) ⊗
A
×
V
φ
⊗f
⊗ S
M×N
).
In the sequel we shall use the natural map (see [Co, p. 104]) π :
∗
(A
×
) →
∗
(B
∞
)
⊗
∗
(B
∞
); we shall still denote by π the induced map
C
∞
(M × N ;
∗
(A
×
) ⊗
A
×
V
φ
⊗f
⊗ S
M×N
)
→ C
∞
(M × N;
∗
(B
∞
)
⊗
∗
(B
∞
) ⊗
A
×
V
φ
⊗f
⊗ S
M×N
).
Using D
M×N
and ∇
M×N
one can define a higher eta-invariant, denoted η(M×N) ∈
∗
(A
×
). The next Lemma will provide a product-formula for η(M × N); it
extends results by Atiyah–Patodi–Singer [APS3, p. 84] and Lott [L2, p. 205].
PROPOSITION 2.1. For the higher eta-invariant η(M × N), the following for-
mula holds modulo graded commutators and d
∗
(B
∞
)
⊗
∗
(B
∞
) +
∗
(B
∞
)
⊗
d
∗
(B
∞
):
π(η(M × N)) = C(m,n)
M
A(M) ∧ ω
φ
∗
(E)
⊗ η(N, s, f, g). (2.2)
In this formula ω
φ
∗
(E)
is the closed biform ∈
∗
(M)
⊗
∗
(B
∞
) introduced by
Lott ([L1] p. 436), and C(m, n) is a nonzero constant depending only on m and n.
Proof. We will treat only the case, where m and n are both odd. Recall that
S
M
, S
N
denote the bundle of spinors over M and N, respectively. Then the spinor
bundle S
M×N
over M × N can be identified with S
M
⊗ S
N
⊗ C
2
.LetY and I be
the two anticommuting involutions of C
2
defined by
Y =
01
10
, I =
0
√
−1
−
√
−10
.
Then for the product metric on M ×N and any α ∈ T
∗
x
M, β ∈ T
∗
y
N, the clifford
action on S
M
⊗S
N
⊗C
2
is given by: cl(α) = cl
M
(α)⊗Id⊗Y, cl(β) = Id⊗cl
N
(β)⊗
I. The (generalized) Dirac operator D
M×N
introduced above acts on the sections
of the A
×
-bundle V
φ
⊗f
⊗S
M
⊗S
N
⊗C
2
over M ×N; we shall endow this bundle
with the Z
2
-grading induced by the decomposition C
2
= C ⊕ C. It’s easy to see
that the operator D
M×N
is equal to D
M
Y + D
N
I (remark that V
φ
⊗f
= V
φ
⊗V
f
).
The higher eta-invariant η(M × N)) is by definition equal to
+∞
0
η(t)dt, where
the local eta-integrand η(t) is defined to be
η(t) = 2/
√
π · STR D
M×N
exp (−(tD
M×N
+∇
M×N
)
2
).
Next we introduce the connection ∇
⊗
M×N
given by
∇
⊗
M×N
=∇
M
⊗ Id
C
2
⊗ Id + Id ⊗ Id
C
2
⊗∇
N