Doc Math J DMV

Higher Index Theorems and

the Boundary Map in Cyclic Cohomology

Victor Nistor

Received June

Communicated by Joachim Cuntz

Abstract We show that the ChernConnes character induces a natural

transformation from the six term exact sequence in lower algebraic K

Theory to the p erio dic cyclic homology exact sequence obtained by Cuntz

and Quillen and we argue that this amounts to a general higher index

theorem In order to compute the b oundary map of the p erio dic cyclic

cohomology exact sequence we show that it satises prop erties similar to

the prop erties satised by the b oundary map of the singular cohomology long

exact sequence As an application we obtain a new pro of of the Connes

Moscovici index theorem for coverings

Mathematics Sub ject Classication Primary K Secondary

D L G

Key Words cyclic cohomology algebraic K theory index morphism etale

group oid higher index theorem

Contents

Introduction

Index theorems and Algebraic K Theory

Pairings with traces and a Fedosov type formula

Higher traces and excision in cyclic cohomology

An abstract higher index theorem

Pro ducts and the b oundary map in p erio dic cyclic cohomology

Cyclic vector spaces

Extensions of algebras and pro ducts

Prop erties of the b oundary map

Relation to the bivariant ChernConnes character

The index theorem for coverings

Group oids and the cyclic cohomology of their algebras

Morita invariance and coverings

The AtiyahSinger exact sequence

The ConnesMoscovici exact sequence and pro of of the theorem

References

1

Partially supp orted by NSF grant DMS a NSF Young Investigator Award DMS

and a Sloan Research Fellowship

Documenta Mathematica

Victor Nistor

Introduction

Index theory and K Theory have b een close sub jects since their app earance

Several recent index theorems that have found applications to Novikovs Conjecture

use algebraic K Theory in an essential way as a natural target for the generalized

indices that they compute Some of these generalized indices are von Neumann

dimensionslike in the L index theorem for coverings that roughly sp eaking

computes the trace of the pro jection on the space of solutions of an elliptic dier

ential op erator on a covering space The von Neumann dimension of the index do es

not fully recover the information contained in the abstract ie algebraic K Theory

index but this situation is remedied by considering higher traces as in the Connes

Moscovici Index Theorem for coverings Since the app earance of this theorem

index theorems that compute the pairing b etween higher traces and the K Theory

class of the index are called higher index theorems

In a general higher index morphism ie a bivariant character was dened

for a class of algebrasor more precisely for a class of extensions of algebrasthat is

large enough to accommo date most applications However the index theorem proved

there was obtained only under some fairly restrictive conditions to o restrictive for

most applications In this pap er we completely remove these restrictions using a

recent breakthrough result of Cuntz and Quillen

In Cuntz and Quillen have shown that p erio dic cyclic homology denoted

HP satises excision and hence that any twosided ideal I of a complex algebra A

gives rise to a p erio dic sixterm exact sequence

HP A HP AI HP I

O

o o

HP AI HP A HP I

similar to the top ological K Theory exact sequence Their result generalizes earlier

results from See also

If M is a smo oth manifold and A C M then HP A is isomorphic to the de

Rham cohomology of M and the ChernConnes character on algebraic K Theory

generalizes the ChernWeil construction of characteristic classes using connection and

curvature In view of this result the excision prop erty equation gives more

evidence that p erio dic cyclic homology is the right extension of de Rham homology

from smo oth manifolds to algebras Indeed if I A is the ideal of functions vanishing

on a closed submanifold N M then

HP I H M N

DR

and the exact sequence for continuous p erio dic cyclic homology coincides with the

exact sequence for de Rham cohomology This result extends to not necessarily

smo oth complex ane algebraic varieties

The central result of this pap er Theorem Section states that the Chern

Connes character

alg

A HP A ch K

i

i

where i is a natural transformation from the six term exact sequence in

lower algebraic K Theory to the p erio dic cyclic homology exact sequence In this

Documenta Mathematica

Higher Index Theorems

formulation Theorem generalizes the corresp onding result for the Chern character

on the K Theory of compact top ological spaces thus extending the list of common

features of de Rham and cyclic cohomology

The new ingredient in Theorem b esides the naturality of the ChernConnes

character is the compatibility b etween the connecting or index morphism in alge

braic K Theory and the b oundary map in the CuntzQuillen exact sequence Theo

rem Because the connecting morphism

alg alg

Ind K AI K I

asso ciated to a twosided ideal I A generalizes the index of Fredholm op erators

Theorem can b e regarded as an abstract higher index theorem and the com

putation of the b oundary map in the p erio dic cyclic cohomology exact sequence can

b e regarded as a cohomological index formula

We now describ e the contents of the pap er in more detail

If is a trace on the twosided ideal I A then induces a morphism

alg

K I C

More generally one canand has toallow to b e a higher trace while still getting

alg

a morphism K I C Our main goal in Section is to identify as explicitly

alg

as p ossible the comp osition Ind K AI C For traces this is done in

Lemma which generalizes a formula of Fedosov In general

Ind

where HP I HP AI is the b oundary map in p erio dic cyclic cohomology

Since is dened purely algebraically it is usually easier to compute it than it is to

alg

I is not known in many interesting compute Ind not to mention that the group K

situations which complicates the computation of Ind even further

In Section we study the prop erties of and show that is compatible with

various pro duct type op erations on cyclic cohomology The pro ofs use cyclic vector

spaces and the external pro duct studied in which generalizes the cross

pro duct in singular homology The most imp ortant prop erty of is with resp ect to

the tensor pro duct of an exact sequence of algebras by another algebra Theorem

We also show that the b oundary map coincides with the morphism induced by the

o dd bivariant character constructed in whenever the later is dened Theorem

As an application in Section we give a new pro of of the ConnesMoscovici

index theorem for coverings The original pro of uses estimates with heat kernels

Our pro of uses the results of the rst two sections to reduce the ConnesMoscovici

index theorem to the AtiyahSinger index theorem for elliptic op erators on compact

manifolds

The main results of this pap er were announced in and a preliminary version

of this pap er has b een circulated as Penn State preprint no PM March

Although this is a completely revised version of that preprint the pro ofs have not

b een changed in any essential way However a few related preprints and pap ers have

app eared since this pap er was rst written they include

I would like to thank Joachim Cuntz for sending me the preprints that have

lead to this work and for several useful discussions Also I would like to thank the

Documenta Mathematica

Victor Nistor

Mathematical Institute of Heidelb erg University for hospitality while parts of this

manuscript were prepared and to the referee for many useful comments

Index theorems and Algebraic K Theory

alg alg

We b egin this section by reviewing the denitions of the groups K and K and of

alg alg

the index morphism Ind K AI K I asso ciated to a twosided ideal I A

There are easy formulas that relate these groups to Ho chschild homology and we

review those as well Then we prove an intermediate result that generalizes a formula

of Fedosov in our Ho chschild homology setting which will serve b oth as a lemma in

the pro of of Theorem and as a motivation for some of the formalisms developed in

this pap er The main result of this section is the compatibility b etween the connecting

or index morphism in algebraic K Theory and the b oundary morphism in cyclic

cohomology Theorem An equivalent form of Theorem states that the Chern

Connes character is a natural transformation from the six term exact sequence in

algebraic K Theory to p erio dic cyclic homology These results extend the results in

in view of Theorem

All algebras considered in this pap er are complex algebras

Pairings with traces and a Fedosov type formula It will b e conve

alg

nient to dene the group K A in terms of idemp otents e M A that is in

terms of matrices e satisfying e e Two idemp otents e and f are called equivalent

in writing e f if there exist x y such that e xy and f y x The direct sum of

two idemp otents e and f is the matrix e f with e in the upp erleft corner and f in

the lowerright corner With the directsum op eration the set of equivalence classes

alg

A is of idemp otents in M A b ecomes a monoid denoted P A The group K

dened to b e the Grothendieck group asso ciated to the monoid P A If e M A

alg

is an idemp otent then the class of e in the group K A will b e denoted e

Let A C b e a trace We extend to a trace M A C still denoted

P

by the formula a a If e f then e xy and f y x for some

ij ii

i

x and y and then the tracial prop erty of implies that e f Moreover

e f e f and hence denes an additive map P A C From the

universal prop erty of the Grothendieck group asso ciated to a monoid it follows that

we obtain a well dened group morphism or pairing with

alg

K A e e e C

The pairing generalizes to not necessarily unital algebras I and traces I

C as follows First we extend to I I C the algebra with adjoint unit to b e

alg

zero on Then we obtain as ab ove a morphism K C The morphism I

alg alg alg

I C is obtained by restricting from K I to K I dened to b e K

alg alg

I K C the kernel of K

alg

The denition of K A is shorter

alg

K A lim GL AGL A GL A

n n n

alg

In words K A is the ab elianization of the group of invertible matrices of the form

a where a M A The pairing with traces is replaced by a pairing with

Ho chschild co cycles as follows

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Higher Index Theorems

If A A is a Ho chschild co cycle then the the functional denes a morphism

alg

K A C by rst extending to matrices over A and then by pairing it

with the Ho chschild cycle u u Explicitly if u a with inverse u b

ij ij

then the morphism is

X

alg

a b C A u u K

ij j i

ij

The morphism dep ends only on the class of in the Ho chschild homology group

HH A of A

If I A AI is an exact sequence of algebras that is if I is a

twosided ideal of A then there exists an exact sequence

Ind

alg alg alg alg alg alg

K I K A K AI K I K A K AI

of Ab elian groups called the algebraic K theory exact sequence The connecting or

index morphism

alg alg

Ind K AI K I

will play an imp ortant role in this pap er and is dened as follows Let u b e an

invertible element in some matrix algebra of AI By replacing AI with M AI for

n

some large n we may assume that u AI Cho ose an invertible element v M A

that pro jects to u u in M AI and let e and e v e v Because

alg

e M I the idemp otent e denes a class in K I Since e e M I

alg

the dierence e e is actually in K I and dep ends only on the class u of u

alg

in K AI Finally we dene

Ind u e e

To obtain an explicit formula for e choose liftings a b A of u and u and let

v the lifting to b e the matrix

a aba ab

v

ba b

as in page Then a short computation gives

ab ab a ba ba

e

bab ba

Continuing the study of the exact sequence I A AI choose an

arbitrary linear lifting l AI A If is a trace on I we let

a b l a l b l a b

Because a xy ax y y a x we have A I and hence is a Ho chschild

co cycle on AI ie ab c a bc ca b The class of in HH AI

denoted turns out to b e indep endent of the lifting l If A is a lo cally convex

algebra then we assume that we can choose the lifting l to b e continuous If

A I then it is enough to consider a lifting of A AI

alg alg

The morphisms K AI C and K I C are related

through the following lemma

Documenta Mathematica

Victor Nistor

Lemma Let be a trace on a twosided ideal I A If

alg alg

I AI K Ind K

is the connecting morphism of the algebraic K Theory exact sequence associated to

the twosided ideal I of A then

Ind

If A I then we may replace I by I

Proof We check that Ind u u for each invertible u M AI

n

By replacing AI with M AI we may assume that n

n

Let l AI A b e the linear lifting used to dene the co cycle representing

equation and choose a l u and b l u in the formula for e equation

Then the left hand side of our formula b ecomes

a b a bab ab Indu ba

Because bab is in I we have a bab a b and hence

Indu e e e e a b

Since the right hand side of our formula is

u u u l u l u l u u a b

the pro of is complete

Lemma generalizes a formula of Fedosov in the following situation Let B H

b e the algebra of b ounded op erators on a xed separable Hilb ert space H and C H

p

B H b e the nonclosed ideal of psummable op erators on H

p

C H fA B H T r A A g

p

We will sometimes omit H and write simply C instead of C H Supp ose now that

p p

the algebra A consists of b ounded op erators that I C and that a is an element

of A whose pro jection u in AI is invertible Then a is a Fredholm op erator and for

a suitable choice of a lifting b of u the op erators ba and ab b ecome the

orthogonal pro jection onto the kernel of a and resp ectively the kernel of a Finally

if T r this shows that

Indu dim kera dim kera T r

and hence that T r Ind recovers the Fredholm index of a The Fredholm index

of a denoted inda is by denition the righthand side of the ab ove formula By

equation we see that we also recover a form of Fedosovs formula

k k

T r ab ba inda T r

if b is an inverse of a mo dulo C H and k p

p

The connecting or b oundary morphism in the algebraic K Theory exact se

quence is usually denoted by However in the present pap er this notation b e

comes unsuitable b ecause the notation is reserved for the b oundary morphism in

the p erio dic cyclic cohomology exact sequence Besides the notation Ind is supp osed

to suggest the name index morphism for the connecting morphism in the algebraic

K Theory exact sequence a name justied by the relation that exists b etween Ind

and the indices of Fredholm op erators as explained ab ove

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Higher Index Theorems

Higher traces and excision in cyclic cohomology The example of

A C M for M a compact smo oth manifold shows that in general few mor

alg

phisms K A C are given by pairings with traces This situation is corrected by

considering highertraces

Let A b e a unital algebra and

n

X

i

a a a a b a a

i i n n

i

n

ba a b a a a a a

n n n n

for a A The Hochschild homology groups of A denoted HH A are the homology

i

n

groups of the complex A AC b The cyclic homology groups

of a unital algebra A denoted HC A are the homology groups of the complex

n

C A b B where

M

nk

C A A AC

n

k

b is the Ho chschild homology b oundary map equation and B is dened by

n

X

k

B a a s t a a

n n

k

Here we have used the notation of that sa a a a and

n n

n

ta a a a a

n n n

More generally Ho chschild and cyclic homology groups can b e dened for mixed

complexes A mixed complex X b B is a graded vector space X en

n n

dowed with two dierentials b and B b X X and B X X satisfying

n n n n

the compatibility relation b B bB B b The cyclic complex denoted C X

asso ciated to a mixed complex X b B is the complex

M

C X X X X X

n n n n nk

k

with dierential b B The cyclic homology groups of the mixed complex X are the

homology groups of the cyclic complex of X

HC X H C X b B

n n

Cyclic cohomology is dened to b e the homology of the complex

C X HomC X C b B

dual to C X From the form of the cyclic complex it is clear that there exists a

morphism S C X C X We let

n n

C X lim C X

n nk

as k the inverse system b eing with resp ect to the p erio dicity op erator S

Then the periodic cyclic homology of X resp ectively the periodic cyclic cohomology

of X denoted HP X resp ectively HP X is the homology resp ectively the

cohomology of C X resp ectively of the complex lim C X

n nk

If A is a unital algebra we denote by X A the mixed complex obtained by

n

letting X A A AC with dierentials b and B given by and The

n

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Victor Nistor

various homologies of X A will not include X as part of notation For example the

p erio dic cyclic homology of X is denoted HP A

For a top ological algebra A we may also consider continuous versions of the

ab ove homologies by replacing the ordinary tensor pro duct with the pro jective tensor

pro duct We shall b e esp ecially interested in the continuous cyclic cohomology of A

denoted HP A An imp ortant example is A C M for a compact smo oth

cont

manifold M Then the Ho chschildKostantRosenberg map

n n

A a a a n a da da M

n n

to smo oth forms gives an isomorphism

M

cont ik

HP C M H M

i

DR

k

of continuous p erio dic cyclic homology with the de Rham cohomology of M

made Z p erio dic The normalization factor n is convenient b ecause it trans

forms B into the de Rham dierential d It is also the right normalization as far

DR

as Chern characters are involved and it is also compatible with pro ducts Theorem

From now on we shall use the de Rhams Theorem

i i

H M H M

DR

to identify de Rham cohomology and singular cohomology with complex coecients

of the compact manifold M

Sometimes we will use a version of continuous p erio dic cyclic cohomology for

algebras A that have a lo cally convex space structure but for which the multiplication

n

is only partially continuous In that case however the tensor pro ducts A come

with natural top ologies for which the dierentials b and B are continuous This is

the case for some of the group oid algebras considered in the last section The p erio dic

cyclic cohomology is then dened using continuous multilinear co chains

One of the original descriptions of cyclic cohomology was in terms of higher

traces A higher traceor cyclic co cycleis a continuous multilinear map

n n

A C satisfying b and a a a a a Thus

n n

cyclic co cycles are in particular Ho chschild co cycles The last prop erty the cyclic

invariance justies the name cyclic co cycles The other name higher traces is

justied since cyclic co cycles on A dene traces on the universal dierential graded

algebra of A

If I A is a twosided ideal we denote by C A I the kernel of C A C AI

For p ossibly nonunital algebras I we dene the cyclic homology of I using the

complex C I I The cyclic cohomology and the p erio dic versions of these groups are

dened analogously using C I I For top ological algebras we replace the algebraic

tensor pro duct by the pro jective tensor pro duct

An equivalent form of the excision theorem in p erio dic cyclic cohomology is the

following result

Theorem CuntzQuillen The inclusion C I I C A I induces an iso

morphism HP A I HP I of periodic cyclic cohomology groups

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Higher Index Theorems

This theorem is implicit in and follows directly from the pro of there of the

Excision Theorem by a sequence of commutative diagrams using the Five Lemma

each time

This alternative denition of excision sometimes leads to explicit formulae for

We b egin by observing that the short exact sequence of complexes C A I

C A C AI denes a long exact sequence

n n n n n

HP A I HP A HP AI HP A I HP A

in cyclic cohomology that maps naturally to the long exact sequence in p erio dic cyclic

cohomology

n n

Most imp ortant for us the b oundary map HP A I HP AI is

determined by a standard algebraic construction We now want to prove that this

b oundary morphism recovers a previous construction equation in the particular

case n As we have already observed a trace I C satises A I

and hence denes by restriction an element of HC A I The traces are the cycles of

the group HC I and thus we obtain a linear map HC I HC A I From the

denition of HP A I HP AI it follows that is the class of the co cycle

a b l a l b l a b which is cyclically invariant by construction Since

our previous notation for the class of was we have thus obtained the paradoxical

relation we hop e this will not cause any confusions

Below we shall also use the natural map transformation

n n nk

HC HP lim HC

k

Lemma The diagram

o

HC A I HC AI HC AI HC I

o

HP I HP A I HP AI HP AI

commutes Consequently if HC I is a trace on I and HP I is its class

in periodic cyclic homology then HP AI where HC AI is

given by the class of the cocycle dened in equation see also above

Proof The commutativity of the diagram follows from denitions If we start with a

trace HC I and follow counterclockwise through the diagram from the upp er

left corner to the lowerright corner we obtain if we follow clo ckwise we obtain

the description for indicated in the statement

An abstract higher index theorem We now generalize Lemma to

p erio dic cyclic cohomology Recall that the pairings and have b een generalized

to pairings

alg

ni

A HC A C i K

i

ni

Thus if b e a higher trace representing a class HC A then using the

alg

ab ove pairing denes morphisms K A C where i The explicit

i

n

n

e e e if i and e formulae for these morphisms are e

n

2

I am indebted to Joachim Cuntz for p ointing out this fact to me

Documenta Mathematica

Victor Nistor

n

is an idemp otent and u n u u u u if i and u is an

invertible element The constants in these pairings are meaningful and are chosen so

that these pairings are compatible with the p erio dicity op erator

Consider the standard orthonormal basis e of the space l N of square

n n

summable sequences of complex numbers the shift op erator S is dened by S e

n

e The adjoint S of S then acts by S e and S e e for n The

n n n

op erators S and S are related by S S and SS p where p is the orthogonal

pro jection onto the vector space generated by e

Let T b e the algebra generated by S and S and C w w b e the algebra of

P

N

k

a w a C g C Z Then Laurent series in the variable w C w w f

k k

nN

there exists an exact sequence

M C T C w w

called the Toeplitz extension which sends S to w and S to w

Let C h a b i b e the free noncommutative unital algebra generated by the symbols

a and b and J kerC h a b i C w w the kernel of the unital morphism that

sends a w and b w Then there exists a morphism C h a b i T uniquely

determined by a S and b S which denes by restriction a morphism

J M C and hence a commutative diagram

C h a b i

C w w

J

0

M C

C w w

T

Lemma Using the above notations we have that HC J is singly generated by

the trace T r

i

Proof We know that HP C w w C see Then Lemma Lemma

and the exact sequence in p erio dic cyclic cohomology prove the vanishing of the

reduced p erio dic cyclic cohomology groups

g

HC T kerHP T HP C

The algebra C h a b i is the tensor algebra of the vector space C a C b and hence

g

the groups HC T V also vanish It follows that the morphism induces

trivially an isomorphism in cyclic cohomology The comparison morphism b etween

the CuntzQuillen exact sequences asso ciated to the two extensions shows using

the Five Lemma that the induced morphisms HP M C HP J is

also an isomorphism This proves the result since the canonical trace T r generates

HP M C

We are now ready to state the main result of this section the compatibility of the

b oundary map in the p erio dic cyclic cohomology exact sequence with the index ie

connecting map in the algebraic K Theory exact sequence The following theorem

generalizes Theorem from

Theorem Let I A AI be an exact sequence of complex

alg alg

algebras and let Ind K AI K I and HP I HP AI be the

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Higher Index Theorems

connecting morphisms in algebraic K Theory and respectively in periodic cyclic co

alg

homology Then for any HP I and u K AI we have

Indu u

Proof We b egin by observing that if the class of can b e represented by a trace

that is if is the equivalence class of a trace in the group HP I then the b oundary

map in p erio dic cyclic cohomology is computed using the recip e we have indicated

Lemma and hence the result follows from Lemma In particular the theorem

is true for the exact sequence

J C h a b i C w w

b ecause all classes in HP J are dened by traces as shown in Lemma We will

now show that this particular case is enough to prove the general case by universal

ity

Let u b e an invertible element in M AI After replacing the algebras involved

n

by matrix algebras if necessary we may assume that n and hence that u is

an invertible element in AI This invertible element then gives rise to a morphism

C w w AI that sends w to u A choice of liftings a b A of u and

u denes a morphism C h a b i A uniquely determined by a a and

b b which restricts to a morphism J I In this way we obtain a

commutative diagram

C h a b i

C w w

J

0

AI

I A

of algebras and morphisms

We claim that the naturality of the index morphism in algebraic K Theory and

the naturality of the b oundary map in p erio dic cyclic cohomology when applied to

the ab ove exact sequence prove the theorem Indeed we have

alg alg

Ind Ind K C w w K I and

HP I HP C w w

As observed in the b eginning of the pro of the theorem is true for the co cycle

on J and hence Ind w w Finally from denition we have

that w u Combining these relations we obtain

Ind u Ind w Ind w Ind w

w w w u

The pro of is complete

The theorem we have just proved can b e extended to top ological algebras and

top ological K Theory If the top ological algebras considered satisfy Bott p erio dicity

then an analogous compatibility with the other connecting morphism can b e proved

and one gets a natural transformation from the sixterm exact sequence in top ological

K Theory to the sixterm exact sequence in p erio dic cyclic homology However a

factor of has to b e taken into account b ecause the ChernConnes character is not

Documenta Mathematica

Victor Nistor

directly compatible with p erio dicity but introduces a factor of See for

details

homology rather So far all our results have b een formulated in terms of cyclic co

than cyclic homology This is justied by the application in Section that will use this

form of the results This is not p ossible however for the following theorem which

states that the Chern character in p erio dic cyclic homology ie the ChernConnes

character is a natural transformation from the six term exact sequence in lower

algebraic K Theory to the exact sequence in cyclic homology

Theorem The diagram

Ind

alg alg alg alg alg alg

K A K AI K I K A K AI K I

HP I HP A HP AI HP I HP A HP AI

alg

in which the vertical arrows are induced by the Chern characters ch K HP

i

i

for i commutes

Proof Only the relation ch Ind ch needs to b e proved and this is dual to

Theorem

Products and the boundary map in periodic cyclic cohomology

Cyclic vector spaces are a generalization of simplicial vector spaces with which they

share many features most notably for us a similar b ehavior with resp ect to pro ducts

Cyclic vector spaces We b egin this section with a review of a few needed

facts ab out the cyclic category from and We will b e esp ecially interested

in the pro duct in bivariant cyclic cohomology More results can b e found in

Definition The cyclic category denoted is the category whose objects are

f ng where n and whose morphisms Hom are

n n m

the homotopy classes of increasing degree one continuous functions S S

satisfying Z Z

n m

A cyclic vector space is a contravariant functor from to the category of complex

vector spaces Explicitly a cyclic vector space X is a graded vector space X

i i

X X for X X s X with structural morphisms d

n n n n n n

n n

i i

i n and t X X such that X d s is a simplicial vector space

n n n n

n n

Chapter VI I Ix and t denes an action of the cyclic group Z satisfying

n n

n n i i i i

d t d and s t t s d t t d and s t t s for

n n n n n n

n n n n n n n n n

i n Cyclic vector spaces form a category

The cyclic vector space asso ciated to a unital lo cally convex complex algebra A

n

is A A with the structural morphisms

n

i

s a a a a a a

n i i n

n

i

a a a a a a for i n and d

n i i n

n

n

d a a a a a a a

n n i i n

n

t a a a a a a

n n n n

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Higher Index Theorems

If X X and Y Y are cyclic vector spaces then we can dene

n n n n

on X Y the structure of a cyclic space with structural morphisms given by

n n n

i i

the diagonal action of the corresp onding structural morphisms s d and t of

n

n n

X and Y The resulting cyclic vector space will b e denoted X Y and called the

external product of X and Y In particular we obtain that AB A B for all

unital algebras A and B and that X C X for all cyclic vector spaces X There

is an obvious variant of these constructions for lo cally convex algebras obtained by

using the complete pro jective tensor pro duct

The cyclic cohomology groups of an algebra A can b e recovered as E xtgroups

For us the most convenient denition of E xt is using exact sequences or resolutions

n

Consider the set E M of resolutions of length n of X by cyclic vector

k

k

spaces such that M Y Thus we consider exact sequences

n

E Y M M M X

n n

of cyclic vector spaces For two such resolutions E and E we write E E whenever

there exists a morphism of complexes E E that induces the identity on X and

n

Y Then Ext X Y is by denition the set of equivalence classes of resolutions

n

E M with resp ect to the equivalence relation generated by The set

k

k

n n

X Y of X Y has a natural group structure The equivalence class in Ext Ext

n

a resolution E M is denoted E This denition of E xt coincides with the

k

k

usual oneusing resolutions by pro jective mo dulesb ecause cyclic vector spaces form

an Ab elian category with enough pro jectives

Given a cyclic vector space X X dene b b X X by

n n n n

P

n

j n n

t b e the extra degeneracy of b d b b d Let s s

n j n

n

j

P

n

j

n nj

X which satises s b b s Also let t N t

n

n

j

and B s N Then X b B is a mixed complex and hence HC X the cyclic ho

mology of X is the homology of X b B by denition Cyclic cohomology

k nk

is obtained by dualization as b efore

The E xtgroups recover the cyclic cohomology of an algebra A via a natural

isomorphism

n n

HC A Ext A C

This isomorphism allows us to use the theory of derived functors to study cyclic

cohomology esp ecially pro ducts

The Yoneda product

n m nm

X Y Ext Y Z Ext X Z Ext

n m

is dened by splicing If E M is a resolution of X and E M a

k

k k k

resolution of Y such that M Y and M Z then E E is represented by

n

m

Z M M

M

M

M X

n

m m

x

x

x

x

x

x

x

x

x

Y

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Victor Nistor

The resulting pro duct generalizes the comp osition of functions Using the same no

tation the external product E E is the resolution

nm

X

A

E E M M

j

k

k j l

l

Passing to equivalence classes we obtain a pro duct

m n mn

Ext X Y Ext X Y Ext X X Y Y

If f X X is a morphism of cyclic vector spaces then we shall sometimes denote

n

E f f E for E Ext X C

The Yoneda pro duct and the external pro duct are b oth asso ciative

and are related by the following identities Lemma

n m

Lemma Let x Ext X Y y Ext X Y and be the natural transfor

mn mn

mation Ext X X Y Y Ext X X Y Y that interchanges the

factors Then

mn

id y x id x y id y x id

X Y Y X

1 1

id y z id y id z

X X X

mn

x y y x and x id x id x

C C

We now turn to the denition of the p erio dicity op erator A choice of a generator

of the group Ext C C denes a p erio dicity op erator

n n

X Y X Y x Sx x Ext Ext

In the following we shall choose the standard generator that is dened over Z

and then the ab ove denition extends the p erio dicity op erator in cyclic cohomology

This and other prop erties of the p erio dicity op erator are summarized in the following

Corollary Corollary

n m

Corollary a Let x Ext X Y and y Ext X Y Then Sx y

Sx y x Sy

n

b If x Ext C X then Sx x

m

Y C then Sy y c If y Ext

d For any extension x we have Sx x

Using the p erio dicity op erator we extend the denition of p erio dic cyclic coho

mology groups from algebras to cyclic vector spaces by

i in

X C HP X lim Ext

i i

the inductive limit b eing with resp ect to S clearly HP A HP A Then Corol

lary a shows that the external pro duct is compatible with the p erio dicity

morphism and hence denes an external pro duct

i j ij

HP A HP B HP A B

on p erio dic cyclic cohomology

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Higher Index Theorems

Extensions of algebras and products Cyclic vector spaces will b e used

to study exact sequences of algebras Let I A b e a twosided ideal of a complex

unital algebra A recall that in this pap er all algebras are complex algebras Denote

by A I the kernel of the map A AI and by A I Ext AI A I

the equivalence class of the exact sequence

A I A AI

of cyclic vector spaces

i i

Let HC A I Ext A I C then the long exact sequence of E xtgroups

asso ciated to the short exact sequence reads

i i i i i

HC AI HC A HC A I HC AI HC A

By standard homological algebra the b oundary map of this long exact sequence is

given by the pro duct

i i

HC A I A I HC AI

For an arbitrary algebra I p ossibly without unit we let I I I Then

n n

the isomorphism b ecomes HC I Ext I C and the excision theorem in

p erio dic cyclic cohomology for cyclic vector spaces takes the following form

Theorem CuntzQuillen The inclusion j I A I of cyclic vector

I A

spaces induces an isomorphism HP A I HP I

It follows that every element HP I is of the form j and that

I A

the b oundary morphism HP I HP AI satises

AI

j A I

AI I A

i i

A I C Formula then uniquely determines for all HC A I Ext

I A

We shall need in what follows a few prop erties of the isomorphisms j Let B

I A

b e an arbitrary unital algebra and I an arbitrary p ossibly nonunital algebra The

inclusion I B I B of unital algebras denes a commutative diagram

I B

I B

C

I B

I B

I B

B

with exact lines The morphism dened for p ossibly nonunital algebras I will

I B

replace the identication A B A B valid only for unital algebras A

+

Using the notation of Theorem we see that j and hence

I B

I B I B

by the same theorem it follows that induces an isomorphism

I B

HP I B HP I B

I B

Using this isomorphism we extend the external pro duct

HP I HP B HP I B

to a p ossibly nonunital algebra I by

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Victor Nistor

j m

i j in

HP I HP B lim Ext I C lim Ext B C

ij l

ij

lim Ext I B C HP I B HP I B

This extension of the external tensor pro duct to p ossibly nonunital algebras will b e

used to study the tensor pro duct by B of an exact sequence I A AI

of algebras

Tensoring by B is an exact functor and hence we obtain an exact sequence

I B A B AI B

Lemma Using the notation introduced above we have the relation

A B I B A I id Ext AI B A B I B

B

Proof We need only observe that the relation A B A B and the exactness

of the functor X X B imply that A I B A B I B

Properties of the boundary map The following theorem is a key to ol in

establishing further prop erties of the b oundary map in p erio dic cyclic homology

Theorem Let A and B be complex unital algebras and I A be a twosided

ideal Then the boundary maps

HP I HP AI

I A

and

HP I B HP AI B

I B AB

satisfy

I B AB I A

for al l HP I and HP B

k k n

Proof The groups HP I is the inductive limit of the groups Ext I C so

will b e the image of an element in one of these Extgroups By abuse of notation we

k

I C shall still denote that element by and thus we may assume that Ext

j

B C Moreover by for some large k Similarly we may assume that Ext

i

Theorem we may assume that j for some Ext A I C

I A

We then have

j

I A I A

A I by equation

id A I id by Lemma

B

C

id id A I id by Lemma

B B

C

A B I B by Lemma and Corollary

j by equation

AB I B I B AB

By denition the morphism j introduced in Theorem satises

I A

j j id

I B AB I A B I B

Equation then gives

I A I B AB I B

ij

AI B C This completes the pro of in view of the denition of the in Ext

external pro duct in the nonunital case

I B

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Higher Index Theorems

We now consider crossed pro ducts Let A b e a unital algebra and a discrete

group acting on A by A a a A Then the algebraic crossed

pro duct A o consists of nite linear combinations of elements of the form a with

the pro duct rule a b a b Let a a which denes a morphism

A o A o C Using we dene on HP A o a HP C mo dule

structure by



HP A o HP C HP A o C HP A o

A invariant twosided ideal I A gives rise to a crossed pro duct exact se

quence

I o A o AI o

of algebras The following theorem describ es the b ehavior of the b oundary map of this

exact sequence with resp ect to the HP C mo dule structure on the corresp onding

p erio dic cyclic cohomology groups

Theorem Let be a discrete group acting on the unital algebra A and let I

be a invariant ideal Then the boundary map

HP I o HP AI o

I oAo

is HP C li near

Proof The pro of is based on the previous theorem Theorem and the naturality

of the b oundary morphism in p erio dic cyclic cohomology

From the commutative diagram

AI o

I o A o

I o C A o C AI o C

we obtain that we have omitted the subscripts Then for each x

HP C and HP I o we have x x and hence using also

Theorem we obtain

x x x x x

The pro of is complete

For the rest of this subsection it will b e convenient to work with continuous

p erio dic cyclic homology Recall that this means that all algebras have compatible

lo cally convex top ologies that we use complete pro jective tensor pro ducts and that

the pro jections A AI have continuous linear splittings which implies that A

AI I as lo cally convex vector spaces Moreover since the excision theorem

is known only for malgebras we shall also assume that our algebras are m

algebras that is that their top ology is generated by a family of submultiplicative

seminorms Slightly weaker results hold for general top ological algebras and discrete

p erio dic cyclic cohomology

There is an analog of Theorem for actions of compact Lie groups If G is

a compact Lie group acting smo othly on a complete lo cally convex algebra A by

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Victor Nistor

G A A then the smooth crossed product algebra is A o G C G A with

the convolution pro duct

Z

f h f h g dh f f g

h

G

the integration b eing with resp ect to the normalized Haar measure on G As b efore

if I A is a complemented Ginvariant ideal of A we get an exact sequence of smo oth

crossed pro ducts

I o G A o G AI o G

Still assuming that G is compact let R G b e the representation ring of G Then

the group HP A o G has a natural R Gmo dule structure dened as follows see

also The diagonal inclusion A o G M A o G induces an isomorphism in

n

cyclic cohomology with inverse induced by the morphism

T r M A o G A o G

n

n

of cyclic ob jects Then for any representation G M C we obtain a unit

n

preserving morphism

A o G M A o G

n

dened by f g f g g C G M A for any f C G A Finally if

n

R G we dene the multiplication by to b e the morphism

A o G A o G HP T r HP

cont cont

Thus x x T r

Theorem Let A be a locally convex malgebra and I A a complemented

Ginvariant twosided ideal Then the boundary morphism associated to the exact

sequence

AI o G I o G HP HP

I oGAoG

cont cont

is R Glinear

Proof First we observe that the morphism T r M A A is functorial and

n

consequently that it gives a commutative diagram

M A o G M AI o G

X

n n

A o G I o G A o G AI o G

where X M A o G M I o G and whose vertical arrows are given by T r

n n

Regarding this commutative diagram as a morphism of extensions we obtain

that

T r M A o G M I o G A o G I o G T r

n n

Then using a similar reasoning we also obtain that

M A o G M I o G A o G I o G

n n

Now let HP I o G which we may assume by Theorem to b e an

cont

i

A o G I o G C Using element of the form j for some Ext

I oGAoG

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Higher Index Theorems

equations and and that the inclusion j j by the naturality of

I oGAoG

is R Glinear we nally get

j j

T r j T r A o G I o G

M A o G M I o G T r T r

n n

The pro of is now complete

In the same spirit and in the same framework as in Theorem we now con

sider the action of Lie algebra cohomology on the p erio dic cyclic cohomology exact

sequence

Assume that G is compact and connected and denote by g its Lie algebra and

by H g the Lie algebra homology of g Since G is compact and connected we

can identify H g with the biinvariant currents on G Let G G G b e

the multiplication Then one can alternatively dene the pro duct on H g as the

comp osition

H g H g HP C G HP C G

cont cont



HP C G G HP C G H g

cont cont

We now recall the denition of the pro duct H g HP A HP A

cont cont

Denote by A C G A the morphism ag a where this time

g

b

C G A is endowed with the pointwise pro duct Then x HP C GA

cont

b

is a continuous co cycle on C G A C GA and we dene x x

The asso ciativity of the pro duct shows that HP A b ecomes a H gmo dule

cont

with resp ect to this action

Theorem Suppose that a compact connected Lie group G acts smoothly on a

complete locally convex algebra A and that I is a closed invariant twosided ideal of

A complemented as a topological vector space Then

x x

for any x H g and HP I

cont

Proof The pro of is similar to the pro of of Theorem using the morphism of exact

sequences

A I AI

A

C G A C G AI

X

where X C G A C G I

Relation to the bivariant ChernConnes character A dierent type

of prop erty of the b oundary morphism in p erio dic cyclic cohomology is its compat

ibility eectively an identication with the bivariant ChernConnes character

Before we can state this result need to recall a few constructions from

Let A and B b e unital lo cally convex algebras and assume that a continuous

linear map

b

A B H B

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Victor Nistor

is given such that the co cycle a a a a a a factors as a comp o

b b b

sition AA C H B B H B of continuous maps Recall that C H is the

p p

b

ideal of psummable op erators and that is the complete pro jective tensor pro duct

Using the co cycle we dene on E A C H B an asso ciative pro duct by the

p

formula

a x a x a a a x x a a a

Then the algebra E ts into the exact sequence

b

C H B E A

p

An exact sequence

b

E C H B E A

p

that is isomorphic to an exact sequence of the form will b e called an admissible

exact sequence If E is an admissible exact sequence and n p then

Theorem asso ciates to E an element

n

n

ch E Ext A B

cont

which for B C recovers Connes Chern character in K homology The sub

script cont stresses that we are considering the version of the Yoneda Ext dened

for lo cally convex cyclic ob jects

P

T e e for any Let T r C H C b e the ordinary trace ie T r T

n n

n

orthonormal basis e of the Hilb ert space H Using the trace T r we dene

n n

n

T r HC C H for n p to b e the class of the cyclic co cycle

n p

n

n

T r a a a T r a a a

n n n

n

n n

The normalization factor was chosen such that T r S T r S T r on C H We

n

have the following compatibility b etween the bivariant ChernConnes character and

the CuntzQuillen b oundary morphism

HP b e the natural transformation that for HP Let HP

disc

disc cont

gets continuity from continuous to ordinary or discrete p erio dic cyclic cohomology

We include the subscript disc only when we need to stress that discrete homology

is used By contrast the subscript cont will always b e included

b

Theorem Let C H B E A be an admissible exact se

p

n

n

A B be its bivariant ChernConnes character quence and ch E Ext

cont

equation If T r is as in equation and n p then

n

q

n

T r ch E HP A

n disc disc

q

B for each HP

cont

This theorem provides usat least in principlewith formul to compute the

b oundary morphism in p erio dic cyclic cohomology see and Prop osition

Before pro ceeding with the pro of we recall a construction implicit in The

j

algebra RA A is the tensor algebra of A and r A is the kernel of the map

j

RA A Because A has a unit we have a canonical isomorphism A C A

We do not consider any top ology on RA but in addition to RA the cyclic ob ject

asso ciated to RA we consider a completion of it in a natural top ology with resp ect

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Higher Index Theorems

to which all structural maps are continuous The new completed cyclic ob ject is

j k

denoted RA A Then and is obtained as follows Let R A

k

cont

j

n

RA lim R A

k

contn

k

with the inductive limit top ology

Proof We b egin with a series of reductions that reduce the pro of of the Theorem to

the pro of of

Since E is an admissible extension there exists by denition a continuous linear

section s A E of the pro jection E A ie s id Then s denes a

commutative diagram

r A RA

A

A

b

C B

E A

p

where the right hand vertical map is the pro jection A C A A

q

By increasing q if necessary we may assume that the co cycle HP B

cont

q

comes from a co cycle also denoted in HC B Let

cont

q n

q n

b b

T r HC C B HC C B

n disc p p

disc

b e as in the statement of the theorem

We claim that it is enough to show that

n

j ch E

A disc

where j A A is the inclusion

A

Indeed assuming and using the ab ove commutative diagram and the natu

rality of the b oundary morphism we obtain

n

ch E j j j

disc A A A A

A

as stated in theorem b ecause j id

A A

Let j r A R A r A b e the morphism inclusion considered in The

r ARA

n n

R A r A C satisfy R A r A Ext orem Also let HC

disc

n n

r A C r A Ext j HC

r ARA

disc

In words restricts to on r A Then using equation we have

R A r A

The rest of the pro of consists of showing that the construction of the o dd bivariant

ChernConnes character provides us with satisfying equations and

n

R A r A j ch E

A disc

This is enough to complete the pro of b ecause equations and imply and

as we have already shown equation implies equation So to complete the

pro of we now pro ceed to construct satisfying and

Recall from that the ideal r A denes a natural increasing ltration of

by cyclic vector spaces RA

cont

F RA F RA RA

cont cont cont

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Victor Nistor

such that r A F RA R A r A If r A is the k th comp onent of the

cont

k

cyclic vector space r A and if in general the lower index stands for the Z grading

of a cyclic vector space then we have the more precise relation

r A F RA for k n

n k

k cont

It follows that the morphism of cyclic vector spaces

T r F F RA B

n n n

cont

for k n p Fix then dened in page satises T r on r A

n

k

q n

b

k q n and conclude that T r HC C B satises

n disc p

n

T r S

n disc n

n

on r A F RA b ecause T r restricts to S T r on C H Now recall the

n n

cont

k

crucial fact that there exists an extension

n

n

C RA Ext F RA F RA

n

cont cont cont

n n

that has the prop erty that C RA i S if i F RA F RA is

n

cont cont

the inclusion see Corollary Using this extension we nally dene

n

n

C RA Ext F RA C

n disc

cont

Since has order k q n n n we obtain from the equations and

F RA that satises ie that it restricts to on r A

n

cont

k

as desired

The last thing that needs to b e checked for the pro of to b e complete is that

satises equation By denition the o dd bivariant ChernConnes character

page is

n n

ch E ch RA j

n A

n n

n

where ch RA C RA C RA q RA and j A A is the

A

inclusion see page denition page and the discussion on page

Moreover q RA is nothing but a continuous version of R A r A that is

q RA R A r A

disc

and hence

n

n

R A r A j C RA q RA j ch E

A n A disc disc

Since satises equation and which imply equation the pro of is

complete

For any lo cally convex algebra B and HP B the discrete p erio dic cyclic

cohomology of B we say that is a continuous class if it can b e represented by

a continuous co cycle on B Put dierently this means that for some

disc

HP B Since the bivariant ChernConnes character can at least in principle

cont

b e expressed by an explicit formula it preserves continuity This gives the following

corollary

Corollary The periodic cyclic cohomology boundary map associated to

an admissible extension maps a class of the form T r for a continuous class

n

to a continuous class

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Higher Index Theorems

It is likely that recent results of Cuntz see will give the ab ove result for

all continuous classes in HP C B not just the ones of the form T r

p n

Using the ab ove corollary we obtain the compatibility b etween the bivariant

ChernConnes character and the index morphism in full generality This result had

b een known b efore only in particular cases

b

Theorem Let C H B E A be an admissible exact sequence

p

n n

and ch E Ext A B be its bivariant ChernConnes character equation

alg alg

b

If T r is as in equation and Ind K A K C H B is the

n p

connecting morphism in algebraic K Theory then for any HP B and u

cont

alg

K A we have

n

h T r Indu i h ch E u i

n

The index theorem for coverings

Using the metho ds we have developed we now give a new pro of of ConnesMoscovicis

f

index theorem for coverings To a covering M M with covering group Connes

and Moscovici asso ciated an extension

C C E C S M n dim M

n CM

the ConnesMoscovici exact sequence dened using invariant pseudo dierential

f

op erators on M see equation If H HP C C is an even

n

cont

cyclic co cycle then the ConnesMoscovici index theorem computes the morphisms

alg

Ind K C S M C

where Ind is the index morphism asso ciated to the ConnesMoscovici exact sequence

Our metho d of pro of then is to use the compatibility b etween the connecting mor

phisms in algebraic K Theory and the connecting morphism in p erio dic cyclic

cohomology Theorem to reduce the pro of to the computation of This com

putation is now a problem to which the prop erties of established in Section can

b e applied

We rst show how to obtain the ConnesMoscovici exact sequence from another

exact sequence the AtiyahSinger exact sequence by a purely algebraic construc

tion Then using the naturality of and Theorem we determine the connecting

morphism of the ConnesMoscovici exact sequence in terms of the connecting

CM

morphism of the AtiyahSinger exact sequence For the AtiyahSinger exact

AS

sequence the pro cedure can b e reversed and we now use the AtiyahSinger Index

Theorem and Theorem to compute

AS

A comment ab out the interplay of continuous and discrete p erio dic cyclic co

homology in the pro of b elow is in order We have to use continuous p erio dic cyclic

cohomology whenever we want explicit computations with the p erio dic cyclic coho

mology of group oid algebras b ecause only the continuous version of p erio dic cyclic

cohomology is known for group oid algebras asso ciated to etale group oids On the

other hand in order to b e able to use Theorem we have to consider ordinary or

discrete p erio dic cyclic cohomology as well This is not an essential diculty b ecause

using Corollary we know that the index classes are represented by continuous

co cycles

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Victor Nistor

Groupoids and the cyclic cohomology of their algebras Our com

putations are based on group oids so we rst recall a few facts ab out group oids

A groupoid is a small category in which every morphism is invertible Think of

a group oid as a set of p oints joined arrows the following examples should clarify this

abstract denition of group oids A smooth etale group oid is a group oid whose set of

morphisms also called arrows and whose set of ob jects also called units are smo oth

manifolds such that the domain and range maps are etale ie lo cal dieomorphisms

To any smo oth etale group oid G assumed Hausdor for simplicity there is asso ciated

the algebra C G of compactly supp orted functions on the set of arrows of G and

c

endowed with the convolution pro duct

X

f f g f f g

r r g

Here r is the range map and r r g is the condition that and g b e comp os

able Whenever dealing with C G we will use continuous cyclic cohomology as

c

in See for more details on etale group oids and for the general theory of

lo cally compact group oids

Etale group oids conveniently accommo date in the same framework smo oth man

ifolds and discrete groups two extreme examples in the following sense the smo oth

etale group oid asso ciated to a smo oth manifold M has only identity morphisms

whereas the smo oth etale group oid asso ciated to the discrete group has only one

ob ject the identity of The algebras C G asso ciated to these group oids are

c

C M and resp ectively the group algebra C Here are other examples used in

c

the pap er

The group oid R asso ciated to an equivalence relation on a discrete set I has I

I

as the set of units and exactly one arrow for any ordered pair of equivalent ob jects

If I is a nite set with k elements and all ob jects of I are equivalent ie if R is

I

the total equivalence relation on I then C R M C and its classifying space

I k

c

in the sense of Grothendieck the space B R is contractable

I

Another example the gluing groupoid G mimics the denition a manifold M

U

in terms of gluing co ordinate charts The group oid G is dened using an op en

U

cover U U of M ie M U Then G has units G U fg

I I U I

U

and arrows

fx I x U U g G

U

If R is the total equivalence relation on I then there is an injective morphism

I

l G M R of etale group oids

U I

Let f G G b e an etale morphism of groupoids that is a morphism of

etale group oids that is a lo cal dieomorphism Then the map f denes a con

tinuous map B f B G B G of classifying spaces and a group morphism

f HP C G HP C G If f is injective when restricted to

T r

cont c cont c

units then there exists an algebra morphism f C G C G such that

c c

f f

TR

The following theorem a generalization of Theorem is based on the

fact that all isomorphisms in the pro of of that theorem are functorial with resp ect to

etale morphisms It is the reason why we use continuous p erio dic cyclic cohomology

when working with group oid algebras Note that the cyclic ob ject asso ciated to

G for G an etale group oid is an inductive limit of lo cally convex nuclear spaces C

c

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Higher Index Theorems

Theorem If G is a Hausdor etale groupoid of dimension n and if o is

the complexied orientation sheaf of B G then there exists a natural embedding

n

H B G o HP C G Here natural means that if f G G

cont c

is an etale morphism of groupoids then the diagram

n

HP C G

H B G o

cont c



f

B f

T r

n

HP C G

H B G o

cont c

whose horizontal lines are the morphisms commutes

For discrete groups Theorem recovers the embedding

H H B C HP C

cont

of

For smo oth manifolds the embedding of Theorem is just the Poincare

dualityan isomorphism This isomorphism has a very concrete form Indeed let

ni

H M o b e an element of the singular cohomology of M with co ecients in

i

M b e an element of the singular cohomology of M the orientation sheaf let H

c

with compact supp orts all cohomology groups have complex co ecients and let

cont ik ik

HP C M H M H M

k k

i c c

cD R

b e the canonical isomorphism induced by the Ho chschildKostantRosenberg map

equation Then the isomorphism is determined by

h i h M i C

cont

M C and the M HP C where the rst pairing is the map HP C

c c cont

second pairing is the evaluation on the fundamental class

Typically we shall use these results for the manifold S M for which there is

an isomorphism H S M HP C S M b ecause S M is oriented The

cont

orientation of S M is the one induced from that of T M as in More precisely

B M the disk bundle of M is given the orientation in which the the horizontal part

is real and the vertical part is imaginary and S M is oriented as the b oundary of

an oriented manifold The shift in the Z degree is due to the fact that S M is o dd

dimensional

Morita invariance and coverings Let M b e a smo oth compact manifold

f f

and q M M b e a covering with Galois group said dierently M is a principal

bundle over M We x a nite cover U U of M by trivializing op en

I

U U and M U The transition functions b etween two sets ie q

trivializing isomorphisms on their common domain the op en set U U denes a

f

co cycle that completely determines the covering q M M

f

In what follows we shall need to lift the covering q M M to a covering

f

q S M S M using the canonical pro jection p S M M All constructions

then lift from M to S M canonically In particular V p U is a nite

covering of S M with trivializing op en sets and the asso ciated co cycle is still

f

Moreover if f M B classies the covering q M M then f f p

f

M S M classies the covering S

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Victor Nistor

Supp ose that the trivializing cover V V of S M consists of k op en sets

I

P

and let b e a partition of unity sub ordinated to V The co cycle identity

ensures then that the matrix

p M C M C

I k

is an idemp otent called the Mishchenko idempotent a dierent choice of a trivializing

cover and of a partition of unity gives an equivalent idemp otent

Using the Mishchenko idemp otent p we now dene the morphism

C S M M C S M C

k

by a ap for a C S M explicitly

ax axpx ax x x

Because the morphism is used to dene the ConnesMoscovici extension equation

b elow we need to identify the induced morphism

HP C S M C HP C S M

cont cont

The identication of Prop osition is based on writing as a comp osition of

three simpler morphisms morphisms that will play an auxiliary role The next few

paragraphs b efore Prop osition will deal with the denition and prop erties of these

morphisms

We dene the rst auxiliary morphism g to b e induced by an etale morphism

of group oids Let G b e the gluing group oid asso ciated to the cover V V

V I

of S M Using the co cycle asso ciated to V that identies the covering

I

f

S M S M we dene the etale morphism of group oids g by

g

G x x G

V V

which induces a morphism g C G C G C and a continuous map

V V

c c

B g B G BG B G B

V V V

The pro jection t G S M is an etale morphism of group oids that induces a

V

homotopy equivalence B G S M and hence also an isomorphism

V

t HP C S M HP C G

T r V

cont cont c

By denition t T r l where l G S M R is the natural inclu

T r V I

sion considered also b efore and T r is the generic notation for the isomorphisms

T r HP M A HP A induced by the trace In particular l is also an

n

isomorphism

Using the homotopy equivalence B t of B G and S M we obtain a continuous

V

map

h S M S M B

uniquely determined by the condition h B t B t id B g

Lemma The map h dened above coincides up to homotopy with the product

f



function id f where f S M B classies S M S M

S M

Proof Denote by p and p the pro jections of S M B onto comp onents The map

p h is easily seen to b e the identity so h id h where h S M B is

S M

induced by the nonetalemorphism of top ological group oids G x

V

In order to show that h coincides with f up to homotopy it is enough to show

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that the principal bundle ie covering that h pulls back from B to S M is

f f

isomorphic to the covering S M M

Let G b e the gluing group oid asso ciated to the cover U U of M It is

U I

seen from the denition that G factors as G G where the function

V V U

G acts as m Thus we may replace S M by M everywhere in the

V

pro of

f

Since the the covering M M is determined by its restriction to lo ops we

may assume that M is the circle S Cover M S by two contractable intervals

I I which intersect in two small disjoint neighborho o ds of and I I

z z z z where z S and jz j is very small We may also assume that

the transition co cycle is the identity on z z and on z z we have

replaced constant co cycles with lo cally constant co cycles The map h maps

each of the units of G and each of the cells corresp onding to the right hand interval

U

z z to the only cell of B the cell corresp onding to the identity e Recall

that the classifying space of a top ological group oid is the geometrical realization of the

simplicial space of comp osable arrows and that that there is a cell for each unit

a cell for each nonidentity arrow a cell for each pair of nonidentity comp osable

arrows and so on The other cells ie corresp onding to the arrows leaving from

a p oint on the left hand side interval will map to the cell corresp onding This

shows that on homotopy groups the induced map Z S B sends

the generator to This completes the pro of of the lemma

We need to introduce one more auxiliary morphism b efore we can determine

P

sub ordinated to V V we dene Using the partition of unity

I

G by C S M C

V

c

f x f x x x

which turns out to b e a morphism of algebras Because the comp osition

l

C S M C G C S M R M C S M

V I k

c c

is unitarily equivalent to the upp erleft corner embedding we obtain that the mor

phism HP C G HP C S M is the inverse of t

V T r

cont c cont

We are now ready to determine the morphism

HP C S M C HP C S M

cont cont

In order to simplify notation in the statement of the following result we shall identify

HP M C S M C with HP C S M C and we shall do the

k

cont cont

same in the pro of

Proposition The composition



H S M B C HP C S M C

cont

HP C S M H S M C

cont

is id f

Proof Consider as b efore the morphism l G S M R of group oids which

V I

denes an injective morphism of algebras l C G C S M R

V I

M C S M and hence also a morphism

k

l id l id C G C S M R M C S M C

V I k

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Victor Nistor

Then we can write

l id g

where g G G is as dened b efore g x x

V V

Because t we have that B t by Theorem

T r

Also by Theorem we have g B g and l id B l id

This gives then

B g B l id B t B g B l id h

Since Lemma states that h id f up to homotopy the pro of is complete

The AtiyahSinger exact sequence Let M b e a smo oth compact man

k

ifold without b oundary We shall denote by M the space of classical order

at most k pseudo dierential op erators on M Fix a smo oth nowhere vanishing den

sity on M Then M acts on L M by b ounded op erators and if an op erator

T M is compact then it is of order More precisely it is known that order

pseudo dierential op erators satisfy M C C L M for any p n

p p

Recall that C H is the ideal of psummable op erators on H equation

p

It will b e convenient to include all n summable op erators in our calculus

so we let E M C and obtain in this way an extension of algebras

AS n

0

C S M C E

n AS

called the AtiyahSinger exact sequence The b oundary morphisms in p erio dic cyclic

cohomology asso ciated to the AtiyahSinger exact sequence denes a morphism

HP C HP C S M

AS n

Let T r HP C b e as in ie T r a a C T r a a for

n n n n n

cont

some constant C and denote

J M T r HP C S M HP C S M

AS n

cont

which is justied by Corollary

We shall determine J M using Theorem In order to do this we need to

make explicit the relation b etween ch the Chern character in cyclic homology and

C h the classical Chern character as dened for example in Let E M b e a

N

smo oth complex vector bundle embedded in a trivial bundle E M C and

let e M C M b e the orthogonal pro jection on E If we endow E with the

N

N

connection ed e acting on E C M then the curvature of this con

DR

nection turns out to b e ed e The classical Chern character C hE is then

DR

in the even de Rham cohomology the cohomology class of the form T r exp

of M Comparing this denition with the denition of the Chern character in cyclic

cohomology via the Ho chschildKostantRosenberg map we see that the two of them

k

are equalup to a renormalization with a factor of If H M H M

k

k

is a cohomology class we denote by its comp onent in H M Explicitly let

k

ik cont

S M b e the canonical isomorphism induced S M H HP C

k Z

c i

by the Ho chschildKostantRosenberg map equation then

X

mi

m

ch C h H M

mi

k Z

alg

C M Note the i for i f g and K

i

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even

Proposition Let T M H S M be the Todd class of the complexi

even

cation of T M lifted to S M and H S M HP C S M be the

cont

isomorphism of Theorem Then

X

n nk

J M T M HP C S M

k

cont

k

Proof We need to verify the equality of two classes in HP C S M It is

cont

hence enough to check that their pairings with chu are equal for any u

alg

K C S M b ecause of the classical result that the Chern character

alg

cont

ch K C S M HP C S M

is onto

If Ind is the index morphism of the AtiyahSinger exact sequence then the Atiyah

Singer index formula states the equality

n

Indu h C hu T M i

Using equation and Theorem see also the discussion following that theorem

we obtain that Indu h chu J M i Equations and then complete the

pro of

The ConnesMoscovici exact sequence and proof of the theorem

We now extend the constructions leading to the AtiyahSinger exact sequence equa

tion to covering spaces

Let M b e a smo oth compact manifold and let E M E C which ts

k

into the exact sequence

0

M C S M C M C C E

k k n

f

Let M M b e a covering of M with Galois group Using the Mishchenko

idemp otent p asso ciated to this covering and the injective morphism

C S M pM C S M C p

k

equation we dene the ConnesMoscovici algebra E as the b ered pro duct

CM

E fT a pE p C S M T ag

CM

By denition the algebra E ts into the exact sequence

CM

p M C C p E C S M

k n CM

We now take a closer lo ok at the algebra E and the exact sequence it denes

CM

k k

f

Observe rst that p acts on L M l and that pL M l L M

via a invariant isometry Since E can b e regarded as an algebra of op erators on

k

L M l that commute with the right action of we obtain that E

CM

can also b e interpreted as an algebra of op erators commuting with the action of on

f

L M Using also Lemma page this recovers the usual description of

f

E that uses prop erly supp orted invariant pseudo dierential op erators on M

CM

Also observe that M is sup eruous in M C b ecause M C C

k k n k n n

actually even p is sup eruous in p M C C p b ecause

k n

p M C C p C C

k n n

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Victor Nistor

by an isomorphism that is uniquely determined up to an inner automorphism Thus

the ConnesMoscovici extension b ecomes

C C E C S M

n CM

up to an inner automorphism

We now pro ceed as for the AtiyahSinger exact sequence The b oundary mor

phisms in p erio dic cyclic cohomology asso ciated to the ConnesMoscovici extensions

denes a map

HP C C HP C S M

CM n

and the ConnesMoscovici Index Theorem amounts to the identication of the classes

T r HP C S M HP C S M

CM n

cont

for co cycles coming from the cohomology of

In order to determine T r we need the following theorem

CM n

Theorem Let G and G be smooth etale groupoids Then the diagram

n m

C G C G HP HP

H B G o H B G o

cont c cont c

nm

HP C G G

H BG G o

cont c

is commutative Here the left product is the external product in cohomology and o

o and o are the orientation sheaves

Proof The pro of is a long but straightforward verication that the sequence of

isomorphisms in is compatible with pro ducts

Using Prop osition c page which states that the pro ducts are

compatible with the tensor pro ducts of mixed complexes we replace everywhere cyclic

vector spaces by mixed complexes Then we go through the sp ecic steps of the pro of

as in This amounts to verify the following facts

i The Ho chschildKostantRosenberg map equation transforms the

dierential B B into the de Rham dierential of the pro duct

ii By the EilenbergZilb er Theorem the augmentation map Prop osi

tion and the isomorphism it induces are compatible with pro ducts

iii The chain map f in the Mo ore isomorphism see Theorems and

page is compatible with pro ducts This to o involves the EilenbergZilb er

theorem

We remark that the pro of of the ab ove theorem is easier if b oth group oids are of

the same type ie if they are b oth groups or smo oth manifolds in which case our

theorem is part of folklore However in the case we shall use this theoremthat of a

group and a manifoldthere are no signicant simplications one has to go through

all the steps of the pro of given ab ove

Lemma Let C S M M C S M C be as dened in and

k

C be as in Then for any cyclic cocycle HP C we T r HP

n n

cont

have

T r J M HP C S M HP C S M

CM n

cont

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Higher Index Theorems

Proof Denote by HP C C HP C S M C the b oundary

n

cont

morphism of the exact sequence Using Theorem we obtain

T r T r J M HP C S M C

n AS n

cont

HP C S M C

Then the naturality of the b oundary map and Theorem show that

CM

This completes the pro of

even

Let T M H S M b e the Todd class of TM C lifted to S M and C h

b e the classical Chern character on K Theory as b efore Also recall that Theorem

denes an embedding H B H HP C HP C

cont

We are now ready to state ConnesMoscovicis Index Theorem for elliptic sys

tems see Theorem page which computes the higher index of a matrix

of P of prop erly supp orted order zero invariant elliptic pseudo dierential op erators

f

on M with principal symbol the invertible matrix u P M C S M

m

f

Theorem ConnesMoscovici Let M M be a covering with Galois group

of a smooth compact manifold M of dimension n and let f S M B the

f

continuous map that classies the covering S M S M Then for each cohomology

q

class H B and each u K S M we have

n

Ind u h C hu T M f S M i

q

where T r HP C C

n n

Proof All ingredients of the pro of are in place and we just need to put them together

q

Let H B and T r b e as in the statement of the theorem Then

n

n

Ind u

n

u by Theorem

CM

n

J M u by Lemma

n

u by Theorem J M

n

id f J M u by Prop osition

n

h J M f chui

n

h J M f chu S M i by equation

P

k n

hT M f chu S M i by Prop osition

k j

k j nq

P

q

hT M f C h u S M i by equation

k j

k j nq

q

hC hu T M f S M i

The pro of is now complete

For q and H B C we obtain that is the von

P

Neumann trace on C that is a a the co ecient of the identity and the

e

ab ove theorem recovers Atiyahs L index theorem for coverings The reason for

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Victor Nistor

obtaining a dierent constant than in is due to dierent normalizations See

for a discussion on how to obtain the usual index theorems from the index theorems

for elliptic systems

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Victor Nistor

Pennsylvania State University

Math Dept

University Park PA

USA

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