Higher Index Theorems and the Boundary Map in Cyclic Cohomology

We show that the Chern{Connes character induces a natural transformation from the six term exact sequence in (lower) algebraic K{ Theory to the periodic 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 boundary map of the periodic cyclic cohomology exact sequence, we show that it satisses properties similar to the properties satissed by the boundary map of the singular cohomology long exact sequence. As an application, we obtain a new proof of the Connes{ Moscovici index theorem for coverings.


Introduction
1. Index theorems and Algebraic K{Theory 1.1.Pairings with traces and a Fedosov type formula 1.2.\Higher traces" and excision in cyclic cohomology 1.3.An abstract \higher index theorem" 2. Products and the boundary map in periodic cyclic cohomology 2.1.Cyclic vector spaces 2.2.Extensions of algebras and products 2.3.Properties of the boundary map 2.4.Relation to the bivariant Chern{Connes character 3. The index theorem for coverings 3.1.Groupoids and the cyclic cohomology of their algebras 3.2.Morita invariance and coverings 3.3.The Atiyah{Singer exact sequence 3.4.The Connes{Moscovici exact sequence and proof of the theorem References Introduction Index theory and K-Theory have been close subjects since their appearance 1,4].
Several recent index theorems that have found applications to Novikov's 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 dimensions"{like in the L 2 {index theorem for coverings 3] that, roughly speaking, computes the trace of the projection on the space of solutions of an elliptic di erential operator on a covering space.The von Neumann dimension of the index does not fully recover the information contained in the abstract (i.e., algebraic K-Theory index) but this situation is remedied by considering \higher traces," as in the Connes{ Moscovici Index Theorem for coverings 11].(Since the appearance of this theorem, index theorems that compute the pairing between higher traces and the K{Theory class of the index are called \higher index theorems.") In 30], a general higher index morphism (i.e., a bivariant character) was de ned for a class of algebras{or, more precisely, for a class of extensions of algebras{that is large enough to accommodate most applications.However, the index theorem proved there was obtained only under some fairly restrictive conditions, too restrictive for most applications.In this paper we completely remove these restrictions using a recent breakthrough result of Cuntz and Quillen.
In 16], Cuntz and Quillen have shown that periodic cyclic homology, denoted HP , satis es excision, and hence that any two{sided ideal I of a complex algebra A gives rise to a periodic six-term exact sequence HP 0 (I) If M is a smooth manifold and A = C 1 (M), then HP (A) is isomorphic to the de Rham cohomology of M, and the Chern{Connes character on (algebraic) K{Theory generalizes the Chern{Weil construction of characteristic classes using connection and curvature 10].In view of this result, the excision property, equation (1), gives more evidence that periodic cyclic homology is the \right" extension of de Rham homology from smooth manifolds to algebras.Indeed, if I A is the ideal of functions vanishing on a closed submanifold N M, then HP (I) = H DR (M; N) and the exact sequence for continuous periodic cyclic homology coincides with the exact sequence for de Rham cohomology.This result extends to (not necessarily smooth) complex a ne algebraic varieties 22].
The central result of this paper, Theorem 1.6, Section 1, states that the Chern{ Connes character ch : K alg i (A) !HP i (A); where i = 0; 1, is a natural transformation from the six term exact sequence in (lower) algebraic K{Theory to the periodic cyclic homology exact sequence.In this formulation, Theorem 1.6 generalizes the corresponding result for the Chern character on the K{Theory of compact topological spaces, thus extending the list of common features of de Rham and cyclic cohomology.
The new ingredient in Theorem 1.6, besides the naturality of the Chern{Connes character, is the compatibility between the connecting (or index) morphism in algebraic K{Theory and the boundary map in the Cuntz{Quillen exact sequence (Theorem 1.5).Because the connecting morphism Ind : K alg 1 (A=I) !K alg 0 (I) associated to a two-sided ideal I A generalizes the index of Fredholm operators, Theorem 1.5 can be regarded as an abstract \higher index theorem," and the computation of the boundary map in the periodic cyclic cohomology exact sequence can be regarded as a \cohomological index formula." We now describe the contents of the paper in more detail.
If is a trace on the two{sided ideal I A, then induces a morphism : K alg 0 (I) !C : More generally, one can{and has to{allow to be a \higher trace," while still getting a morphism : K alg 1 (I) !C .Our main goal in Section 1 is to identify, as explicitly as possible, the composition Ind : K alg 1 (A=I) !C .For traces this is done in Lemma 1.1, which generalizes a formula of Fedosov.In general, Ind = (@ ) ; where @ : HP 0 (I) !HP 1 (A=I) is the boundary map in periodic cyclic cohomology.
Since @ is de ned purely algebraically, it is usually easier to compute it than it is to compute Ind, not to mention that the group K alg 0 (I) is not known in many interesting situations, which complicates the computation of Ind even further.
In Section 2 we study the properties of @ and show that @ is compatible with various product type operations on cyclic cohomology.The proofs use cyclic vector spaces 9] and the external product studied in 30], which generalizes the crossproduct in singular homology.The most important property of @ is with respect to the tensor product of an exact sequence of algebras by another algebra (Theorem 2.6).
We also show that the boundary map @ coincides with the morphism induced by the odd bivariant character constructed in 30], whenever the later is de ned (Theorem 2.10).
As an application, in Section 3 we give a new proof of the Connes{Moscovici index theorem for coverings 11].The original proof uses estimates with heat kernels.Our proof uses the results of the rst two sections to reduce the Connes{Moscovici index theorem to the Atiyah{Singer index theorem for elliptic operators on compact manifolds.
The main results of this paper were announced in 32], and a preliminary version of this paper has been circulated as \Penn State preprint" no.PM 171, March 1994.Although this is a completely revised version of that preprint, the proofs have not been changed in any essential way.However, a few related preprints and papers have appeared since this paper was rst written; they include 12,13,33].
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 Mathematical Institute of Heidelberg University for hospitality while parts of this manuscript were prepared, and to the referee for many useful comments.

Index theorems and Algebraic K{Theory
We begin this section by reviewing the de nitions of the groups K alg 0 and K alg 1 and of the index morphism Ind : K alg 1 (A=I) !K alg 0 (I) associated to a two-sided ideal I A.
There are easy formulas that relate these groups to Hochschild homology, and we review those as well.Then we prove an intermediate result that generalizes a formula of Fedosov in our Hochschild homology setting, which will serve both as a lemma in the proof of Theorem 1.5, and as a motivation for some of the formalisms developed in this paper.The main result of this section is the compatibility between the connecting (or index) morphism in algebraic K{Theory and the boundary morphism in cyclic cohomology (Theorem 1.5).An equivalent form of Theorem 1.5 states that the Chern{ Connes character is a natural transformation from the six term exact sequence in algebraic K{Theory to periodic cyclic homology.These results extend the results in 30] in view of Theorem 2.10.
All algebras considered in this paper are complex algebras.
1.1.Pairings with traces and a Fedosov type formula.It will be convenient to de ne the group K alg 0 (A) in terms of idempotents e 2 M 1 (A), that is, in terms of matrices e satisfying e 2 = e.Two idempotents, e and f, are called equivalent (in writing, e f) if there exist x; y such that e = xy and f = yx.The direct sum of two idempotents, e and f, is the matrix e f (with e in the upper{left corner and f in the lower{right corner).With the direct{sum operation, the set of equivalence classes of idempotents in M 1 (A) becomes a monoid denoted P(A).The group K alg 0 (A) is de ned to be the Grothendieck group associated to the monoid P(A).If e 2 M 1 (A) is an idempotent, then the class of e in the group K alg 0 (A) will be denoted e].
Let : A ! C be a trace.We extend to a trace M 1 (A) !C , still denoted , by the formula ( a ij ]) = P i (a ii ).If e f, then e = xy and f = yx for some x and y, and then the tracial property of implies that (e) = (f).Moreover (e f) = (e) + (f), and hence de nes an additive map P(A) !C .From the universal property of the Grothendieck group associated to a monoid, it follows that we obtain a well de ned group morphism (or pairing with ) K alg 0 (A) 3 e] ! ( e]) = (e) 2 C : ( The pairing (2) generalizes to not necessarily unital algebras I and traces : I !C as follows.First, we extend to I + = I + C 1, the algebra with adjoint unit, to be zero on 1.Then, we obtain, as above, a morphism : K alg 0 (I + ) ! C .The morphism : K alg 0 (I) !C is obtained by restricting from K alg 0 (I + ) to K alg 0 (I), de ned to be the kernel of K alg 0 (I + ) !K alg 0 (C ).The de nition of K alg 1 (A) is shorter: In words, K alg 1 (A) is the abelianization of the group of invertible matrices of the form 1 + a, where a 2 M 1 (A).The pairing with traces is replaced by a pairing with Hochschild 1{cocycles as follows.

Documenta Mathematica 2 (1997) 263{295
If : A A is a Hochschild 1-cocycle, then the the functional de nes a morphism : K alg 1 (A) !C , by rst extending to matrices over A, and then by pairing it with the Hochschild 1{cycle u u 1 .Explicitly, if u = a ij ], with inverse u 1 = b ij ], then the morphism is K alg The morphism depends only on the class of in the Hochschild homology group HH 1 (A) of A.
If 0 !I !A ! A=I !0 is an exact sequence of algebras, that is, if I is a two{sided ideal of A, then there exists an exact sequence 26], K alg of Abelian groups, called the algebraic K{theory exact sequence.The connecting (or index) morphism will play an important role in this paper and is de ned as follows.Let u be an invertible element in some matrix algebra of A=I.By replacing A=I with M n (A=I), for some large n, we may assume that u 2 A=I.Choose an invertible element v 2 M 2 (A) that projects to u u 1 in M 2 (A=I), and let e 0 = 1 0 and e 1 = ve 0 v 1 .Because e 1 2 M 2 (I + ), the idempotent e 1 de nes a class in K alg 0 (I + ).Since e 1 e 0 2 M 2 (I), the di erence e 1 ] e 0 ] is actually in K alg 0 (I) and depends only on the class u] of u in K alg 1 (A=I).Finally, we de ne To obtain an explicit formula for e 1 , choose liftings a; b 2 A of u and u 1 and let v, the lifting, to be the matrix v = 2a aba ab ( Continuing the study of the exact sequence 0 !I !A ! A=I !0, choose an arbitrary linear lifting, l : A=I 2 ! A. If is a trace on I, we let (a; b) = ( l(a); l(b)] l( a; b])): (6) Because a; xy] = ax; y]+ ya; x], we have ( A; I 2 ]) = 0, and hence is a Hochschild 1{cocycle on A=I 2 (i.e., (ab; c) (a; bc)+ (ca; b)).The class of in HH 1 (A=I 2 ), denoted @ , turns out to be independent of the lifting l.If A is a locally convex algebra, then we assume that we can choose the lifting l to be continuous.If ( A; I]) = 0, then it is enough to consider a lifting of A ! A=I.
Documenta Mathematica 2 (1997) 263{295 Lemma.1.1.Let be a trace on a two-sided ideal I A. If is the connecting morphism of the algebraic K{Theory exact sequence associated to the two-sided ideal I 2 of A, then Ind = (@ ) : If ( A; I]) = 0, then we may replace I 2 by I.
By replacing A=I 2 with M n (A=I 2 ), we may assume that n = 1.
Lemma 1.1 generalizes a formula of Fedosov in the following situation.Let B(H) be the algebra of bounded operators on a xed separable Hilbert space H and C p (H) B(H) be the (non-closed) ideal of p{summable operators 36] on H: C p (H) = fA 2 B(H); Tr(A A) p=2 < 1g: (We will sometimes omit H and write simply C p instead of C p (H).) Suppose now that the algebra A consists of bounded operators, that I C 1 , and that a is an element of A whose projection u in A=I is invertible.Then a is a Fredholm operator, and, for a suitable choice of a lifting b of u 1 , the operators 1 ba and 1 ab become the orthogonal projection onto the kernel of a and, respectively, the kernel of a .Finally, if = Tr, this shows that Tr Ind( u]) = dim ker(a) dim ker(a ) and hence that Tr Ind recovers the Fredholm index of a. (The Fredholm index of a, denoted ind(a), is by de nition the right-hand side of the above formula.)By equation (7), we see that we also recover a form of Fedosov's formula: ind(a) = Tr (1 ba) k Tr (1 ab) k if b is an inverse of a modulo C p (H) and k p.
The connecting (or boundary) morphism in the algebraic K{Theory exact sequence is usually denoted by `@'.However, in the present paper, this notation becomes unsuitable because the notation `@' is reserved for the boundary morphism in the periodic cyclic cohomology exact sequence.Besides, the notation `Ind' is supposed to suggest the name `index morphism' for the connecting morphism in the algebraic K{Theory exact sequence, a name justi ed by the relation that exists between Ind and the indices of Fredholm operators, as explained above.Documenta Mathematica 2 (1997) 263{295 1.2.\Higher traces" and excision in cyclic cohomology.The example of A = C 1 (M), for M a compact smooth manifold, shows that, in general, few morphisms K alg 0 (A) !C are given by pairings with traces.This situation is corrected by considering `higher-traces, ' 10].
Let A be a unital algebra and b 0 (a 0 : : : ( 1) i a 0 : : : a i a i+1 : : : a n ; b(a 0 : : : a n ) = b 0 (a 0 : : : a n ) + ( 1) n a n a 0 : : : a n 1 ; (9) for a i 2 A. The Hochschild homology groups of A, denoted HH (A), are the homology groups of the complex (A (A=C 1) n ; b).The cyclic homology groups 10, 24, 37] of a unital algebra A; denoted HC n (A); are the homology groups of the complex (C(A); b + B), where C n (A) = M k 0 A (A=C 1) n 2k : (10) b is the Hochschild homology boundary map, equation (9), and B is de ned by B(a 0 : : : a n ) = s n X k=0 t k (a 0 : : : a n ): (11) Here we have used the notation of 10], that s(a 0 : : : a n ) = 1 a 0 : : : a n and t(a 0 : : : a n ) = ( 1) n a n a 0 : : : a n 1 : More generally, Hochschild and cyclic homology groups can be de ned for \mixed complexes," 21].A mixed complex (X ; b; B) is a graded vector space (X n ) n 0 , endowed with two di erentials b and B, b : X n !X n 1 and B : X n !X n+1 , satisfying the compatibility relation b 2 = B 2 = bB+Bb = 0.The cyclic complex, denoted C(X), associated to a mixed complex (X ; b; B) is the complex C n (X ) = X n X n 2 X n 4 : : : with di erential b + B. The cyclic homology groups of the mixed complex X are the homology groups of the cyclic complex of X: HC n (X ) = H n (C(X ); b + B): Cyclic cohomology is de ned to be the homology of the complex (C(X ) 0 = Hom(C(X ); C ); (b + B) 0 ); dual to C(X).From the form of the cyclic complex it is clear that there exists a morphism S : C n (X ) ! C n 2 (X ).We let C n (X ) = lim C n+2k (X ) as k ! 1, the inverse system being with respect to the periodicity operator S. Then the periodic cyclic homology of X (respectively, the periodic cyclic cohomology of X), denoted HP (X ) (respectively, HP (X )) is the homology (respectively, the cohomology) of C n (X ) (respectively, of the complex lim !C n+2k (X ) 0 ).
If A is a unital algebra, we denote by X(A) the mixed complex obtained by letting X n (A) = A (A=C 1) n with di erentials b and B given by ( 9) and (11).The various homologies of X(A) will not include X as part of notation.For example, the periodic cyclic homology of X is denoted HP (A).
For a topological algebra A we may also consider continuous versions of the above homologies by replacing the ordinary tensor product with the projective tensor product.We shall be especially interested in the continuous cyclic cohomology of A, denoted HP cont (A).An important example is A = C 1 (M), for a compact smooth manifold M. Then the Hochschild-Kostant-Rosenberg map : A ^ n+1 3 a 0 a 1 : : : a n !(n!) 1 a 0 da 1 : : : da n 2 n (M) (12) to smooth forms gives an isomorphism of continuous periodic cyclic homology with the de Rham cohomology of M 10, 24] made Z 2 {periodic.The normalization factor (n!) 1 is convenient because it transforms B into the de Rham di erential d DR .It is also the right normalization as far as Chern characters are involved, and it is also compatible with products, Theorem 3.5.From now on, we shall use the de Rham's Theorem to identify de Rham cohomology and singular cohomology with complex coe cients of the compact manifold M.
Sometimes we will use a version of continuous periodic cyclic cohomology for algebras A that have a locally convex space structure, but for which the multiplication is only partially continuous.In that case, however, the tensor products A n+1 come with natural topologies, for which the di erentials b and B are continuous.This is the case for some of the groupoid algebras considered in the last section.The periodic cyclic cohomology is then de ned using continuous multi-linear cochains.
One of the original descriptions of cyclic cohomology was in terms of \higher traces" 10].A higher trace{or cyclic cocycle{is a continuous multilinear map : A n+1 !C satisfying b = 0 and (a 1 ; : : : ; a n ; a 0 ) = ( 1) n (a 0 ; : : : ; a n ).Thus cyclic cocycles are, in particular, Hochschild cocycles.The last property, the cyclic invariance, justi es the name \cyclic cocycles."The other name, \higher traces" is justi ed since cyclic cocycles on A de ne traces on the universal di erential graded algebra of A.

If I A is a two{sided ideal, we denote by C(A; I) the kernel of C(A) ! C(A=I).
For possibly non-unital algebras I, we de ne the cyclic homology of I using the complex C(I + ; I).The cyclic cohomology and the periodic versions of these groups are de ned analogously, using C(I + ; I).For topological algebras we replace the algebraic tensor product by the projective tensor product.
An equivalent form of the excision theorem in periodic cyclic cohomology is the following result.
Theorem.1.2 (Cuntz{Quillen).The inclusion C(I + ; I) , !C(A; I) induces an isomorphism, HP (A; I) ' HP (I), of periodic cyclic cohomology groups.This theorem is implicit in 16], and follows directly from the proof there of the Excision Theorem by a sequence of commutative diagrams, using the Five Lemma each time. 2his alternative de nition of excision sometimes leads to explicit formulae for @.We begin by observing that the short exact sequence of complexes 0 !C(A; I) !C(A) !C(A=I) !0 de nes a long exact sequence :: HP n (A; I) HP n (A) HP n (A=I) @ HP n 1 (A; I) HP n 1 (A) :: in cyclic cohomology that maps naturally to the long exact sequence in periodic cyclic cohomology.
Most important for us, the boundary map @ : HP n (A; I) !HP n+1 (A=I) is determined by a standard algebraic construction.We now want to prove that this boundary morphism recovers a previous construction, equation (6), in the particular case n = 0.As we have already observed, a trace : I !C satis es ( A; I 2 ]) = 0, and hence de nes by restriction an element of HC 0 (A; I 2 ).The traces are the cycles of the group HC 0 (I), and thus we obtain a linear map HC 0 (I) !HC 0 (A; I 2 ).From the de nition of @ : HP 0 (A; I) !HP 1 (A=I), it follows that @ ] is the class of the cocycle (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 hope this will not cause any confusions.) Below we shall also use the natural map (transformation) commutes.Consequently, if 2 HC 0 (I) is a trace on I and ] 2 HP 0 (I) is its class in periodic cyclic homology, then @ ] = @ ] 2 HP 1 (A=I), where @ 2 HC 1 (A=I 2 ) is given by the class of the cocycle de ned in equation ( 6) (see also above).
Proof.The commutativity of the diagram follows from de nitions.If we start with a trace 2 HC 0 (I) and follow counterclockwise through the diagram from the upper{ left corner to the lower{right corner we obtain @ ]; if we follow clockwise, we obtain the description for @ ] indicated in the statement. 1.
3. An abstract \higher index theorem".We now generalize Lemma 1.1 to periodic cyclic cohomology.Recall that the pairings (2) and ( 3) have been generalized to pairings K alg i (A) HC 2n+i (A) !C ; i = 0; 1: 10].Thus, if be a higher trace representing a class ] 2 HC 2n+i (A), then, using the above pairing, de nes morphisms : K alg i (A) !C , where i = 0; 1.The explicit formulae for these morphisms are ( e]) = ( 1) n (2n)! n! (e; e; : : : ; e), if i = 0 and e is an idempotent, and ( u]) = ( 1) n n! (u; u1 ; u; : : : ; u 1 ), if i = 1 and u is an invertible element.The constants in these pairings are meaningful and are chosen so that these pairings are compatible with the periodicity operator.
Consider the standard orthonormal basis (e n ) n 0 of the space l2 (N) of square summable sequences of complex numbers; the shift operator S is de ned by Se n = e n+1 .The adjoint S of S then acts by S e 0 = 0 and S e n+1 = e n , for n 0. The operators S and S are related by S S = 1 and SS = 1 p, where p is the orthogonal projection onto the vector space generated by e 0 .
Let T be the algebra generated by S and S and C w; w 1 ] be the algebra of Laurent series in the variable w, C w; Then there exists an exact sequence 0 !M 1 (C ) ! T !C w; w 1 ] !0; called the Toeplitz extension, which sends S to w and S to w 1 .Let C h a; b i be the free non-commutative unital algebra generated by the symbols a and b and J = ker(C h a; b i !C w; w 1 ]), the kernel of the unital morphism that sends a !w and b !w 1 .Then there exists a morphism 0 : C h a; b i !T , uniquely determined by 0 (a) = S and 0 (b) = S , which de nes, by restriction, a morphism The algebra C h a; b i is the tensor algebra of the vector space C a C b, and hence the groups g HC (T (V )) also vanish 24].It follows that the morphism 0 induces (trivially) an isomorphism in cyclic cohomology.The comparison morphism between the Cuntz{Quillen exact sequences associated to the two extensions shows, using \the Five Lemma," that the induced morphisms : HP (M 1 (C )) !HP (J) is also an isomorphism.This proves the result since the canonical trace Tr generates HP (M 1 (C )).
We are now ready to state the main result of this section, the compatibility of the boundary map in the periodic cyclic cohomology exact sequence with the index (i.e., connecting) map in the algebraic K{Theory exact sequence.The following theorem generalizes Theorem 5.4 from 30].
Theorem.1.5.Let 0 !I !A ! A=I !0 be an exact sequence of complex algebras, and let Ind : K alg connecting morphisms in algebraic K{Theory and, respectively, in periodic cyclic cohomology.Then, for any ' 2 HP 0 (I) and u] 2 K alg 1 (A=I), we have ' (Ind u]) = (@') u] : (13) Proof.We begin by observing that if the class of ' can be represented by a trace (that is, if ' is the equivalence class of a trace in the group HP 0 (I)) then the boundary map in periodic cyclic cohomology is computed using the recipe we have indicated, Lemma 1.3, and hence the result follows from Lemma 1.1.In particular, the theorem is true for the exact sequence 0 !J ! C h a; b i !C w; w 1 ] !0; because all classes in HP 0 (J) are de ned by traces, as shown in Lemma 1.4.We will now show that this particular case is enough to prove the general case \by universality." Let u be an invertible element in M n (A=I).After replacing the algebras involved by matrix algebras, if necessary, we may assume that n = 1, and hence that u is an invertible element in A=I.This invertible element then gives rise to a morphism : C w; w 1 ] !A=I that sends w to u.A choice of liftings a 0 ; b 0 2 A of u and u 1 de nes a morphism 0 : C h a; b i !A, uniquely determined by 0 (a) = a 0 and 0 (b) = b 0 , which restricts to a morphism : J ! I.In this way we obtain a We claim that the naturality of the index morphism in algebraic K{Theory and the naturality of the boundary map in periodic cyclic cohomology, when applied to the above exact sequence, prove the theorem.Indeed, we have Ind = Ind : K alg 1 (C w; w 1 ]) !K alg 0 (I); and @ = @ : HP (I) !HP +1 (C w; w 1 ]): As observed in the beginning of the proof, the theorem is true for the cocycle (') on J, and hence ( (')) (Ind w]) = (@ (')) w].Finally, from de nition, we have that w] = u].Combining these relations we obtain The proof is complete.The theorem we have just proved can be extended to topological algebras and topological K{Theory.If the topological algebras considered satisfy Bott periodicity, then an analogous compatibility with the other connecting morphism can be proved and one gets a natural transformation from the six-term exact sequence in topological K{Theory to the six-term exact sequence in periodic cyclic homology.However, a factor of 2 { has to be taken into account because the Chern-Connes character is not directly compatible with periodicity 30], but introduces a factor of 2 {.See 12] for details.So far all our results have been formulated in terms of cyclic cohomology, rather than cyclic homology.This is justi ed by the application in Section 3 that will use this form of the results.This is not possible, however, for the following theorem, which states that the Chern character in periodic cyclic homology (i.e., the Chern{Connes character) is a natural transformation from the six term exact sequence in (lower) algebraic K{Theory to the exact sequence in cyclic homology.Proof.Only the relation ch Ind = @ ch needs to be proved, and this is dual to Theorem 1.5.
2. 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 behavior with respect to products.2.1.Cyclic vector spaces.We begin this section with a review of a few needed facts about the cyclic category from 9] and 30].We will be especially interested in the {product in bivariant cyclic cohomology.More results can be found in 23].Definition.2.1.The cyclic category, denoted , is the category whose objects are n = f0; 1; : : : ; ng, where n = 0; 1; : : : and whose morphisms Hom ( n ; m ) are the homotopy classes of increasing, degree one, continuous functions ' : S 1 !S 1 satisfying '(Z n+1 ) Z m+1 .
A cyclic vector space is a contravariant functor from to the category of complex vector spaces 9].Explicitly, a cyclic vector space X is a graded vector space, X = (X n ) n 0 , with structural morphisms d i n : X n !X n 1 , s i n : X n !X n+1 , for 0 i n, and t n+1 : X n !X n such that (X n ; d i n ; s i n ) is a simplicial vector space ( 25], Chapter VIII,x5) and t n+1 de nes an action of the cyclic group Z n+1 satisfying d 0 n t n+1 = d n n and s 0 n t n+1 = t 2 n+2 s n n , d i n t n+1 = t n d i 1 n , and s i n t n+1 = t n+2 s i 1 n for 1 i n.Cyclic vector spaces form a category.
The cyclic vector space associated to a unital locally convex complex algebra A is A \ = (A n+1 ) n 0 , with the structural morphisms s i n (a 0 : : : a n ) = a 0 : : : a i 1 a i+1 : : : a n ; d i n (a 0 : : : a n ) = a 0 : : : a i a i+1 : : : a n ; for 0 i < n; and d n n (a 0 : : : a n ) = a n a 0 : : : a i a i+1 : : : a n 1 ; t n+1 (a 0 : : : a n ) = a n a 0 a 1 : : : a n 1 : If X = (X n ) n 0 and Y = (Y n ) n 0 are cyclic vector spaces, then we can de ne on (X n Y n ) n 0 the structure of a cyclic space with structural morphisms given by the diagonal action of the corresponding structural morphisms, s i n ; d i n , and t n+1 , of X and Y .The resulting cyclic vector space will be denoted X Y and called the external product of X and Y .In particular, we obtain that (A B) \ = 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 locally convex algebras, obtained by using the complete projective tensor product.
The cyclic cohomology groups of an algebra A can be recovered as Ext{groups.For us, the most convenient de nition of Ext is using exact sequences (or resolutions).Consider the set E = (M k ) n k=0 of resolutions of length n + 1 of X by cyclic vector spaces, such that M n = Y .Thus we consider exact sequences of cyclic vector spaces.For two such resolutions, E and E 0 , we write E ' E 0 whenever there exists a morphism of complexes E !E 0 that induces the identity on X and Y .Then Ext n (X; Y ) is, by de nition, the set of equivalence classes of resolutions E = (M k ) n k=0 with respect to the equivalence relation generated by '.The set Ext n (X; Y ) has a natural group structure.The equivalence class in Ext n (X; Y ) of a resolution E = (M k ) n k=0 is denoted E].This de nition of Ext coincides with the usual one{using resolutions by projective modules{because cyclic vector spaces form an Abelian category with enough projectives.
Given a cyclic vector space X = (X n ) n 0 de ne b; b 0 : X n !X n 1 by b 0 = P n 1 j=0 ( 1) j d j ; b = b 0 +( 1) n d n .Let s 1 = s n n t n+1 be the `extra degeneracy' of X, which satis es s 1 b 0 +b 0 s 1 = 1.Also let = 1 ( 1) n t n+1 , N = P n j=0 ( 1) nj t j n+1 and B = s 1 N. Then (X; b; B) is a mixed complex and hence HC (X), the cyclic homology of X, is the homology of ( k 0 X n 2k ; b+B), by de nition.Cyclic cohomology is obtained by dualization, as before.
The Ext{groups recover the cyclic cohomology of an algebra A via a natural isomorphism, HC n (A) ' Ext n (A \ ; C \ ); (14) 9].This isomorphism allows us to use the theory of derived functors to study cyclic cohomology, especially products.
The Yoneda product, The resulting product generalizes the composition of functions.Using the same notation, the external product E E 0 is the resolution Passing to equivalence classes, we obtain a product Ext m (X; Y ) Ext n (X 1 ; Y 1 ) ! Ext m+n (X X 1 ; Y Y 1 ): If f : X !X 0 is a morphism of cyclic vector spaces then we shall sometimes denote E 0 f = f (E 0 ), for E 0 2 Ext n (X 0 ; C \ ).
The Yoneda product, \ ," and the external product, \ ," are both associative and are related by the following identities, 30], Lemma 1.2.
Lemma.2.2.Let x 2 Ext n (X; Y ), y 2 Ext m (X 1 ; Y 1 ), and be the natural transformation Ext m+n (X 1 X; Y 1 Y ) ! Ext m+n (X X 1 ; Y Y 1 ) that interchanges the factors.Then x y = (id Y y) (x id X1 ) = ( 1) mn (x id Y1 ) (id X y); id X (y z) = (id X y) (id X z); x y = ( 1) mn (y x); and x id C \ = x = id C \ x: We now turn to the de nition of the periodicity operator.A choice of a generator of the group Ext 2 (C \ ; C \ ), de nes a periodicity operator Ext n (X; Y ) 3 x !Sx = x 2 Ext n+2 (X; Y ): (15) In the following we shall choose the standard generator that is de ned `over Z', and then the above de nition extends the periodicity operator in cyclic cohomology.This and other properties of the periodicity operator are summarized in the following Corollary ( 30] Using the periodicity operator, we extend the de nition of periodic cyclic cohomology groups from algebras to cyclic vector spaces by HP i (X) = lim !Ext i+2n (X; C \ ); ( 16) the inductive limit being with respect to S; clearly, HP i (A \ ) = HP i (A).Then Corollary 2.3 a) shows that the external product is compatible with the periodicity morphism, and hence de nes an external product, HP i (A) HP j (B) !HP i+j (A B); (17) on periodic cyclic cohomology.Documenta Mathematica 2 (1997) 263{295 2.2.Extensions of algebras and products.Cyclic vector spaces will be used to study exact sequences of algebras.Let I A be a two{sided ideal of a complex unital algebra A (recall that in this paper all algebras are complex algebras.)Denote by (A; I) \ the kernel of the map A \ !(A=I) \ , and by A; I] 2 Ext 1 ((A=I) \ ; (A; I) \ ) the (equivalence class of the) exact sequence 0 !(A; I) \ !A \ !(A=I) \ !0 (18) of cyclic vector spaces.
Let HC i (A; I) = Ext i ((A; I) \ ; C \ ), then the long exact sequence of Ext{groups associated to the short exact sequence (18) reads !HC i (A=I) !HC i (A) !HC i (A; I) !HC i+1 (A=I) !HC i+1 (A) !By standard homological algebra, the boundary map of this long exact sequence is given by the product HC i (A; I) 3 !A; I] 2 HC i+1 (A=I): For an arbitrary algebra I, possibly without unit, we let I = (I + ; I) \ : Then the isomorphism ( 14) becomes HC n (I) ' Ext n (I ; C \ ), and the excision theorem in periodic cyclic cohomology for cyclic vector spaces takes the following form.
It follows that every element 2 HP (I) is of the form = 0 j I;A , and that the boundary morphism @ A;I : HP (I) !HP +1 (A=I) satis es @ A;I ( 0 j I;A ) = 0 A; I] (19) for all 0 2 HC i (A; I) = Ext i ((A; I) \ ; C \ ).Formula (19) then uniquely determines @ I;A .
We shall need in what follows a few properties of the isomorphisms j I;A .Let B be an arbitrary unital algebra and I an arbitrary, possibly non{unital algebra.The 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) \ : 2.3.Properties of the boundary map.The following theorem is a key tool in establishing further properties of the boundary map in periodic cyclic homology.
Theorem.2.6.Let A and B be complex unital algebras and I A be a two-sided ideal.Then the boundary maps @ I;A : HP (I) !HP +1 (A=I) and @ I B;A B : HP (I B) !HP +1 ((A=I) B) satisfy @ I B;A B ( ) = @ I;A ( ) for all 2 HP (I) and 2 HP (B).
Proof.The groups HP k (I) is the inductive limit of the groups Ext k+2n (I ; C \ ) so will be the image of an element in one of these Ext{groups.By abuse of notation, we shall still denote that element by , and thus we may assume that 2 Ext k (I ; C \ ), for some large k.Similarly, we may assume that 2 Ext j (B \ ; C \ ).Moreover, by Theorem 2.4, we may assume that = 0 j I;A , for some 0 2 Ext i ((A; I) \ ; C \ ).
For the rest of this subsection it will be convenient to work with continuous periodic cyclic homology.Recall that this means that all algebras have compatible locally convex topologies, that we use complete projective tensor products, and that the projections A ! A=I have continuous linear splittings, which implies that A ' A=I I as locally convex vector spaces.Moreover, since the excision theorem is known only for m{algebras 13], we shall also assume that our algebras are m{ algebras, that is, that their topology is generated by a family of sub-multiplicative seminorms.Slightly weaker results hold for general topological algebras and discrete periodic cyclic cohomology.
There is an analog of Theorem 2.7 for actions of compact Lie groups.If G is a compact Lie group acting smoothly on a complete locally convex algebra A by equations ( 23) and (24) and that the inclusion j = j IoG;AoG , by the naturality of , is R(G)-linear, we nally get @( ) = @( ( 0 j)) = @(( 0 ) j) = = @( 0 Tr j The proof is now complete.
In the same spirit and in the same framework as in Theorem 2.8, we now consider the action of Lie algebra cohomology on the periodic 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 bi-invariant currents on G. Let : G G ! G be the multiplication.Then one can alternatively de ne the product on H (g) as the composition We now recall the de nition of the product H (g) HP cont (A) !HP cont (A): Denote by ' : A ! C 1 (G; A) the morphism '(a)(g) = g (a), where, this time, C 1 (G; A) is endowed with the pointwise product.Then x 2 HP cont (C 1 (G) b A) is a (continuous) cocycle on C 1 (G; A) ' C 1 (G) b A, and we de ne x = ' (x ).
The associativity of the -product shows that HP cont (A) becomes a H (g){module with respect to this action.Theorem.2.9.Suppose that a compact connected Lie group G acts smoothly on a complete locally convex algebra A and that I is a closed invariant two-sided ideal of A, complemented as a topological vector space.Then @(x ) = x(@ ); for any x 2 H (g) and 2 HP cont (I) .
Proof.The proof is similar to the proof of Theorem 2.8, using the morphism of exact sequences 0 is given, such that the cocycle `(a 0 ; a 1 ) = (a 0 ) (a 1 ) (a 0 a 1 ) factors as a composition A b A ! C p (H) b B ! B(H) b B of continuous maps.(Recall that C p (H) is the ideal of p{summable operators and that b is the complete projective tensor product.) Using the cocycle `, we de ne on E = A C p (H) ^ B an associative product by the formula (a 1 ; x 1 )(a 2 ; x 2 ) = (a 1 a 2 ; (a 1 )x 2 + x 1 (a 2 ) + `(a 1 ; a 2 )): Then the algebra E ts into the exact sequence that is isomorphic to an exact sequence of the form (25) will be called an admissible exact sequence.If E] is an admissible exact sequence and n p 1, then 30, Theorem Tr n (a 0 ; a 1 ; : : : ; a 2n ) = ( 1) n n! (2n)!Tr(a 0 a 1 : : : a 2n ): The normalization factor was chosen such that Tr n = S n Tr 1 = S n Tr on C 1 (H).We have the following compatibility between the bivariant Chern-Connes character and the Cuntz{Quillen boundary morphism.
Let HP cont 3 !disc 2 HP disc := HP be the natural transformation that \forgets continuity" from continuous to ordinary (or discrete) periodic 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 be included.
This theorem provides us{at least in principle{with formul to compute the boundary morphism in periodic cyclic cohomology, see 29] and 30], Proposition 2.3.
Before proceeding with the proof, we recall a construction implicit in 30].The algebra RA = j 0 A ^ j is the tensor algebra of A, and rA is the kernel of the map RA !A + .Because A has a unit, we have a canonical isomorphism A + ' C A.
We do not consider any topology on RA, but in addition to (RA) \ ; the cyclic object associated to RA, we consider a completion of it in a natural topology with respect to which all structural maps are continuous.The new, completed, cyclic object is denoted (RA) \ cont and is obtained as follows.Let R k A = k j=0 A ^ j .Then with the inductive limit topology.Proof.We begin with a series of reductions that reduce the proof of the Theorem to the proof of (29).
Since E] is an admissible extension, there exists by de nition a continuous linear section s : A ! E of the projection : E !A (i.e., s = id).where the right hand vertical map is the projection A + ' C A ! A.
By increasing q if necessary, we may assume that the cocycle 2 HP q cont (B) comes from a cocycle, also denoted , in HC q cont (B).Let be as in the statement of the theorem.We claim that it is enough to show that @(' 1 ) j A = ( ch 2n+1 1 ( E])) disc ; (29) where j A = A \ !(A + ) \ is the inclusion.
Recall from 30] that the ideal rA de nes a natural increasing ltration of (RA) \ cont by cyclic vector spaces: (RA) \ such that (rA) F 1 (RA) \ cont = (RA; rA) \ .If (rA) k is the k{th component of the cyclic vector space (rA) (and if, in general, the lower index stands for the Z + {grading of a cyclic vector space) then we have the more precise relation (rA) k (F n 1 (RA) \ cont ) k ; for k n: (33) It follows that the morphism of cyclic vector spaces ~ n = Tr F n 1 ( ) : F n 1 (RA) \ cont !B \ (de ned in 30], page 579) satis es ~ n = Tr ' on (rA) k , for k n p 1. Fix then k = q + 2n, and conclude that 1 = Tr n disc 2 HC q+2n (C p b B) satis es on (rA) k F n 1 (RA) \ cont , because Tr n restricts to S n Tr on C 1 (H).Now recall the crucial fact that there exists an extension that has the property that C 2n 0 (RA) i = S n , if i : F n 1 (RA) \ cont !F 1 (RA) \ cont is the inclusion (see 30], Corollary 2.2).Using this extension, we nally de ne Since 2 has order k = q + 2n 2n n, we obtain from the equations ( 33) and (34) that 2 satis es (30) (i.e., that it restricts to ' 1 on (rA) k F n 1 (RA) \ cont ), as desired.
For any locally convex algebra B and 2 HP (B), the discrete periodic cyclic cohomology of B, we say that is a continuous class if it can be represented by a continuous cocycle on B. Put di erently, this means that = disc , for some 2 HP cont (B).Since the bivariant Chern{Connes character can, at least in principle, be expressed by an explicit formula, it preserves continuity.This gives the following corollary.
Corollary.2.11.The periodic cyclic cohomology boundary map @ associated to an admissible extension maps a class of the form Tr n , for a continuous class, to a continuous class.

Documenta Mathematica 2 (1997) 263{295
It is likely that recent results of Cuntz, see 12, 13], will give the above result for all continuous classes in HP (C p ^ B) (not just the ones of the form Tr n ).
Using the above corollary, we obtain the compatibility between the bivariant Chern{Connes character and the index morphism in full generality.This result had been known before only in particular cases 30].
Theorem.2.12.Let 0 !C p (H) b B ! E !A ! 0 be an admissible exact sequence and ch 2n+1 1 ( E]) 2 Ext 2n+1 (A \ ; B \ ) be its bivariant Chern{Connes character, equation (27).If Tr n is as in equation ( 28) and Ind : K alg 1 (A) !K alg 0 (C p (H) b B) is the connecting morphism in algebraic K{Theory then, for any ' 2 HP 0 cont (B) and u] 2 3. The index theorem for coverings Using the methods we have developed, we now give a new proof of Connes{Moscovici's index theorem for coverings.To a covering f M !M with covering group , Connes and Moscovici associated an extension (the Connes{Moscovici exact sequence), de ned using invariant pseudodi erential operators on f M; see equation ( 45).If ' 2 H ( ) HP cont (C n+1 C ]) is an even cyclic cocycle, then the Connes{Moscovici index theorem computes the morphisms where Ind is the index morphism associated to the Connes{Moscovici exact sequence.Our method of proof then is to use the compatibility between the connecting morphisms in algebraic K{Theory and @, the connecting morphism in periodic cyclic cohomology (Theorem 1.5), to reduce the proof to the computation of @.This computation is now a problem to which the properties of @ established in Section 2 can be applied.
We rst show how to obtain the Connes{Moscovici exact sequence from another exact sequence, the Atiyah{Singer exact sequence, by a purely algebraic construction.Then, using the naturality of @ and Theorem 2.6, we determine the connecting morphism @ CM of the Connes{Moscovici exact sequence in terms of the connecting morphism @ AS of the Atiyah{Singer exact sequence.For the Atiyah{Singer exact sequence the procedure can be reversed and we now use the Atiyah-Singer Index Theorem and Theorem 1.5 to compute @ AS .
A comment about the interplay of continuous and discrete periodic cyclic cohomology in the proof below is in order.We have to use continuous periodic cyclic cohomology whenever we want explicit computations with the periodic cyclic cohomology of groupoid algebras, because only the continuous version of periodic cyclic cohomology is known for groupoid algebras associated to etale groupoids 7].On the other hand, in order to be able to use Theorem 1.5, we have to consider ordinary (or discrete) periodic cyclic cohomology as well.This is not an essential di culty because, using Corollary 2.11, we know that the index classes are represented by continuous cocycles.
3.1.Groupoids and the cyclic cohomology of their algebras.Our computations are based on groupoids, so we rst recall a few facts about groupoids.
A groupoid is a small category in which every morphism is invertible.(Think of a groupoid as a set of points joined arrows; the following examples should clarify this abstract de nition of groupoids.)A smooth etale groupoid is a groupoid whose set of morphisms (also called arrows) and whose set of objects (also called units) are smooth manifolds such that the domain and range maps are etale (i.e., local di eomorphisms).
To any smooth etale groupoid G, assumed Hausdor for simplicity, there is associated the algebra C 1 c (G) of compactly supported functions on the set of arrows of G and endowed with the convolution product , Here r is the range map and r( ) = r(g) is the condition that 1 and g be composable.Whenever dealing with C 1 c (G), we will use continuous cyclic cohomology, as in 7].See 7] for more details on etale groupoids, and 35] for the general theory of locally compact groupoids.
Etale groupoids conveniently accommodate in the same framework smooth manifolds and (discrete) groups, two extreme examples in the following sense: the smooth etale groupoid associated to a smooth manifold M has only identity morphisms, whereas the smooth etale groupoid associated to the (discrete) group has only one object, the identity of .The algebras C 1 G (1)  U = f(x; ; ); ; 2 I; x 2 U \ U g: If R I is the total equivalence relation on I, then there is an injective morphism l : G U , !M R I of etale groupoids.Let f : G 1 !G 2 be an etale morphism of groupoids, that is, a morphism of etale groupoids that is a local di eomorphism.Then the map f de nes a continuous map, Bf : BG 2 !BG 1 , of classifying spaces and a group morphism, ).If f is injective when restricted to units, then there exists an algebra morphism (f The following theorem, a generalization of 7], Theorem 5.7.(2), is based on the fact that all isomorphisms in the proof of that theorem are functorial with respect to etale morphisms.It is the reason why we use continuous periodic cyclic cohomology when working with groupoid algebras.Note that the cyclic object associated to C 1 c (G), for G an etale groupoid, is an inductive limit of locally convex nuclear spaces.For smooth manifolds, the embedding of Theorem 3.1 is just the Poincar e duality{an isomorphism.This isomorphism has a very concrete form.Indeed, let 2 H n i (M; o) be an element of the singular cohomology of M with coe cients in the orientation sheaf, let 2 H i c (M) be an element of the singular cohomology of M with compact supports (all cohomology groups have complex coe cients), and let : be the canonical isomorphism induced by the Hochschild-Kostant-Rosenberg map , equation (12).Then the isomorphism is determined by h ( ); i = h ^ ( ); M]i 2 C ; (37) where the rst pairing is the map HP cont (C 1 c (M)) HP cont (C 1 c (M)) !C and the 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 1 (S M) ' HP cont (C 1 (S M)), because S M is oriented.(The orientation of S M is the one induced from that of T M as in 5].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 boundary of an oriented manifold.)The shift in the Z 2 -degree is due to the fact that S M is odd dimensional.
3.2.Morita invariance and coverings.Let M be a smooth compact manifold and q : f M !M be a covering with Galois group ; said di erently, f M is a principal {bundle over M. We x a nite cover U = (U ) 2I of M by trivializing open sets, i.e., q 1 (U ) ' U and M = U .The transition functions between two trivializing isomorphisms on their common domain, the open set U \ U , de nes a 1{cocycle that completely determines the covering q : f M !M.
In what follows, we shall need to lift the covering q : f M !M to a covering q : S f M !S M, using the canonical projection p : S M !M. All constructions then lift, from M to S M, canonically.In particular, V = p 1 (U ) is a nite covering of S M with trivializing open sets, and the associated 1{cocycle is (still) .Moreover, if f 0 : M !B classi es the covering q : f M !M, then f = f 0 p classi es the covering S f M !S M. Documenta Mathematica 2 (1997) 263{295 that the principal -bundle (i.e., covering) that h 1 pulls back from B to S M is isomorphic to the covering S f M !f M.
Let G U be the gluing groupoid associated to the cover U = (U ) 2I of M. It is seen from the de nition that G V !factors as G V !G U ! , where the function G V !acts as (m; ; ) ! .Thus we may replace S M by M everywhere in the proof.
Since the the covering f M !M is determined by its restriction to loops, we may assume that M is the circle S 1 .Cover M = S 1 by two contractable intervals I 0 \ I 1 which intersect in two small disjoint neighborhoods of 1 and 1: I 0 \ I 1 = (z; z 1 ) ( z; z 1 ) where z 2 S 0 and jz 1j is very small.We may also assume that the transition cocycle is the identity on (z; z 1 ) and 2 on ( z; z 1 ) (we have replaced constant {cocycles with locally constant {cocycles).The map h 1 maps each of the units of G U and each of the 1-cells corresponding to the right hand interval (z; z 1 ) to the only 0-cell of B , the cell corresponding to the identity e 2 .(Recall that the classifying space of a topological groupoid is the geometrical realization of the simplicial space of composable arrows 34], and that that there is a 0 cell for each unit, a 1-cell for each non-identity arrow, a 2-cell for each pair of non-identity composable arrows, and so on).The other 1-cells (i.e., corresponding to the arrows leaving from a point on the left hand side interval) will map to the 1-cell corresponding .This shows that, on homotopy groups, the induced map Z = 1 (S 1 ) ! = 1 (B ) sends the generator 1 to .This completes the proof of the lemma.
We need to introduce one more auxiliary morphism before we can determine .
Using the partition of unity P ' 2 = 1 subordinated to V = (V ) 2I , we de ne which turns out to be a morphism of algebras.Because the composition is (unitarily equivalent to) the upper{left corner embedding, we obtain that the morphism : HP cont (C 1 c (G V )) !HP cont (C 1 (S M)) is the inverse of t Tr .
We are now ready to determine the morphism : HP cont (C 1 (S M) C ]) !HP cont (C 1 (S M)): In order to simplify notation, in the statement of the following result we shall identify HP cont (M k (C 1 (S M)) C ]) with HP cont (C 1 (S M) C ]), and we shall do the same in the proof.Proposition.3.3.The composition Proof.Consider as before the morphism l : G V !S M R I of groupoids, which de nes an injective morphism of algebras (l) : C 1 (G V ) ! C 1 (S M R I ) = M k (C 1 (S M)), and hence also a morphism Then we can write = (l id) (g) ; where g : G V !G V is as de ned before: g(x; ; ) = (x; ; ; ).Because = (t Tr ) 1 , we have that 1 = (B t) 1 , by Theorem 3.1.
Also by Theorem 3.1, we have (g) = (B g) and (l id) = (B l id) .
This gives then 1 = 1 (B g) (B l id) = (B t) 1 (B g) (B l id) = h 0 : Since Lemma 3.2 states that h 0 = id f, up to homotopy, the proof is complete.3.3.The Atiyah{Singer exact sequence.Let M be a smooth compact manifold (without boundary).We shall denote by k (M) the space of classical, order at most k pseudodi erential operators on M. Fix a smooth, nowhere vanishing density on M. Then 0 (M) acts on L 2 (M) by bounded operators and, if an operator T 2 0 (M) is compact, then it is of order 1.More precisely, it is known that order 1 pseudodi erential operators satisfy 1 (M) C p = C p (L 2 (M)) for any p > n. (Recall that C p (H) is the ideal of p{summable operators on H, equation ( 8)).
We shall determine J (M) using Theorem 1.5.In order to do this, we need to make explicit the relation between ch, the Chern character in cyclic homology, and Ch, the classical Chern character as de ned, for example, in 27].Let E !M be a smooth complex vector bundle, embedded in a trivial bundle: E M C N , and let e 2 M N (C 1 (M)) be the orthogonal projection on E. If we endow E with the connection ed DR e, acting on 1 (E) C 1 (M) N , then the curvature of this connection turns out to be = e(d DR e) 2 .The classical Chern character Ch(E) is then the cohomology class of the form Tr(exp( 2 { )) in the even (de Rham) cohomology of M. Comparing this de nition with the de nition of the Chern character in cyclic cohomology via the Hochschild-Kostant-Rosenberg map, we see that the two of them are equal{up to a renormalization with a factor of 2 {.We now take a closer look at the algebra E CM and the exact sequence it de nes.
Observe rst that p acts on (L 2 (M) l 2 ( )) k and that p(L 2 (M) l 2 ( )) k ' L 2 ( f M) via a {invariant isometry.Since E 1 can be regarded as an algebra of operators on (L 2 (M) l 2 ( )) k that commute with the (right) action of , we obtain that E CM can also be interpreted as an algebra of operators commuting with the action of on L 2 ( f M).Using also 11], Lemma 5.1, page 376, this recovers the usual description of E CM that uses properly supported {invariant pseudodi erential operators on f M. We now proceed as for the Atiyah{Singer exact sequence.The boundary morphisms in periodic cyclic cohomology associated to the Connes{Moscovici extensions de nes a map @ CM : HP (C n+1 C ]) !HP +1 (C 1 (S M)); and the Connes{Moscovici Index Theorem amounts to the identi cation of the classes @ CM (T r n ) 2 HP +1 cont (C 1 (S M)) HP +1 (C 1 (S M)); for cocycles coming from the cohomology of .
In order to determine @ CM (T r n ); we need the following theorem.Proof.The proof is a long but straightforward veri cation that the sequence of isomorphisms in 7] is compatible with products.
Using 30], Proposition 1.5.(c), page 563, which states that the -products are compatible with the tensor products of mixed complexes, we replace everywhere cyclic vector spaces by mixed complexes.Then we go through the speci c steps of the proof as in 7].This amounts to verify the following facts: (i) The Hochschild-Kostant-Rosenberg map (equation ( 12)) transforms the di erential B 1 + 1 B into the de Rham di erential of the product.
(iii) The chain map f in the Moore isomorphism (see 6], Theorems 4.1 and 4.2, page 32) is compatible with products.This too involves the Eilenberg-Zilber theorem.
We remark that the proof of the above theorem is easier if both groupoids are of the same \type," i.e., if they are both groups or smooth manifolds, in which case our theorem is part of folklore.However, in the case we shall use this theorem{that of a group and a manifold{there are no signi cant simpli cations: one has to go through all the steps of the proof given above.Lemma.3.6.Let : C 1 (S M) !M k (C 1 (S M)) C ] be as de ned in (39) and Tr n 2 HP 0 (C n+1 ) be as in (28).Then, for any cyclic cocycle 2 HP cont (C ]), we have @ CM (T r n ) = (J (M) ) 2 HP +1 Proof.Denote by @ 1 : HP cont (C n+1 C ]) !HP +1 (C 1 (S M C ])) the boundary morphism of the exact sequence (44).Using Theorem 2.6, we obtain @ 1 (T r n ) = @ AS (T r n ) = J (M) 2 HP +1 cont (C 1 (S M) C ]) HP +1 (C 1 (S M) C ]): Then, the naturality of the boundary map and Theorem 2.10 show that @ CM = @ 1 .
This completes the proof.
Let T (M) 2 H even (S M) be the Todd class of TM C lifted to S M and Ch be the classical Chern character on K{Theory, as before.Also, recall that Theorem We are now ready to state Connes{Moscovici's Index Theorem for elliptic systems, see 11] Theorem 5.4], page 379, which computes the \higher index" of a matrix of P of properly supported, order zero, -invariant elliptic pseudodi erential operators on f M, with principal symbol the invertible matrix u = 0 (P ) 2 M m (C 1 (S M)).
Theorem.3.7 (Connes{Moscovici).Let f M !M be a covering with Galois group of a smooth compact manifold M of dimension n, and let f : S M !B the continuous map that classi es the covering S f M !S M.Then, for each cohomology class 2 H 2q (B ) and each u] 2 K 1 (S M), we have ~ (Ind u]) = ( 1) n (2 {) q h Ch(u) ^T (M) ^f ; S M] i; where ~ = Tr n ( ) 2 HP 0 (C n+1 C ]).
Proof.All ingredients of the proof are in place, and we just need to put them together.(2 {) q hT (M) 2k ^f ^Ch 2j 1 u]; S M]i by equation (42) = (2 {) q hCh u] ^T (M) ^f ; S M]i: The proof is now complete.
For q = 0 and = 1 2 H 0 (B ) ' C , we obtain that = ( ) is the von Neumann trace on C ], that is ( P a ) = a e , the coe cient of the identity, and the above theorem recovers Atiyah's L 2 {index theorem for coverings 2].The reason for obtaining a di erent constant than in 11] is due to di erent normalizations.See 19] for a discussion on how to obtain the usual index theorems from the index theorems for elliptic systems.
inclusion (I B) + !I + B, of unital algebras, de nes a commutative diagram .The morphism I;B , de ned for possibly non-unital algebras I, will replace the identi cation A \ B \ = (A B) \ , valid only for unital algebras A. Using the notation of Theorem 2.4, we see that I;B = j I B;I + B , and hence, by the same theorem, it follows that I;B induces an isomorphism HP (I B \ ) 3 !I;B 2 HP (I B): Using this isomorphism, we extend the external product : HP (I) HP (B) !HP (I B) to a possibly non-unital algebra I by Documenta Mathematica 2 (1997) 263{295 HP i (I) HP j (B) = lim !Ext i+2n (I ; C \ ) lim !Ext j+2m (B \ ; C \ ) ! lim !Ext i+j+2l (I B \ ; C \ ) = HP (I B \ ) ' HP i+j (I B): This extension of the external tensor product to possibly non-unital algebras will be used to study the tensor product by B of an exact sequence 0 !I !A ! A=I !0 of algebras.Tensoring by B is an exact functor, and hence we obtain an exact sequence 0 !I B !A B ! (A=I) B ! 0: (20) Lemma.2.5.Using the notation introduced above, we have the relation A B; I B] = A; I] id B 2 Ext 1 ((A=I B) \ ; (A B; I B) \ ):

2 =
\ ) (( 0 A; I]) id B ) by Lemma 2.2 = (id C \ ) ( 0 id B ) ( A; I] id B ) by Lemma 2.;I B (( 0 ) j I B;A B ) by equation (19).By de nition, the morphism j I;A introduced in Theorem 2.4 satis es j I B;A B = (j I;A id B ) I;B : ((A=I B) \ ; C \ ).This completes the proof in view of the de nition of the external product in the non-unital case: = ( ) I;B .Documenta Mathematica 2 (1997) 263{295 We now consider crossed products.Let A be a unital algebra and a discrete group acting on A by A 3 ( ; a) !(a) 2 A. Then the (algebraic) crossed product A o consists of nite linear combinations of elements of the form a , with the product rule (a )(b 1 ) = a (b) 1 .Let (a ) = a , which de nes a morphism : A o !A o C ]. Using , we de ne on HP (A o ) a HP (C ]){module structure 28] by HP (A o ) HP (C ]) !HP ((A o ) C ]) !HP (A o ): A {invariant two-sided ideal I A gives rise to a \crossed product exact sequence" 0 !I o !A o !(A=I) o !0 of algebras.The following theorem describes the behavior of the boundary map of this exact sequence with respect to the HP (C ]){module structure on the corresponding periodic cyclic cohomology groups.Theorem.2.7.Let be a discrete group acting on the unital algebra A, and let I be a -invariant ideal.Then the boundary map @ Io ;Ao : HP (I o ) ! HP +1 ((A=I) o ) is HP (C ])-linear.Proof.The proof is based on the previous theorem, Theorem 2.6, and the naturality of the boundary morphism in periodic cyclic cohomology.

4 .
Relation to the bivariant Chern{Connes character.A di erent type of property of the boundary morphism in periodic cyclic cohomology is its compatibility (e ectively an identi cation) with the bivariant Chern-Connes character 30].Before we can state this result, need to recall a few constructions from 30].Let A and B be unital locally convex algebras and assume that a continuous linear map : A ! B(H) b B Documenta Mathematica 2 (1997) 263{295

1 (
3.5] associates to E] an element ch 2n+1 E]) 2 Ext 2n+1 ;cont (A \ ; B \ ); (27) which for B = C recovers Connes' Chern character in K-homology 10].(The subscript \cont" stresses that we are considering the version of the Yoneda Ext de ned for locally convex cyclic objects.)Let Tr : C 1 (H) !C be the ordinary trace, i.e., Tr(T) = P n (T e n ; e n ) for any orthonormal basis (e n ) n 0 of the Hilbert space H. Using the trace Tr we de ne Tr n 2 HC 2n (C p (H)), for 2n p 1, to be the class of the cyclic cocycle c (G) associated to these groupoids are C 1 c (M) and, respectively, the group algebra C ].Here are other examples used in the paper.The groupoid R I associated to an equivalence relation on a discrete set I has I as the set of units and exactly one arrow for any ordered pair of equivalent objects.If I is a nite set with k elements and all objects of I are equivalent (i.e., if R I is the total equivalence relation on I) then C 1 c (R I ) ' M k (C ) and its classifying space in the sense of Grothendieck 34], the space B R I , is contractable 17, 34].Another example, the gluing groupoid G U , mimics the de nition a manifold M in terms of \gluing coordinate charts."The groupoid G U is de ned 7] using an open cover U = (U ) 2I of M, i.e., M = 2I U .Then G U has units G 0 U = 2I U f g and arrows (If 2 H (M) = k H k (M)is a cohomology class, we denote by k its component in H k (M).)Explicitly, let : HP cont i (C 1 c (S M)) ' k2Z H i+2k (S M) be the canonical isomorphism induced by the Hochschild-Kostant-Rosenberg map , equation(12); 1g and 2 K alg i (C 1 (M)).(Note the ` i').Documenta Mathematica 2 (1997) 263{295Proposition.3.4.Let T (M) 2 H even (S M) be the Todd class of the complexication of T M, lifted to S M, and : H even (S M) !HP 1 cont (C 1 (S M)) be the isomorphism of Theorem 3.1.ThenJ (M) = ( 1) n X k (2 {) n k (T (M) 2k ) 2 HP 1 cont (C 1 (S M)):Proof.We need to verify the equality of two classes in HP 1 cont (C 1 (S M)).It is hence enough to check that their pairings with ch( u]) are equal, for any u] 2K alg 1 (C 1 (S M)), because of the classical result that the Chern character ch : K alg1 (C 1 (S M)) !HP cont 1 (C 1 (S M)) is onto.If Ind is the index morphism of the Atiyah{Singer exact sequence then the Atiyah-Singer index formula 5] states the equality Ind u] = ( 1) n h Ch u]; T (M) i:(43) Using equation (41) and Theorem 1.5 (see also the discussion following that theorem), we obtain that Ind u] = h ch u]; J (M) i. Equations (37) and (43) then complete the proof.3.4.The Connes{Moscovici exact sequence and proof of the theorem.We now extend the constructions leading to the Atiyah{Singer exact sequence, equation (40), to covering spaces.Let M be a smooth compact manifold and let E 1 = M k (E) C ], which ts into the exact sequence 0 !M k (C n+1 ) C ] !E 1 0 !M k (C 1 (S M)) C ] be a covering of M with Galois group .Using the Mishchenko idempotent p associated to this covering and the injective morphism : C 1 (S M) !p(M k (C 1 (S M)) C ])p; equation 39, we de ne the Connes{Moscovici algebra E CM as the bered product E CM = f(T; a) 2 pE 1 p C 1 (S M); 0 (T ) = (a)g: By de nition, the algebra E CM ts into the exact sequence 0 !p M k (C n+1 ) C ] p ! E CM !C 1 (S M) !0: Also observe that \Mk " is super uous in M k (C n+1 ) because M k (C n+1 ) ' C n+1 ; actually, even \p" is super uous in p M k (C n+1 ) C ] p because p M k (C n+1 ) C ] p ' C n+1 C ] Documenta Mathematica 2 (1997) 263{295by an isomorphism that is uniquely determined up to an inner automorphism.Thus the Connes{Moscovici extension becomes 0 !C n+1 C ] !E CM !C 1 (S M)) !0;(45) up to an inner automorphism.

Theorem. 3 . 5 .
Let G 1 and G 2 be smooth etale groupoids.Then the diagram H +n (B G 1 ; o 1 ) H +m (B G 2 ; o 2 ) Here the left product is the external product in cohomology and o 1 , o 2 , and o are the orientation sheaves.