the novikov conjecture for low degree cohomology classes
TRANSCRIPT
The Novikov Conjecture for Low Degree
Cohomology Classes
VARGHESE MATHAIDepartment of Mathematics, University of Adelaide, Adelaide 5005, Australia.e-mail: [email protected] and Department of Mathematics, MIT,
Cambridge, MA 02139, U.S.A. e-mail: [email protected]
(Received: 7 March 2001; accepted in final form: 28 January 2002)
Abstract. We outline a twisted analogue of the Mishchenko–Kasparov approach to prove theNovikov conjecture on the homotopy invariance of the higher signatures. Using our approach,we give a new and simple proof of the homotopy invariance of the higher signatures associated
to all cohomology classes of the classifying space that belong to the subring of the cohomologyring of the classifying space that is generated by cohomology classes of degree less than orequal to 2, a result that was first established by Connes and Gromov and Moscovici using
other methods. A key new ingredient is the construction of a tautological C�r ðG;sÞ-bundle
and connection, which can be used to construct a C�r ðG; sÞ-index that lies in the Grothendieck
group of C�r ðG;sÞ, where s is a multiplier on the discrete group G corresponding to a degree 2
cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our
theorem.
Mathematics Subject Classifications (2000). 19K56, 46L80, 58G12, 57R20, 53C15.
Key words. higher signatures, homotopy invariance, index theory, Novikov conjecture, opera-
tor K-theory, twisted group C� algebras.
Introduction
Let K be a closed connected, oriented smooth manifold and LðKÞ 2 H4�ðK;RÞ denote
the Hirzebruch L-class of K. Let BG denote the classifying space of a discrete
group G and f : K ! BG be a continuous map. The characteristic numbers,
SignuðK; f Þ ¼ hLðKÞ [ f �ðuÞ; ½K�i for u 2 H�ðBG;RÞ are called the higher signatures
of K. Recall that by the Hirzebruch signature theorem, one knows that
Sign1ðK; f Þ ¼ hLðKÞ; ½K�i is the ordinary signature of the manifold K (of course in
this case, the map f is irrelevant).
Then one has
CONJECTURE 1 (The Novikov Conjecture). Let h: K 0 ! K be a smooth, orien-
tation preserving homotopy equivalence of smooth, closed, oriented manifolds K 0;K.
Then for any discrete group G and for any continuous map f : K ! BG, one hasSignuðK; f Þ ¼ SignuðK
0; f ohÞ for all u 2 H�ðBG;RÞ, that is, all of the higher signatures
SignuðK; f Þ of K are homotopy invariants of ðK; f Þ.
Geometriae Dedicata 99: 1–15, 2003. 1# 2003 Kluwer Academic Publishers. Printed in the Netherlands.
It is more common to formulate the Novikov conjecture when the map
f : K ! BG classifies the universal cover of K, and G ¼ p1ðKÞ, but the formula-
tion given above is equivalent to it, cf. the survey article [FPR], the book [W].
This fundamental conjecture has been open for more than 30 years and has
generated an enormous amount of exciting research including the recent papers
[Hi, Y]. The purpose of this short paper is to present further evidence of its
truth by settling the conjecture in the affirmative in the special case of the coho-
mology classes u 2 HjðBG;RÞ; j ¼ 0; 1; 2 and the cohomology ring generated by
these.
We shall be concerned with an equivalent formulation of the Novikov conjecture
using K-theory, which we will will now describe. First of all, we need only consider
the even-dimensional case as there is a standard reduction of the odd-dimensional
case to the even-dimensional case by considering K� S1 instead. Let OpðKÞ denotethe space of smooth differential forms of degree p on a smooth closed, oriented
manifold K of dimension 2n. Choose a Riemannian metric on K and let
�: OpðKÞ ! O2n�pðKÞ denote the Hodge �-operator. Let Z: OpðKÞ ! OpðKÞ be
defined as ZðoÞ ¼ ipð p�1Þþno. Then there is a grading E ¼ �Z on O�ðKÞ ¼L2n
p¼0 OpðKÞ. Let
OþðKÞ ¼
1� E2
� �O�
ðKÞ
and
O�ðKÞ ¼
1þ E2
� �O�
ðKÞ
be the þ1 and �1 eigenspaces of the grading E, respectively. Let
dK: OpðKÞ ! Opþ1ðKÞ denote the deRham differential and dK ¼ �EdKE its formal
adjoint with respect to the inner product
ðo1;o2Þ ¼
ZK
ð�o1Þ ^ o2 ð1Þ
Let dK þ dK denote the signature operator
dK þ dK: OþðKÞ ! O�
ðKÞ:
Now let Oþð2ÞðKÞ, O
�ð2ÞðKÞ and O�
ð2ÞðKÞ denote the L2 completion of the spaces Oþ
ðKÞ,
O�ðKÞ and O�
ðKÞ respectively, with respect to the inner product in (1). Define the
Laplacian D ¼ dKdK þ dKdK. Since ð1þ DÞ�1=2 is a norm bounded operator on
O�ð2ÞðKÞ, we see that ðd
K þ dKÞð1þ DÞ�1=2: Oþð2ÞðKÞ ! O�
ð2ÞðKÞ is a pseudodifferential
operator of order zero, which is therefore a bounded operator [Ho]. Moreover, it
2 VARGHESE MATHAI
is Fredholm since dK þ dK is an elliptic operator [Ho]. That is, the signature operatordK þ dK defines a class
½dK þ dK� ¼ ðO�ð2ÞðKÞ; ðd
K þ dKÞð1þ DÞ�1=2Þ 2 K0ðKÞ ¼ KK0ðCðKÞ;CÞ:
in the K-homology of the manifold K. The equivalent K-theory formulation of the
Novikov conjecture is
CONJECTURE 2 (K-theory version of the Novikov Conjecture). Let h: K0 ! K be
a smooth, orientation preserving homotopy equivalence of smooth, closed, oriented
manifolds K0;K. Then for any discrete group G and for any continuous map
f : K ! BG, one has
f�½dK þ dK� ¼ f�h�½d
K0
þ dK0
� 2 K0ðBGÞ � Q:
That is, f�½dK þ dK� is a homotopy invariant of ðK; f Þ.
Let s be a multiplier on the discrete group G with trivial Dixmier–Douady invari-
ant, that is s ¼ e2pic, where c is an R-valued group 2-cocycle on G, cf. Section 1. In
Section 2, we define a C�r ðG; sÞ-vector bundle VBGs ! BG over BG which defines a
canonical class ½VBGs � 2 K0ðC0ðBGÞ � C�r ðG; sÞÞ. Some standard modifications have
to be made when BG is not compact, which are done in Section 2. Using this, one
can define a twisted Kasparov map ms: K�ðBGÞ ! K�ðC�r ðG; sÞÞ which is given by
the Kasparov intersection product msð�Þ ¼ ½VBGs � �C0ðBGÞ �. In the special case when
the multiplier s is trivial, ms reduces to the standard Kasparov map, [Kas]. This con-struction and its variants can be viewed as our main contribution. Using it and a
result in [HS], we will prove the following theorem.
THEOREM 0.1 (Homotopy invariance). Let h: K0 ! K be a smooth, orientation
preserving homotopy equivalence of smooth, closed, oriented manifolds K0;K. Then for
any discrete group G and for any multiplier s on G with trivial Dixmier–Douady
invariant and for any continuous map f : K ! BG, there exists e > 0 such that
mssð f�½dK þ dK�Þ ¼ mss ð f�h�½d
K0
þ dK0
�Þ 2 K0ðC�r ðG; s
sÞÞ; ð0:1Þ
for all s 2 ½0; eÞ. Here ss is a homotopy through multipliers from the trivial multiplier tos as in Lemma 3:3.That is, mssð f�½d
K þ dK�Þ is a homotopy invariant of ðK; f Þ for all s sufficiently small.
Remarks 1. In the special case when the multiplier s is trivial, the theorem above
reduces to the one established by Kaminker and Miller [KM], Kasparov [Kas] and
Mishchenko [Mis].
We also deduce the following important corollaries that are mentioned in the
abstract, from Theorem 0.1 together with the twisted L2-index theorem that is
proved in the appendix of [Ma].
THE NOVIKOV CONJECTURE 3
COROLLARY 0.2. Under the same hypotheses as in Theorem 0:1, one has that
hLðKÞ [ f �ðChðVBGss ÞÞ; ½K�i 2 K0ðC�r ðG; s
sÞÞ ð0:2Þ
is a homotopy invariant of ðK; f Þ for all s sufficiently small, where Ch denotes the Chern
character of ½MiSo�.
COROLLARY 0.3 (Novikov conjecture for low degree cohomology classes). Let
h: K0 ! K be a smooth, orientation preserving homotopy equivalence of smooth,
closed, oriented manifolds K0;K. Then for any discrete group G and for any continuousmap f : K ! BG, one has SignuðK; f Þ ¼ SignuðK
0; f ohÞ for all u belonging to the
subring R of the cohomology ring of BG, H�ðBG;RÞ, where R is generated by
HjðBG;RÞ for j ¼ 0; 1; 2. That is, all the higher signatures SignuðK; f Þ for u 2 R are
homotopy invariants of ðK; f Þ.
TWISTED APPROACH TO THE NOVIKOV CONJECTURE
Let mQs denote the map ms over the rational numbers, i.e. m
Qs ¼ ms � Q. Then we see
from Theorem 0.1 above that the Novikov conjecture is valid whenever the map mQs
is injective for some multiplier s on G that is sufficiently close to the trivial multiplier.
The point is that it may be easier to prove injectivity of mQs for a multiplier other than
the trivial one, and is an outline of our new approach to the Novikov conjecture. In
the special case when the multiplier s is trivial, this reduces to the Kasparov–
Mishchenko approach to the Novikov conjecture. We also suspect that Theorem 0.1
remains valid for all s 2 ½0; 1�, as this is the case at least after one applies the von
Neumann trace to Equation (0.1). Another result that supports this is in [Ma2],
which says in particular that K0ðC�r ðG; s
sÞÞ ffi K0ðC�r ðGÞÞ for all s 2 ½0; 1�.
Theorem 0.1 and its corollaries are proved in Section 3 of the paper.
1. Preliminaries
We include some preliminary material from [Ma, MS] so as to make the paper more
self-contained.
1.1. TWISTED GROUP ALGEBRAS
Let G be a discrete group and s be a multiplier on G, that is, s: G� G ! Uð1Þ
satisfies
. sðg; eÞ ¼ sðe; gÞ ¼ 1; 8 g 2 G;
. sðg1; g2Þsðg1g2; g3Þ ¼ sðg1; g2g3Þsðg2; g3Þ; 8g1; g2; g3 2 G (the cocycle relation).
It follows from these relations that sðg; g�1Þ ¼ sðg�1; gÞ. Note that the complex
conjugate of s, �s is also a multiplier on G.
4 VARGHESE MATHAI
Denote by ‘2ðGÞ the standard Hilbert space of complex-valued L2-functions on the
discrete group G. We will use a left ðG; �sÞ-action on ‘2ðGÞ (or, equivalently, a ðG; �sÞ-unitary representation in ‘2ðGÞ) which is given explicitly by
TLg fðg0Þ ¼ fðg�1g0Þ �sðg; g�1g0Þ; g; g0 2 G:
It is easy to see that this is indeed a ðG; �sÞ-action, i.e.
TLe ¼ Id and TLg1TLg2¼ �sðg1; g2ÞT
Lg1g2
; 8g1; g2 2 G:
Also
ðTLg Þ�¼ sðg; g�1ÞTLg�1 :
Let us define the following operators in ‘2ðGÞ:
TRg fðg0Þ ¼ fðg0gÞsðg0; gÞ; g; g0 2 G:
It is easy to check that they form a right ðG; sÞ-action in ‘2ðGÞ, i.e.
TRe ¼ Id and TRg1TRg2¼ sðg1; g2ÞT
Rg1g2
; 8g1; g2 2 G;
and also
ðTRg Þ�¼ �sðg; g�1ÞTRg�1 :
This action commutes with the left ðG; �sÞ-action defined above, i.e.
TLg TRg0 ¼ TRg0T
Lg ; 8g; g0 2 G:
Recall the definition of the twisted group algebra CðG; sÞ, consisting of complex
valued functions with finite support on G and with the twisted convolution operation
ð f � gÞðgÞ ¼Xg1g2¼g
fðg1Þgðg2Þsðg1; g2Þ:
The basis of CðG; sÞ as a vector space is obtained by d-functions fdggg2G, dgðg0Þ ¼ 1 if
g ¼ g0 and 0 otherwise. Then
dg1 � dg2 ¼ sðg1; g2Þdg1g2 :
Associativity of this multiplication is equivalent to the cocycle condition for s.Note also that the d-functions fdggg2G form an orthonormal basis in ‘2ðGÞ. It is
easy to check that
TLg dg0 ¼ �sðg; g0Þdgg0 ; TRg dg0 ¼ sðg0g�1; gÞdg0g�1:
It is clear that the correspondences dg 7!TLg and dg 7!TRg define representations of
CðG; �sÞ and CðG; sÞ on ‘2ðGÞ, respectively. In both cases the norm closure of the
image of the twisted group algebra is called reduced twisted group C�-algebras which
are denoted C�r ðG; �sÞ and C
�r ðG; sÞ respectively. There is a finite von Neumann trace
trG; �s: C�r ðG; �sÞ ! C on C�
r ðG; �sÞ which is defined by the formula trG; �sA ¼ ðAde; deÞ.
THE NOVIKOV CONJECTURE 5
It can also be written as trG; �sA ¼ Ag;g ¼ Adg; dg� �
for any g 2 G because the right-
hand side does not depend of g.Suppose that the multipliers s0 and s are cohomologous, that is,
s0ðg1; g2Þ ¼ sðg1; g2Þ@yðg1; g2Þ;
where y: G ! Uð1Þ is a group 1-cochain and recalling that the boundary of this
group cochain is @yðg1; g2Þ ¼ yðg1Þyðg2Þyðg1; g2Þ�1. Then there is a �-isomorphism
of reduced twisted group C�-algebras C�r ðG; s
0Þ and C�r ðG; sÞ defined on the d-
functions fdggg2G as dg ! yðgÞ�1dg.
1.2. PROJECTIVE ðG;sÞ-ACTION OF G-VECTOR BUNDLES ON THE COVERING SPACE
Let K be a compact connected Riemannian manifold, G be its fundamental group
and ~K be its universal cover, i.e. one has the principal bundle G ! ~K !pK.
Let f : K ! BG denote a continuous map classifying the universal cover of K, and
let ½c� 2 H2ðBG;RÞ. Let o be a closed two form on K such that the cohomology class
defined by it, ½o�, is equal to f �ð½c�Þ. Note that eo ¼ p�o ¼ dZ is exact on the universalcovering space ~K, since the lift of c to EG is trivial. For s 2 R, define Hs ¼ dþ s iZ.Then Hs is a Hermitean connection on the trivial line bundle over ~K, and the curva-
ture of Hs is ðHsÞ2 ¼ i seo. Then Hs defines a projective action of G on any G-vectorbundle as follows:
Observe that since eo is G-invariant, one has 0 ¼ g�eo� eo ¼ dðg�Z� ZÞ 8g 2 G. Sog�Z� Z is a closed 1-form on the simply connected manifold ~K, therefore
g�Z� Z ¼ dcg; 8g 2 G;
where cg is a smooth function on ~K. We can assume without loss of generality that it
satisfies the following normalization condition:
. cgðx0Þ ¼ 0 for a fixed x0 2 eM; 8g 2 G.
It follows that cg is real-valued and ceðxÞ � 0, where e denotes the identity element
of G. It is also easy to verify that
. cgðxÞ þ cg0 ðgxÞ � cg0gðxÞ is independent of x 2 eM; 8g; g0 2 G.
Then sðg; g0Þ ¼ expð�icgðg0 � x0ÞÞ defines a multiplier on G. The complex conjugate
multiplier is �sðg; g0Þ ¼ expðicgðg0 � x0ÞÞ.
Let E be a vector bundle over K and ~E be the induced G-invariant vector bundleover ~K. For u 2 ~E and g 2 G define
Ugu ¼ ðg�1Þ�u; Sgu ¼ expð�icgðlðuÞÞÞ u;
where l: ~E ! ~K is the projection. Then the operators Tg ¼ Ug � Sg satisfy
Te ¼ Id; Tg1Tg2 ¼ sðg1; g2ÞTg1g2 ;
6 VARGHESE MATHAI
for all g1; g2 2 G. That is, the map T : G ! Autð ~EÞ, g 7!Tg, is a projective ðG; sÞ-action on ~E. Moreover, T : G ! Autð ~EÞ is a free projective ðG; sÞ-action on ~E since
Ugu ¼ u if and only if g ¼ e.
It is also easy to verify that the adjoint operator to Tg in ~E (with respect to a
smooth G-invariant inner product) is T �g ¼ �sðg; g�1ÞTg�1 .
1.3. DEPENDENCE ON Z
If we make another choice Z0 such that dZ0 ¼ eo, then it follows that Z0 � Z is a closed1-form on a simply connected manifold ~K, and therefore it is exact, i.e. Z0 ¼ Zþ df,where f 2 C1ð ~KÞ. We will again assume without loss of generality that f is normal-
ized by the condition fðx0Þ ¼ 0.
Then one sees that the associated Hermitean connection on the trivial line bundle
over ~K, H0s ¼ dþ isZ0, s 2 R, is unitarily conjugate to Hs, i.e. H0s ¼ e�isfHseisf. There-fore H0s� ¼ e�isfHs�eisf.Define the function c0
g from dc0g ¼ g�Z0 � Z0, c0
gðx0Þ ¼ 0 (with the same point
x0 2M as above). Then
c0g ¼ cg þ g�f� f� fðgx0Þ:
The new multiplier will be
s0ðg1; g2Þ ¼ sðg1; g2Þ@yðg1; g2Þ:
where yðgÞ ¼ exp i½fðgx0Þð Þ and recalling that the boundary of this group cochain is
@yðg1; g2Þ ¼ yðg1Þyðg2Þyðg1; g2Þ�1. That is, the multipliers s0 and s are cohomologous.
The modified translations are defined by T 0g ¼ U0
gS0g, where U
0g ¼ Ug ¼ ðg�1Þ� and
S0g ¼ expð�ic0gÞ. Then T
0g1T 0
g2¼ s0ðg1; g2ÞT
0g1g2
.
The relation between projective ðG; sÞ-action and the projective ðG; s0Þ-action is
T 0g ¼ eifðg�x0Þ e�ifTge
if� �; ð1:1Þ
which is again the same unitary conjugation up to a constant unitary factor.
1.4. DEPENDENCE ON o
If we choose another closed 2-form o0 on K such that the cohomology class defined
by it , ½o0� is also equal to f �ð½c�Þ, then o0 � o ¼ dl, where l is a 1-form on K. Then a
similar calculation as in Section 1.3 shows that the multipliers s0 and s obtained
using o0 and o are cohomologous, and that the relation between projective ðG; sÞ-action and the projective ðG; s0Þ-action is again unitary conjugation up to a constant
unitary factor.
1.5. DIXMIER–DOUADY INVARIANT
Let s be a multiplier on a discrete group G. Recall that s is a normalized Uð1Þ-valued
group 2-cocycle on G, so it defines a cohomology class ½s� 2 H2ðG;Uð1ÞÞ. Consider
the short exact sequence of coefficient groups
THE NOVIKOV CONJECTURE 7
1 ! Z !i
R �!e2p
ffiffiffiffi�1
p
Uð1Þ ! 1;
which gives rise to a long exact sequence of cohomology groups (the change of coef-
ficient groups sequence)
� � � ! H2ðG;ZÞ !i�H2ðG;RÞ �!
e2pffiffiffiffi�1
p
�H2ðG;Uð1ÞÞ !
dH3ðG;ZÞ ! � � � :
Then one definition of the Dixmier–Douady invariant of s is dðsÞ 2 H3ðG;ZÞ. The
following easy lemma will be useful later on.
LEMMA 1.1. Let s be a multiplier on G such that dðsÞ ¼ 0. Then there is a homotopy
through multipliers ss; s 2 ½0; 1� such that s0 ¼ 1 and s1 ¼ s.Proof. Since dðsÞ ¼ 0, there is a R-valued 2-cocycle c on G such that e2pic ¼ s.
Then define the homotopy through multipliers ss ¼ e2pisc for all s 2 ½0; 1�. &
1.6. CLASSIFYING SPACE FOR DISCRETE GROUPS
Let G be a discrete group, and EG ! BG be a locally trivial, principal G-bundle suchthat EG is contractible and BG is paracompact. Then BG is unique up to homotopy
and is called the classifying space of G. It classifies locally trivial, principal G-bundles,and one can obtain an explicit construction of BG as the quotient of the infinite join
of G by G, cf. [MiSt].
2. Canonical C ��r ðG;sÞ-Line Bundle and the Twisted Kasparov Map
2.1. CANONICAL C�r ðG; sÞ-LINE BUNDLE
We will now construct a C�r ðG; sÞ-line bundle over K. Let pr1: L ¼ ~K� C ! ~K be
the trivial line bundle over ~K. Let f : K ! BG denote a continuous map classifying
the universal cover of K, and let ½c� 2 H2ðBG;RÞ. Let o be a closed two form on K
such that the cohomology class defined by it, ½o�, is equal to f �ð½c�Þ. Recall fromSection 1.2 that this determines a multiplier s on G and a free ðG; �sÞ-action on L.
Moreover, any other choice of closed two form on K that is cohomologous to odetermines a multiplier that is cohomologous to s, and therefore determines a pro-
jective action that is unitarily equivalent to the free ðG; �sÞ-action on L.
Consider the tensor product C�r ðG; sÞ �C L ! ~K, which is a trivial C�
r ðG; sÞ-vec-tor bundle over ~K. We recall that for any unital C�-algebra A, an A-vector bundle
over a locally compact space X is a locally trivial Banach vector bundle whose fibres
have the structure of a finitely generated projective left A-module. Since the fibres of
C�r ðG; sÞ �C L are free modules of rank one, it is known as a C�
r ðG; sÞ-line bundle.We next observe that the tensor product of the left ðG; �sÞ-action on L and the right
ðG; sÞ-action on C�r ðG; sÞ yields a G-action on C
�r ðG; sÞ �C L that covers the G-action
on ~K. That is, C�r ðG; sÞ �C L is a G-invariant C�
r ðG; sÞ-line bundle on ~K, so it is the
lift of a C�r ðG; sÞ-line bundle VKs ! K on K.
8 VARGHESE MATHAI
An equivalent way of defining the C�r ðG; sÞ-line bundle VKs on K is as follows. Let
Gs be equal to G�Uð1Þ as sets, with group law given by ðg; zÞ:ðg0; z0Þ ¼ðgg0; sðg; g0Þzz0Þ. Then ðe; 1Þ is the identity element in Gs, where e is the identity ele-
ment in G and 1 is the identity element in Uð1Þ. The inverse of ðg; zÞ is
ðg�1; �sðg; g�1Þz�1Þ. This makes Gs into a group. Then the inclusion homomorphism
Uð1Þ ! Gs given by z ! ðe; zÞ and the projection homomorphism Gs ! G given
by ðg; zÞ ! g, gives rise to the short exact sequence of groups
1 ! Uð1Þ ! Gs ! G ! 1:
That is, Gs is a Uð1Þ-central extension of G, corresponding to the cocycle s. Conver-sely, given a Uð1Þ-central extension of G
1 ! Uð1Þ ! G0 !p
G ! 1;
by choosing a section s : G ! G0 of p, one can define a multiplier sðg; g0Þ ¼sðgÞsðg0Þsðgg0Þ�1, and verify that G0 ¼ Gs. That is, any Uð1Þ-central extension of G is
of the form Gs for some multiplier s on G. There is a free action of the group Gs
on ~K�Uð1Þ given by
ðg; zÞð p; xÞ ¼ ðg�1p; expð�icgðg�1pÞÞzxÞ; ðg; zÞ 2 Gs; ð p; xÞ 2 ~K�Uð1Þ:
which determines a principal Gs-bundle Gs ! ~K�Uð1Þ �!p�pr1
K. Gs acts on the right
on C�r ðG; sÞ as follows,
ððg; zÞf Þðg0Þ ¼ zfðg0gÞsðg0; gÞ; ðg; zÞ 2 Gs; f 2 C�r ðG; sÞ; g 2 G:
Then VKs is isomorphic to the associated bundle
ðC�r ðG; sÞ � ~K�Uð1ÞÞ
�Gs:
Now the Grothendieck group of formal differences of equivalence classes of
A-vector bundles over a compact space Y with addition given by the Whitney direct
sum, is denoted by K 0ðX;AÞ. It is known that K 0ðX;AÞ is canonically isomorphic to
K0ðCðXÞ �C AÞ, cf. [Ros1]. Therefore the C�r ðG; sÞ-line bundle VKs determines a class
½VKs � 2 K0ðCðKÞ �C C�r ðG; sÞÞ ¼ K0 K;C�
r ðG; sÞ� �
.
The twisted Kasparov map mKs : K�ðKÞ ! K�ðC�r ðG; sÞÞ is then the intersection pro-
duct mKs ðyÞ ¼ ½VKs � �CðKÞ y. In the special case when the cocycle c is trivial and hence
the multiplier s is trivial, this reduces to the standard Kasparov map [Kas].
LEMMA 2.1. In the notation above, given ½c� 2 H2ðBG;RÞ, if s0 is another mutiplieron G obtained as in Section 1:2, then s0 is cohomologous to s, and after identifyingC�r ðG; sÞ and C
�r ðG; s
0Þ, one has ½VKs � ¼ ½VKs0 � 2 K0ðCðKÞ �C C�r ðG; sÞÞ .
Proof. Section 1.4 shows that s0 and s are cohomologous and we need to show
that VKs and VKs0 are isomorphic in this case. An isomorphism between C�r ðG; sÞ and
C�r ðG; s
0Þ (which exists cf. Section 1.1), induces an isomorphism between the
C�r ðG; sÞ-line bundle C
�r ðG; sÞ �C L and the C�
r ðG; s0Þ-line bundle C�
r ðG; s0Þ �C L that
THE NOVIKOV CONJECTURE 9
is G-equivariant, therefore inducing an isomorphism between the C�r ðG; sÞ-line
bundle VKs and the C�r ðG; s
0Þ-line bundle VKs0 . &
LEMMA 2.2. In the sense above, ½VKs � 2 K0ðCðKÞ �C C�r ðG; sÞÞ depends only on
½c� 2 H2ðBG;RÞ, f : K ! BG and K, and is therefore canonical.
2.2. UNIVERSAL TWISTED KASPAROV MAP
Let G be a countable discrete group and let R be a finite subcomplex of some reali-
zation of BG ¼ KðG; 1Þ as a CW complex. Let G ! ~R ! R denote the restriction of
G ! EG ! BG to R. Then as above we can form the C�r ðG; sÞ-vector bundle
VRs ! R overR as in Proposition 2.1, except that neither BG norR is smooth in gene-
ral, and BG need not be compact. Therefore we will first briefly define the projective
actions in this more general case.
We represent the class ½c� 2 H2ðBG;RÞ by an Alexander–Spanier 2-cocycle. Then
since EG is contractible, the lift of c, ~c is a coboundary dZ, where Z is an Alexander–Spanier 1-cochain. Since ec is G-invariant, we see that 0 ¼ g�ec� ec ¼ dðg�Z� ZÞ, that isg�Z� Z is an Alexander–Spanier 1-cocycle on EG. Since EG is contractible, it follows
that g�Z� Z ¼ dcg, where cg is an Alexander–Spanier 0-cochain on EG, i.e. a con-
tinuous function on EG. Since df ¼ 0; f a 0-cochain implies that f is a constant,
we can assume without loss of generality that cg satisfies the following normalization
condition:
. cgðx0Þ ¼ 0 for a fixed x0 2 EG; 8g 2 G.
It follows that cg is real-valued and ceðxÞ � 0, where e denotes the identity element
of G. It is also easy to verify that
. cgðxÞ þ cg0 ðgxÞ � cg0gðxÞ is independent of x 2 EG; 8g; g0 2 G.
Then sðg; g0Þ ¼ expð�icgðg0 � x0ÞÞ defines a multiplier on G. The rest of the con-
struction remains the same as in Section 2.1. It follows that we can define a
C�r ðG; sÞ-line bundle VRs over R that determines a canonical class ½VRs � 2
K0ðCðRÞ � C�r ðG; sÞÞ and therefore, a morphism mRs : K�ðRÞ ! K�ðC
�r ðG; sÞÞ which
is defined by the Kasparov intersection product mRs ð�Þ ¼ VRs� �
�CðRÞ �. Passing to
the limit as R ranges over all finite subcomplexes of the realization of BG as a
CW complex, we obtain the universal twisted Kasparov map
ms: K�ðBGÞ ! K� C�r ðG; sÞ
� �:
Let K be a finite CW complex, f : K ! BG be a continuous map and
G ! eK !qK be the principal G-bundle which is the pullback via f of the universal
principal G-bundle G ! EG ! BG. Let f�: K�ðKÞ ! K�ðBGÞ, j ¼ 0; 1, be the
induced map on K-theory. Since ½VKs � ¼ ½ f �ðVBGs Þ� and since mKs and ms are defined
using these C�r ðG; sÞ-vector bundles, then one easily sees using the functoriality of
the Kasparov intersection product that one has the following basic relationship
between the Kasparov maps mKs ¼ ms � f�:
10 VARGHESE MATHAI
2.3. TWISTED INDEX MAP
Let ½½VKs �� 2 KKðCðKÞ;CðKÞ � C�r ðG; sÞÞ be defined as the class of the graded module
VKs � 0 and the zero operator. The diagonal embedding D: K ! K� K induces a
�-homomorphism D�: CðKÞ � CðKÞ ! CðKÞ which defines an element ½D�� 2
KK0ðCðKÞ � CðKÞ;CðKÞÞ. Then by definition of the Kasparov intersection product,
one has ½½VKs �� ¼ ½VKs � � ½D��. For ½P� 2 K�ðKÞ, define the induced element
½PVKs � 2 KK�ðCðKÞ;C�r ðG; sÞÞ as the Kasparov intersection product
½PVKs � ¼ ½½VKs �� �CðKÞ ½P� 2 KK�ðCðKÞ;C�r ðG; sÞÞ:
Let p: K ! pt be the projection to a point. Then the C�r ðG; sÞ-index of ½PVKs � is by
definition
indC�r ðG;sÞð½PVKs �Þ ¼ p�ð½PVKs �Þ 2 K�ðC
�r ðG; sÞÞ:
By associativity and functoriality of the Kasparov intersection product, cf. [Ros],
one has
indC�r ðG;sÞð½PVKs �Þ ¼ p�ð½PVKs �Þ
¼ p�ð½½VKs ��Þ �CðKÞ ½P�
¼ ½½VKs �� � p�ð½D��Þ �CðKÞ ½P�
¼ ½½VKs �� � 1CðKÞ �CðKÞ ½P�
¼ ½VKs � �CðKÞ ½P�
¼ mKs ð½P�Þ
¼ msð f�½P�Þ:
3. Homotopy Invariance
We will define a natural connection Hs on the bundle VKs , which has central cur-
vature. Here K is a closed connected, oriented smooth manifold. We will show
that there is a multiplier s associated to any degree two cohomology class
½c� 2 H2ðBG;RÞ such that the curvature of the connection Hs on the bundle
VKs is as small as one desires. This enables us to apply the Hilsum–Skandalis
Theorem 4.2 to conclude that the C�r ðG; sÞ-index of the signature operator
BK ¼ dK þ dK twisted by Hs is a homotopy invariant of the manifold. Finally,
we use the twisted L2-index theorem, cf. [Ma], to deduce Theorem 0.1, Corollary
0.1 and Corollary 0.2 on the Novikov conjecture for low degree cohomology
classes.
We follow the notation of Sections 1.2 and 2.1. First define a Hermitean connec-
tion eHs on the C�r ðG; sÞ-line bundle C
�r ðG; sÞ �C L on ~K as follows, where L is the
trivial line bundle on ~K.
~Hsða� f Þ ¼ a� ðdf� ifZÞ; f 2 C1ð ~KÞ; a 2 C�r ðG; sÞ
THE NOVIKOV CONJECTURE 11
i.e. ~Hs ¼ I� ðd� iZÞ, where Z is a 1-form on ~K as in Section 1, that is, dZ ¼ eo, and Iis the identity operator. We will now verify that d� iZ commutes with the ðG; �sÞ-action on L. Recall that this action is given by Tg ¼ Ug � �Sg. Therefore
ðd� iZÞUg � �Sg ¼ eiðg�1Þ
�cg idðg�1Þ�cg þ d� iZ� �
and also
Ug � �Sgðd� iZÞ ¼ eiðg�1Þ
�cg d� iðg�1Þ�Z� �
:
Therefore
½Tg; ðd� iZÞ� ¼ �ieiðg�1Þ
�cg ðg�1Þ�Z� Zþ dðg�1Þ�cg
� �¼ �ieiðg
�1Þ�cg dcg�1 þ dðg
�1Þ�cg
� �¼ 0:
Since the identity operator commutes with the ðG; sÞ-action on C�r ðG; sÞ, it follows
that eHs commutes with the G-action on C�r ðG; sÞ �C L. It therefore induces a connec-
tion Hs on the quotient VKs . Observe that the connection Hs is central, since it has
curvature Hsð Þ2¼ I� io.
We will assume without loss of generality that our manifolds are of even dimen-
sion. We next recall the following theorem:
THEOREM 3.1 (Theorem 4.2, [HS]). Given a smooth oriented homotopy equivalence
h: K0 ! K between smooth, closed, oriented Riemannian manifolds, then there is a
positive constant C ¼ CðK;K0Þ such that for all A-bundles x over K with connection Hhaving curvature O such that jjOjjð1Þ 4C, one has
indexAðBK � HÞ ¼ indexAðBK0 � h�HÞ 2 K0ðAÞ;
where A is a unital C�-algebra and B� is the signature operator of the manifold �.
Now given a closed 2-form o as in Section 1, we can consider the closed 2-form
so; s 2 R, which defines a multiplier ss. Then the following theorem is an immediate
consequence of Theorem 3.1.
THEOREM 3.2. Given a smooth oriented homotopy equivalence h: K0 ! K between
smooth, closed, oriented Riemannian manifolds, then there is a positive constant
C ¼ CðK;K0Þ such that for all C�r ðG; s
sÞ-bundles VKss over K with connection Hss having
curvature so such that sjjojjð1Þ 4C, one has
indexC�r ðG;ssÞ BK � Hss� �
¼ indexC�r ðG;ssÞðBK0 � h�Hss Þ 2 K0ðC
�r ðG; s
sÞÞ;
where B� is the signature operator of the manifold �.
Theorem 0.1 is deduced from Theorem 3.2 above by using the discussion in Section
2.3. More precisely, the signature operator defines a class ½BK� 2 K0ðKÞ, and the
induced element ½BVKs � 2 KK�ðCðKÞ;C�r ðG; sÞÞ is just the class of ½BK � Hs�. Therefore
by the discussion in Section 2.3, we deduce that mss ð f�½BK�Þ ¼ indexC�r ðG;ssÞ BK � Hss
� �.
12 VARGHESE MATHAI
By Theorem 3.2, we deduce that mss ð f�½BK�Þ is a homotopy invariant of K for all s
sufficiently small.
3.1. THE CHERN CHARACTER OF C�r ðG; sÞ-VECTOR BUNDLES
Recall that a C�r ðG; sÞ-vector bundle over K is a locally trivial Banach vector bundle
E ! K where the fibres have the structure of finitely generated projective left
C�r ðG; sÞ modules, [MiFo]. Morphisms of such bundles are required to in addition
preserve the C�r ðG; sÞ module structure of the fibres. The Grothendieck group of for-
mal differences of equivalence classes of such bundles, with addition given by
the Whitney direct sum, is denoted by K0 K;C�r ðG; sÞ
� �, and it coincides
with K0ðCðKÞ � C�r ðG; sÞÞ. Therefore by the Kunneth theorem, there is a natural
isomorphism.
K0 K;C�r ðG; sÞ
� �� Q ffi K0ðKÞ � K0ðC
�r ðG; sÞÞ � Q
� ��
� K1ðKÞ � K1ðC�r ðG; sÞÞ � Q
� �:
Using the ordinary Chern character ch: K0ðKÞ ! HevenðK;QÞ and ch: K1ðKÞ !
HoddðK;QÞ, one obtains the Chern character of Mishchenko–Solov’ev [MiSo]:
Ch: K0 K;C�r ðG; sÞ
� �! HevenðK;QÞ � K0ðC
�r ðG; sÞÞ
� �� HoddðK;QÞ � K1ðC
�r ðG; sÞÞ
� �;
which is an isomorphism modulo torsion.
Corollary 0.2 follows by applying the Mishchenko–Formenko C�-index theorem
[MiFo] to Theorem 3.2. More precisely, as argued earlier,
mssð f�½BK�Þ ¼ indexC�r ðG;ssÞðBVKss
Þ:
The C�-index theorem [MiFo] asserts that the Chern character of indexC�r ðG;ssÞðBVKss
Þ
equals hLðKÞ [ ChðVKssÞ; ½K�i ¼ hLðKÞ [ f �ðChðVBGss ÞÞ; ½K�i. Therefore we deduce
Corollary 0.2.
As in Section 6, [CM], a choice of Borel, bounded, almost everywhere smooth
cross-section b: K ! ~K of p: ~K ! K, yields an isomorphism of C�r ðG; s
sÞ modules
U: O�ð2Þð
~KÞ ! O�ð2ÞðK;VKssÞ
such that BK � Hss ¼ Uð ~BK � ~Hss ÞU�, where O�ð2Þð
~KÞ denotes the L2 differential forms
on the covering space ~K, O�ð2ÞðK;VKssÞ denotes the VKss-valued L2 differential forms on
K, ~BK � ~Hss is the signature operator on ~K twisted by the connection ~Hss which was
defined earlier in the section. Therefore
indexC�r ðG;ssÞ BK � Hss� �
¼ indexC�r ðG;ssÞð
eBK � ~HssÞ:
It is known cf. [Ma2] that the right-hand side can be expressed in terms of heat ker-
nels, and in particular that trG;ssðindexC�r ðG;ssÞð
~BK � ~Hss ÞÞ is equal to the L2 index,
THE NOVIKOV CONJECTURE 13
indexL2 ð ~BK � ~Hss Þ. Therefore by the L2 index theorem of Gromov, as proved in
[Ma, Ma2], one have
trG;ss ðindexC�r ðG;ssÞð
~BK � ~HssÞÞ ¼ hLðKÞ [ ½eso�; ½K�i
¼Xmj¼0
sj
j!hLðKÞ [ ½o� j; ½K�i
where n ¼ 2m is the dimension of K, are homotopy invariants for all s small enough.
Therefore hLðKÞ [ f �ð½c�Þ j; ½K�i are homotopy invariants of K for j ¼ 0; 1; . . . ;m,
since ½o� ¼ f �ð½c�Þ. Let S be the subring of H�ðBG;RÞ generated by elements in
HjðBG;RÞ for j ¼ 0; 2. If u 2 S \H2jðBG;RÞ, then u ¼Pj
jaj¼0 aa½ca�, where aa 2 R,
a ¼ ða1; . . . ; a‘Þ 2 N‘ and ½ca� ¼ ½ca11 � [ � � � [ ½ca‘‘ �. Here ½c1�; . . . ; ½c‘� are elements in
H2ðBG;RÞ and ½cajj � ¼ ½cj� [ � � � [ ½cj�, where the cup product is taken aj times. So it
suffices to establish the homotopy invariance of hLðKÞ [ f �ð½ca�Þ; ½K�i for each
a 2 N‘. For each j ¼ 1; . . . ‘, let oj be the smooth 2 form on K such that
½oj� ¼ f �ð½cj�Þ. In the argument above, if we replace so byP‘
j¼1 sjoj, where sj > 0,
we deduce that
LðKÞ [ e
P‘
j¼1sjoj
� �; ½K�
�¼
Xmjaj¼0
sa
a!hLðKÞ [ ½oa�; ½K�i
are homotopy invariants for all sj small enough. Therefore hLðKÞ [ ½oa�; ½K�i are are
homotopy invariants of K for jaj ¼ 0; 1; . . . ;m, as asserted. Finally, by taking the
product of K with the circle, and applying the above, Corollary 0.3 is deduced.
Alternately, if the Chern character of C�r ðG; sÞ-vector bundles could be expressed
using Chern–Weil theory, then trG;ss ðChðVKssÞÞ ¼ trG;sðeðHss Þ
2
Þ ¼ eso, modulo factors
of i ¼ffiffiffiffiffiffiffi�1
p, since ðHss Þ
2¼ iso. Applying the von Neumann trace to Equation
(0.2) in Corollary 0.2, we see that
trG;ss hLðKÞ [ f�ðChðVBGss ÞÞ; ½K�i
� �¼ hLðKÞ [ ½eso�; ½K�i;
and Corollary 0.3 can be deduced.
Acknowledgement
The author thanks the referee for numerous helpful comments and suggestions and
acknowledges support from the Clay Mathematical Institute.
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THE NOVIKOV CONJECTURE 15