Nicolae Teleman
email: teleman@dipmat.univpm.it

Professor Enzo Martinelli

Nicolae Teleman

Noncommutative Topology and Prospects in Index Theory

The index formula is a local statement made on global and local data; for this reason we introduce local Alexander -Spanier cohomology, local periodic cyclic homology, local Chern character and local T * -theory. Index theory should be done: Case 1: for arbitrary rings, Case 2: for rings of functions over topological manifolds. Case 1 produces general index theorems, as for example, over pseudo-manifolds. Case 2 gives a general treatment of classical and non-commutative index theorems. All existing index theorems belong to the second category. The tools of the theory would contain: local T * -theory, local periodic cyclic homology, local Chern character. These tools are extended to non-commutative topology. The index

formula has three stages: Stage I is done in T loc i -theory, Stage II is done in the local periodic cyclic homology and Stage III involves products of distributions, or restriction to the diagonal. For each stage there corresponds a topological index and an analytical index. The construction of T * -theory involves the T -completion. It involves also the need to work with half integers; this should have important consequences.

La formule de l'indice est une f ormule locale faite sur les donnes globales et locales; pour cette raison, nous introduisons local Alexander -Spanier co-homologie, homologie cyclique priodique locale, caractre de Chern locale et la thorie local T * . La thorie de l'indice doit tre fait: Case 1: pour anneaux arbitraires, Cas 2: pour anneaux de fonctions sur les varietes topologiques. Cas 1 est le cas des thormes de l'indice gnral, comme par exemple, par example pseudovarietees. Cas 2 donne un traitement gnral des thormes d'index classiques et de la geometrie non-commutatives. Tous les thormes d'index existants appartiennent la deuxime catgorie. Le outils de la thorie contiendra: local T * -thorie, homologie cyclique priodique locale, caractre de Chern locale. Ces outils sont tendus topologie non-commutative. La formule de l'indice a ha trois tapes: Etape I est fait dans T loc i -thorie, Etape II est fait dans l'homologie cyclique priodique local et Etape III implique des produits de distributions, ou la restriction la diagonale. Pour chaque tape correspond un index topologique et une index analytique. La construction de T * -thorie utilises le T -completion. Elle implique galement la ncessit de travailler avec demi entiers; cela devrait avoir des consquences importantes.

Part I

Local algebraic structures. Let A be an unital associative ring. The ring A is called localised ring provided it is endowed with an additional structure satisfying the axioms (1) -(3) below.

Axiom 1. The underlying space A has a decreasing filtration by sub-spaces {A µ } n∈N ⊂ A.

Axiom 2. C.1 ⊂ A µ , f or any µ ∈ N Axiom 3. For any µ, µ ′ ∈ N + , A µ . A µ ′ ⊂ A Min(µ,µ ′ )-1 , (A 0 . A 0 ⊂ A 0 ). Remark 2 A ring A could have different localisations.

3 Local Alexander -Spanier co-homology. Definition 3 Let A = A µ be a localised unital ring and G an Abelian group. Define

C p AS (A µ , G) = { i g i a i 0 ⊗a i 1 ⊗• • •⊗a i p , g i ∈ G, a i ∈ A µ } p=0,1,...∞ .
(1) The boundary map d is defined as in the classical definition of Alexander -Spanier co-homology

d : C p AS (A µ , G) -→ C p+1 AS (A µ , G) (2)
d(g a 0 ⊗ a i 1 ⊗ • • • ⊗ a p ) = (3) g [1 ⊗ a 0 ⊗ a i 1 ⊗ • • • ⊗ a p -a 0 ⊗ 1 ⊗ a i 1 ⊗ • • • ⊗ a p + (4)
• • • + (-1) p+1 a 0 ⊗ a i 1 ⊗ • • • ⊗ a p ⊗ 1]. ( 5
)
The local Alexander -Spanier co-homology is

H loc,p AS (A) = P rojLim µ H p (C * AS (A µ , G), d). (6)
4 Local periodic cyclic homology. Long exact sequence.

We assume that A µ is a localised unital ring. The operators, see [START_REF] Connes | Noncommutative differential Geometry[END_REF], T , N, B, I, S pass to localised rings. Therefore local cyclic homology, local periodic cyclic homology may be extended to localised rings.

Theorem 4

The local Hochschild and cyclic homology are well defined for a localised ring.

One has the exact sequence (analogue of Connes' exact sequence, see Connes [START_REF] Connes | Noncommutative differential Geometry[END_REF]) . . .

Theorem 5 For A = C ∞ (M), one has

H loc p (C ∞(M )) = H loc p (C ∞(M )) . (8)
The local bi-complex (b, B)(A µ ) is well defined for localised rings too. The general term of the (b, B)(A µ ) bi-complex is C p,q = ⊗ p-q+1 A µ , q ≤ p (9) b : C p,q -→ C p,q-1 (10) B : C p,q -→ C p+1,q .

(11)

Definition 6

The homology of the bi-complex (b, B)(A µ ) has two components: the even, resp. odd, component corresponding to the p -q = even number, resp. p -q = odd number.

The local periodic cyclic homology of the localised ring A is H loc,per,λ ev,odd (A) := LimP roj µ H ev,odd H * (A µ ). [START_REF] Blackadar | K-Theory for Operator Algebras[END_REF] 5 Local periodic Chern character

In this section we localise the periodic even/odd Chern character. All operations involved in the construction of the cyclic homology may be localised. Here we use the bi-complrx (b, B)(A µ ) only onto non degenerate elements, i.e. onto the range of the idempotent Π.

Definition 7 local periodic cyclic Chern character.

Let A µ be a localised ring. Ch ev,Aµ (e) is an even cycle in the (b, B)(A µ ) cyclic complex. Its homology class is the periodic cyclic Connes Chern character of e.

Let e ∈ T loc 0 (A); define Ch ev (e) = P rojLim µ Ch ev,Aµ (e) ∈ H loc,per,λ ev (A). ( 14)

2. For any invertible u ∈ M n (A µ ), define

Ch odd (u) = p (-1) 2p+1 (2p+1)! T r (u -1 du).(du -1 du). . . . (du -1 du).

(15) Let u ∈ T loc 1 (A); define Ch odd (u) = P rojLim µ Ch odd,Aµ (u) ∈ H loc,per,λ odd (A). ( 16)

In non-commutative topology, discussed later in this article, we will present a definition of the Chern character of idempotents.

Local index theorem

In this section we consider an elliptic pseudo-differential operator A. We are going to define the Chern character of differences of idempotents; the skew-symmetrisation is not necessary. For more details about this topic see Teleman [START_REF] Teleman | Local 3 Index Theorem[END_REF], [START_REF] Teleman | The Local Index Theorem[END_REF].

Lemma 8 Let R(A) := P -e, where P and e are idempotents. Then R(A) satisfies the identity

R(A) 2 = R(A) -(e.R(A) + R(A).e). (17)
Definition 9 Let R(A) := P -e, where P and e are idempotents. Let Λ = C + e.C. Then,

C q (R(A)) := T r ⊗ 2q+1 Λ R(A) (18)
is the Chern character of R(A). For the definition of R(A) see §23, Definition 57.

Lemma 10 C q (R(A)) is a cycle in the complex {⊗ * Λ M N (A)} * , such that λ.a 0 ⊗ Λ a 1 ⊗ Λ • • • ⊗ Λ a k = a 0 ⊗ Λ a 1 ⊗ Λ • • • ⊗ Λ a k .λ (19)
for any λ ∈ Λ.

Proof. One has

b ′ C q (R(A)) = T r [ (R(A) -(e.R(A) + R(A).e)) ⊗ Λ • • • ⊗ Λ R(A) . . . ( 20
) -R(A) ⊗ Λ R(A) ⊗ Λ • • • ⊗ Λ (R(A) -(e.R(A) + R(A).e)) ] = (21) T r ; [ -(e.R(A) + R(A).e)) ⊗ Λ • • • ⊗ Λ R(A) . . . ( 22
) -R(A) ⊗ Λ R(A) ⊗ Λ • • • ⊗ Λ (-(e.R(A) + R(A).e) ] = (23) -T r [ e.R(A) ⊗ Λ • • • ⊗ Λ R(A) -R(A) ⊗ Λ • • • ⊗ Λ R(A).e ] = 0. ( 24
)
C q (R(A)) is a cyclic cycle. It represents a cyclic homology class in H λ * (⊗ 2q+1 (A µ ).
The ring Λ is separable. Indeed the mapping

µ : Λ⊗Λ op -→ Λ, µ(1⊗1) = 1, µ(1⊗e) = µ(e⊗1) = e, µ(e⊗e) = e
has the Λ-bimodule splitting

s : Λ -→ Λ ⊗ Λ op , s(1) = e ⊗ e + (1 -e) ⊗ (1 -e), s(e) = e ⊗ e.
Theorem 1.2.13 [START_REF] Loday | Cyclic Homology, Grundlehren in mathematischen Wissenschaften 301[END_REF] states that the homology of the Hochschild complex is isomorphic to the homology of the complex C S (A). Using the long exact sequence in homology associated to the complex H λ * (⊗ 2q+1 (A µ ) and the Connes exact sequence, we find that the cyclic homology of H λ * (⊗ 2q+1 (A µ ) is isomorphic to the cyclic homology of the algebra A. Here we have used the localisation of the algebra M N (A) defined by the supports of the operators.

It remains to solve the problem of producing the whole index class. This could be obtained by the formula

Ch ev R(A) -Ch ev e II , (25)
where R(A) is the the operator associated to the signature operator.

Part II

Prospects in Index Theory 7 Local algebraic T i theory.

This section deals with the program presented in [START_REF] Teleman | Local Algebraic K-Theory[END_REF].

For any associative ring A we define the commutative groups T i (A), i = 0, 1. We introduce the notion of localised rings, see Definition 1, A = {A µ }, given by a linear filtration of the algebra A and we associate the commutative groups T loc i (A). Although we define solely T loc i (A) for i = 0, 1 , we expect our construction could be extended in higher degrees.

We stress that our construction of T i (A) and T loc i (A), i = 0, 1, uses exclusively matrices. The projective modules are totally avoided. The role of equivalence of projective modules, used in the classical construction of the algebraic K-theory, is played by conjugation.

The commutative group T 0 (A) is by definition the Grothendieck completion of the space of idempotent matrices factorised through the equivalence relations: -i) stabilisation ∼ s , -2) conjugation ∼ c , and -3) for localised groups, T loc 0 (A), projective limit with respect to the filtration, denoted ∼ p . The non-localised group T 0 (A) coincides with the classical algebraic K 0 -theory group.

The groups T 1 (A) are by definition the quotient of GL(A) through the equivalence relation generated by -1) stabilisation ∼ s , -2) conjugation ∼ c and -3) ∼ O(A) , where O(A) is the sub-group generated by elements of the form u ⊕ u -1 , for any u ∈ GL(A). For any

u 1 , u 2 ∈ GL(A) / (∼ s ∪ ∼ c ), one defines u 1 ∼ O(A) u 2 provided there exist ξ 1 , ξ 2 ∈ O(A) such that u 1 + ξ 1 = u 2 + ξ 2 .
The third relation is a particular case of a new completion procedure which we call T-completion. The operation

GL(A) / (∼ s ∪ ∼ c ) -→ GL(A) / (∼ s ∪ ∼ c ∪ ∼ O(A) ) transforms the commutative semi-group GL(A) / (∼ s ∪ ∼ c ) in the commutative group GL(A) / (∼ s ∪ ∼ c ∪ ∼ O(A) ).
The groups T loc i (A) follow the same construction as that of T i (A), provided the supports of the elements belong to A µ . Our definition of T 1 (A) and T loc 1 (A) does not use the commutator sub-group [GL(A), GL(A)] nor elementary matrices in its construction.

We define short exact sequences of localised rings and we get the corresponding open six terms exact sequence (Theorem ).

We stress that one has to take the tensor product of the expected six terms exact sequence by Z[ 1 2 ] in order to get the open six terms exact sequence. We expect the factor Z[ 1 2 ] to have important implications, among them, Pontrjagin classes, existence of a generator of the K-homology fundamental class and Kirby-Siebenmann obstruction class.

Our work shows that the basic relations which define T 1 and T loc 1 reside in the additive sub-group generated by elements of the form u ⊕ u -1 , u ∈ GL(A), rather than in the multiplicative commutator sub-group [GL(A), GL(A)].

Even for trivialy filtered algebras, A = {A µ }, for all µ ∈ N, the groups T loc 1 (A) provides more information than the classical group T 1 (A). For the computation of the groups T loc i (C) see [START_REF] Teleman | [END_REF].

Motivation

To motivate the next considerations we have in mind the index formula applied onto the algebra of integral operators and pseudodifferential operators. The index formula is a global statement whose ingredients may be computed by local data. Our leading idea is to localise K-theory and periodic cyclic homology along the lines of the Alexander-Spanier co-homology in such a way that the new tools operate naturally with it, see [START_REF] Teleman | Local Algebraic K-Theory[END_REF]. This article goes along the lines of [START_REF] Teleman | Local Algebraic K-Theory[END_REF]. We define localised rings A and we define and their local T -theory, T loc i (A), i = 0, 1. Keeping in mind that both the topological and analytic indices of an elliptic operator are stable under cutting the operators about the diagonal, leads us naturally to the notion of localised rings. Based on this notion we pass to the problem of finding a natural K-theory able to controle these local entities. These are the local algebraic T -theory groups, T loc i (A), i = 0, 1 and local cyclic homology. For the construction of K i -theory groups, in the pure algebraic context, the reader may consult the following books [START_REF] Milnor | Introduction to Algebraic K-Theory[END_REF], [START_REF] Karoubi | Homologie cyclique et K-Théorie[END_REF], [START_REF] Rosenberg | Algebraic K-Theory and its Applications[END_REF], [START_REF] Weibel | K-theory[END_REF], in the Banach algebras or C * -algebras category, [START_REF] Rordam | An Introduction to K-Theory for C * -Algebras[END_REF], [START_REF] Blackadar | K-Theory for Operator Algebras[END_REF].

Our construction of T loc 0 uses exclusively idempotent matrices. The reasons why we chose to avoid projective modules are: -i) projective modules, in comparison with idempotent matrices, contain more ambiguity and -ii) matrices are more suitable for controlling the ring filtration data and are more prone to make calculations. No reference to projective modules is used in our constructions.

Regarding our construction of T loc 1 we recall that the classical algebraic K-theory group K 1 (A) of the algebra A, see [START_REF] Whitehead | Generalised homology theories[END_REF], [START_REF] Bass | Introduction to some Methods of Algebraic K-theory[END_REF], [START_REF] Milnor | Introduction to Algebraic K-Theory[END_REF], [START_REF] Karoubi | Homologie cyclique et K-Théorie[END_REF], is by definition the Whitehead group

K 1 (A) := GL(A)/[GL(A), GL(A)], (26)
where [GL(A), GL(A)] is the commutator normal sub-group of the group of invertible matrices GL(A). Our definition of local T -theory groups needs to keep track of the number of multiplications performed inside the algebra A. In order for our constructions to hold it is necessary to involve a bounded number of multiplications. It is important to state that, in general, the number of multiplications needed to generate the whole commutator sub-group is un-bounded.

It is known that the commutator sub-group is also generated by the elementary matrices. This is the reason why our definition of T loc 1 (A) avoids entirely factorising GL(A) through the commutator sub-group or the sub-group generated by elementary matrices.

T loc 0 (A) is by definition the Grothendieck completion of the semigroup of idempotent matrices in M n (A µ ) modulo three equivalence relations: -i) stabilisation ∼ s , -ii) local conjugation ∼ c by invertible elements u ∈ GL n (A µ ) and -iii) projective limits with respect to µ ∈ N.

To understand the relationship between our definition of the group T loc 1 and K 1 , recall that the commutator sub-group [GL(A, GL(A] is given by arbitrary products of multiplicative commutators [A, B] := ABA -1 B -1 , f or any A, B ∈ GL n (A).

On the other side, supposing that A and B are conjugated, i.e. A = UBU -1 , we have

A = UBU -1 = UBU -1 .B -1 B = [U, B].B.
This shows that if A and B are conjugated, they differ, multiplicatively, by a commutator. To complete this remark, we say that A and B are locally conjugated provided A, B and U belong to some GL n (A µ ); here, A µ denote the terms of the filtration of A.

It is important to note that in the particular case A = C the quotient of GL(M(C)) through the commutator sub-group GL(M(C)), GL(M(C))] gives much less information than T loc 1 (M(C)).

Factorisation of u ⊗ u -1

Proposition 11 In the algebra of matrices one has the identity

u 0 0 u -1 = 1 u 0 1 1 0 -u -1 1 1 u 0 1 0 -1 1 0 . ( 27
)
This formula will play an important role in the construction of T * -theory. 9 Algebraic T i and T loc i -theory.

9.1 Generalities and Notation.

Let A be a ring, with or without unit. If the unit will be needed, the unit will be adjoined.

Definition 12

Given the ring A we denote by M n (A) the space of n × n matrices with entries in A. M n (A) is a bi-lateral A-module.

Let Idemp n ⊂ M n (A) be the subset of idempotents p (p 2 = p) of size n with entries in A.

Suppose A has an unit. We denote by GL n (A) the sub-space of matrices M of size n, with entries in A which are invertible, i.e. there exists the matrix M -1 ∈ M n (A) such that MM -1 = M -1 M = 1. GL n (A) is a non-commutative group under multiplication.

Definition 13 The inclusions -i) Idemp n (A) -→ Idemp n+1 (A) p → p ′ = p 0 0 0 (28) -ii) GL n (A) -→ GL n+1 (A) u → u ′ = u 0 0 1 ( 29
)
are stabilisations.

Stabilisations of idempotents -i) and invertible matrices -ii) define two direct systems with respect to n ∈ N.

-iii) If p or p ′ are idempotents and one of them is an iterated stabilisation of the other we write p ∼ s p ′ . If u or u ′ are invertible matrices and one of them is an iterated stabilisation of the other we write u ∼ s u ′ .

Localised rings

We recall the definition of localised rings given before.

Definition 14 Let A be an unital associative ring. The ring A is called localised ring provided it is endowed with an additional structure satisfying the axioms (1) -( 4) below. Axiom 1. The underlying space A has a decreasing filtration by sub-spaces

{A µ } n∈N ⊂ A. Axiom 2. C.1 ⊂ A µ , f or any µ ∈ N Axiom 3. For any µ, µ ′ ∈ N + , A µ . A µ ′ ⊂ A Min(µ,µ ′ )-1 , (A 0 . A 0 ⊂ A 0 ).
Definition 15 Homomorphisms of localised rings. Induced homomorphism.

-i) A homomorphism from the localised ring A = {A µ } µ∈N to the localised ring B = {B µ } µ∈N is an ring homomorphism φ : A -→ B such that φ : A µ -→ B µ , for any µ ∈ N.

-ii) Let f : A -→ B be a localised ring homomorphism. Let f * : M n (A µ ) -→ M n (B µ ) be the induced homomorphism which replaces any component a ij of the matrix M with the component f (a ij ) of the matrix f * (M).

Remark 16

The notion of localised ring differs from the notion of m-algebras defined by Cuntz [START_REF] Cuntz | Cyclic Theory, Bivariant K-theory and the bivariant Chern-Connes character, Operator Algebras in Noncommutative Geometry II[END_REF], in many respects. The sub-spaces A n are not required to be algebras, or even more, topological algebras. In the Cuntz' definition of localised Banach algebras, the projective limit of these sub-algebras might be the zero Banach algebra. However, even in these cases, the corresponding local T -theory could be not trivial.

Remark 17

In this section any localisation of the algebra of bounded operators on the Hilbert space H := L 2 (M × M) derives from a decreasing filtration {U µ }, towards the diagonal, of the space of bounded operators on M × M. In such a case, if µ ′ > µ, one has the close inclusions

L 2 (U µ ′ ) ⊂ L 2 (U µ ) ( 30
)
and any internal authomorphism of the algebra B(U µ ′ ) is an internal authomorphism of the algebra B(U µ ) (any auto-morphism of the algebra L 2 (U µ ) will be extended by the identity on the complement of the Hilbert space L 2 (U µ ′ )).

Remark 18

The immediate application of this theory regards pseudodifferential operators. The pseudo-differential operators on a compact smooth manifold form a localised (Banach) algebra. The filtration is defined in terms of the support of the operators; the bigger the filtration order is, the smaller the supports of the operators towards the diagonal are.

11 Local Mayer-Vietoris Diagrams.

In this section we adapt Milnor's [START_REF] Milnor | Introduction to Algebraic K-Theory[END_REF] desciption of the first two algebraic K-theory groups to the case of localised rings.

Let Λ, Λ 1 , Λ 2 and Λ ′ be rings with unit 1 and let

Λ i 1 ---→ Λ 1   i 2   j 1 Λ 2 j 2 ---→ Λ ′ (31)
be a commutative diagram of ring homomrphisms. All ring homomorphisms f are assumed to satisfy f (1) = 1. Any module in this paper is a left module.

We assume the diagram satisfies the three conditions below. Hypothesis 1. All rings and ring homomorphisms are localised, see Definition 1.

Hypothesis 2. Λ is a local product of Λ 1 and Λ 2 , i.e. for any pair of elements

λ 1 ∈ Λ 1,µ and λ 2 ∈ Λ 2,µ such that j 1 (λ 1,µ ) = j 2 (λ 2,µ ) = λ ′ ∈ Λ ′ µ , there exists only one element λ n ∈ Λ µ such that i 1 (λ µ ) = λ 1,µ and i 2 (λ µ ) = λ 2,µ .
The ring structure in Λ is defined by

(λ 1 , λ 2 )+(λ ′ 1 , λ ′ 2 ) := (λ 1 +λ ′ 1 , λ 2 +λ ′ 2 ), (λ 1 , λ 2 ).(λ ′ 1 , λ ′ 2 ) := (λ 1 .λ ′ 1 , λ 2 .λ ′ 2 ), (32)
i.e. the ring operations in Λ are performed component-wise.

Hypothesis 3. At least one of the homomorphisms j 1 and j 2 is surjective. (21.6) induce onto the space of double matrices, (resp. the space M(Λ 1 ) × M(Λ 2 )) the following relations

Remark 19 -i) Any matrix M ∈ M n (Λ) consists of a pair of matri- ces (M 1 , M 2 ) ∈ M n (Λ 1 ) × M n (Λ 2 ) subject to the condition j 1, * M 1 = j 2, * M 2 . Any matrix M ∈ M n (Λ) is called double matrix. -ii) if (M 1 , M 2 ), (N 1 , N 2 ) are double matrices, (resp. belong to M(Λ 1 ) × M(Λ 2 )) and (λ 1 , λ 2 ) ∈ Λ, (resp. (λ 1 , λ 2 ) ∈ Λ 1 × Λ 2 ) then relations
(λ 1 , λ 2 ) (M 1 , M 2 ) = (λ 1 M 1 , λ 2 M 2 ) (M 1 , M 2 ) + (N 1 , N 2 ) = (M 1 + N 1 , M 2 + N 2 ) (M 1 , M 2 ) . (N 1 , N 2 ) = (M 1 . N 1 , M 2 . N 2 )
Definition 20 A commutative diagram satisfying Hypotheses 1. 2.

will be called local Mayer-Vietoris diagram.

Standing Hypothesis. In the remaining part of this chapter we assume that the ring A is localised. We assume also that

Λ 1 = Λ 2 = A and that J ⊂ A is a bi-lateral ideal. Define Λ := {(λ 1 , λ 2 ) ∈ A ⊕ A, such that λ 1 -λ 2 ∈ J}. Denote i α (λ 1 , λ 2 ) = λ α , α = 1, or 2.

Denote also Λ

′ := A/J and j α (λ 1 , λ 2 ) := λ α mod. J. One has the ring short exact sequence

0 -→ Λ (i 1 ,i 2 ) -→ Λ 1 ⊕ Λ 2 j 1 -j 2 -→ Λ ′ -→ 0.
We assume that the above scheme is a localised Mayer -Vietoris diagramme.

12 Preparing the definition of T loc 0 (A) and T loc 1 (A).

Definition 21 We assume the ring A is localised. We consider the space of matrices with entries in A µ and we denote it by M n (A µ ).

Let Idemp n (A µ ) denote the space of idempotent matrices of size n with entries in A µ .

Let GL n (A µ ) denote the space of invertible matrices M of size n with the property that the entries of both M and M -1 belong to A µ .

Let

Idemp(A µ ) := inj lim n Idemp n (A µ ). Let GL(A µ ) := inj lim n GL n (A µ ).
Definition 22 -i) Two matrices s, t ∈ M n (A) are called conjugated and we write s ∼ c t, provided they are related through an inner auto-morphism, i.e. there exists u, u -1 ∈ GL n (A) such that s = utu -1 .

-ii) Two matrices s, t ∈ M n (A µ ) will be called locally conjugated and we write s ∼ lc t provided they are related through a local inner auto-morphism defined by u, u -1 ∈ GL n (A µ ) such that s = utu -1 .

In particular, -ii.1) two idempotents p, q ∈ Idemp n (A µ ) are locally conjugated and we write p ∼ lc q provided there exists u, u -1 ∈ GL n (A µ ) such that q = u p u -1 and -ii.2) two invertible matrices s, t ∈ GL n (A µ ) are locally conjugated and we write s ∼ lc t provided there exists u, u

-1 ∈ GL n (A µ ) such that s = utu -1 .
Sometimes, if it is clear from the context, ∼ lc will be simply denoted by ∼ c .

Proposition 23 -i) Idemp n (A µ ) and GL n (A µ ) are semigroups with respect to the direct sum

A + B := A 0 0 B (33)
-ii) The operations of stabilisation, conjugation and local conjugation of idempotents, resp. invertible matrices, commute.

The spaces Idemp n (A), GL n (A µ ) are compatible with stabilisations.

The direct sum addition of idempotents, resp. invertibles, is compatible with the local conjugation equivalence relation. Indeed, if s 1 , s 2 are conjugated through an inner auto-morphism defined by the element u 1 (s 1 ∼ l s 2 ) and t 1 , t 2 are conjugated through an inner auto-morphism defined by the element u 2 (t 1 ∼ l t 2 ), then (s 1 +t 1 ) ∼ l (s 2 +t 2 ) are conjugated through the inner auto-morphism

u 1 ⊕ u 2 .
With this observation, the associativity of the addition is now immediate.

These show that Idemp(A µ ) / ∼ lc , resp. GL(A µ ) / ∼ l , is an associative semi-group.

Proposition 24 The semi-groups

Idemp(A) / ∼ c , GL(A µ ) / ∼ lc are commutative.
Proof. The result follows from the following identity valid for any

two matrices A, B ∈ M n (A µ ) A 0 0 B = 0 -1 1 0 B 0 0 A 0 1 -1 0 = 0 -1 1 0 B 0 0 A 0 -1 1 0 -1 , (34) which tells that (A + B) ∼ c (B + A), resp. (A + B) ∼ lc (B + A),
For more information about the relationship between the classical algebraic K-theory and the local T -theory see 8.

13 Definition of T 0 (A µ ) and T loc 0 (A).

We suppose the stabilisation is involved without being specified.

Definition 25 Suppose A is a localised unital associative ring. We define -i)

T 0 (A µ ) = G (Idemp n (A µ ) / ∼ c ). ( 35
) -ii) T 0 (A) = P rojLim µ∈N T 0 (A µ ). ( 36
)
where G is Grothendieck completion.

Any local inner automorphism induces the identity on T loc 0 (A). Proposition 26 For any unital associative algebra A, trivially localised (A µ = A), T 0 (A) = K 0 (A).

Proof. See Rosenberg [START_REF] Rosenberg | Algebraic K-Theory and its Applications[END_REF] Lemma 1.2.1.

14

T 1 (A µ ) and T loc 1 (A).

As in the previous sub-section, we assume the stabilisation is involved without being specified.

The equivalence class of the invertible element u ∈ GL(A) modulo conjugation, [u] ∼c , will be called the abstract Jordan canonical form of u. The group T 1 (A µ ) we are going to define preserves much of the information provided by the abstract Jordan form. The classical definition of K 1 extracts a minimal part of the abstract Jordan form. As the addition in the semi-group GL(A) is given by the direct sum and the Jordan canonical form J(u) (in the classical case of the algebra GL n (R)) J behaves additively (J(u ⊕ v) = J(u) ⊕ J(V ) modulo permutations of the Jordan blocks), given an arbitrary element u ∈ GL(A), it is not reasonable to expect existence of an element ũ such that [u + ũ] ∼c = [1 2n ] ∼c . Given that we want T 1 (A µ ) to be a group, we introduce the group structure (opposite elements) forcibly. In the case of the classical K 1 -theory, the class of the element u -1 represents the opposite class, -[u] ∈ K 1 (A). In our case, the opposite elements will be introduced by means of the Tcompletion technique, to be explained next.

T -completion

Proposition 27 Let S be an additive commutative semi-group with zero element 0. Let I : S -→ S be an additive involution, (I 2 = Id), such that I(0) = 0.

Define the equivalence relation

∼ O in S: u ∼ O v iff these exist two elements u 0 , u 1 ∈ S, such that the elements u, v, ξ 0 = u 0 + I(u 0 ) and ξ 1 = u 1 + I(u 1 ), satisfy u + ξ 0 = v + ξ 1 . (37)
O(S) := {O(u) | u ∈ S } (40)
Proof. -i) Obvious.

-ii) ∼ is clearly reflexive and symmetric. If

u 1 + ξ 1 = u 2 + ξ 2 and u 2 + ξ 3 = u 3 + ξ 4 , u 1 , u 2 , u 3 , u 4 ∈ O(S), then u 1 + (ξ 1 + ξ 3 ) = u 2 + (ξ 2 + ξ 3 ) = u 3 + (ξ 2 + ξ 4 )
, which shows that ∼ is transitive. Therefore ∼ is an equivalence relation.

-iii) One the other side, for any element u ∈ S, one has

[u] = [u + 0] = [u] + [0] = [0] + [u], which shows that the class of [0] is a the zero element of S/ ∼ O . We have also [u] + [I(u)] = [O(u)] = [0 + O(u)] = [0] because 0 ∼ (0 + O(u)); therefore, -[u] = [I(u)] exists. From this we get further [O(u)] = [u + I(u)] = [u] + [I(u)] = 0.
16 Definition of T 1 (A µ ) and T loc 1 (A)

The construction of the groups T 1 (A µ ) and T loc 1 (A) involves the Tcompletion §15. We take

• S = GL(A µ ) ∼ c • the involution I : GL(A µ ) ∼ cl -→ GL(A µ ) ∼ cl is given by I(u) = u -1
• the identity element I ∈ GL(A µ )/ ∼ c becomes the zero element of S.

Here ∼ l indicates that the invertible elements and their conjugation occurs locally. Note that if

u 1 ∼ cl u 2 , then u -1 1 ∼ cl u -1 2 .

The next proposition summarises the properties of O(A µ )

Proposition 29 -i) The space O(A µ ) is a commutative group; the zero element is the class if the identity -ii) the mapping

O : GL(A µ ) -→ O(A µ ) (41)
is additive and commutes with local conjugation

O(u 1 + u 2 ) = O(u 1 ) + O(u 2 ) (42) O(λ u λ -1 ) = λ O(u) λ -1 , λ ∈ GL(A µ ), (43) i
.e. if u 1 ∼ sl u 2 then O(u 1 ) ∼ sl O(u 2 ) -iii) O(u -1 ) ∼ c O(u). ( 44
) -iv) O(u 1 u 2 ) = O(u 1 )O(u 2 ), (45)
Definition 30

T (A µ ) := T -completion of GL(A µ )/ ∼ lc (46) and T loc (A) := proj lim µ∈N T 1 (A µ ) (47) Proposition 31 -i) T 1 (A µ ) and T loc 1 (A) are commutative groups. -ii) The inverse element of [u] is -[u] = [u -1 ].
Proof. -i) and -ii) follow from the properties of the T-completion, see Proposition 21.5 §15.

For the computation of the local algebraic T -theory, i = 0, 1, of the algebra of complex numbers C see Teleman [START_REF] Teleman | [END_REF].

For the computation of the local cyclic homology of the algebra of Hilbert-Schmidt operators on simplicial spaces see [START_REF] Teleman | Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces[END_REF].

For the local index theorem see [START_REF] Teleman | The Local Index Theorem[END_REF], [START_REF] Teleman | Local 3 Index Theorem[END_REF].

17 Induced homomorphisms.

Definition 32 Let f : A -→ B be a localised ring homomorphism.

Then the ring homomorphism f induces homomorphisms

f * : T 0 (A µ ) -→ T 0 (B µ ), f * : T loc 0 (A) -→ T loc 0 (B) (48)
and

f * : T 0 (A µ ) -→ T 0 (B µ ), f * : T loc 1 (A) -→ T loc 1 (B) (49)
18 Constructing idempotents and invertible matrices over Λ µ .

We come back to the situation presented in the section 11. Here we produce local idempotents and invertibles of the ring Λ. These results will be used in the proof of the six terms exact sequence Theorem 7.

18.1 Constructing idempotents over Λ µ .

Theorem 33 Let p 1 ∈ Idemp n (Λ 1,µ ) be an idempotent matrix with entries in Λ 1,µ , resp. p 2 ∈ Idemp n (Λ 2,µ ), such that

j 1 * (p 1 ) = u j 2 * (p 2 ) u -1 , (50)
where u ∈ GL n (Λ ′ µ ) is an invertible matrix.

-i) Then there exists an idempotent double matrix p ∈ Idemp(Λ µ )

such that i 1 * (p) = p 1 ⊕ 0 n and i 2 * (p) = p (51)
where the idempotent p2

∈ M 2n (Λ 2,µ ) is conjugated to p 2 ⊕0 n through an invertible matrix Ũ (u) ∈ GL 2n (Λ 2,µ ), that is p2 = Ũ(u)(p 2 ⊕ 0 n ) Ũ (u) -1 (52) j 2, * (p 2 ) = (u j 1, * (p 1 ) u -1 ) ⊕ 0 n = j 1, * (p 1 ⊕ 0 n ) ( 53
)
-ii) The corresponding double matrix idempotent is denoted p = (p 1 , p 2 , Ũ (u)).

Definition 34 Let Ũ(u) be the lifting (27

) of u ⊗ u -1 in Λ 2,µ .
Denote by p = (p 1 , p 2 , Ũ (u)) the idempotent over Λ µ produced by Theorem 33.

Condition (50) says that [j 1 * p 1 ] = [j * p 2 ] ∈ T 0 (Λ ′ µ ). Part -i) says that the pair ([p 1 ⊕ 1 n ], [p 2 ]) ∈ T 0 (Λ 1 ) ⊕ T 0 (Λ 2 ) belongs to the image of (i 1 * , i 2 * ).
Proof. To prove this theorem we will operate onto objects related to Λ 2,µ .

Lemma 35 Let p 1 = (a ij ) ∈ Idemp n (Λ 1,µ ) and p 2 = (b ij ) ∈ Idemp n (Λ 2,µ ) be idempotents.
Suppose the idempotents j 1 * (p 1 ), j 2 * (p 2 ) are conjugated through an inner automorphism defined by u ∈ GL n (Λ ′ µ ), i.e.

j 1 * (p 1 ) = u j 2 * (p 2 ) u -1 . (54)
Assume, additionally, that the invertible element u lifts to an invertible element ũ ∈ GL n (Λ 2,µ ) (i.e. j 2 * ũ = u).

Then p = (p 1 , p ′ 2 , u) ∈ Idemp n (Λ µ ) is an idempotent given by the double matrix p = ((a ij , c ij )), (55)
where

(a ij ) = p 1 ∈ Idemp n (Λ 1,µ ) and p ′ 2 = (c ij ) := ũ p 2 ũ-1 ∈ Idemp n (Λ 2,µ ). ( 56
)
remark that in this lemma the size of the double matrix p does not change.

Proof of Lemma 35. We use Remark 19-ii). It is clear that the matrix p given by ( 55) is an idempotent. In fact, to evaluate p 2 amounts to compute separately the square of the first and second component matrices of the matrix p, i.e. the squares of (a ij ) and (c ij ). These are

(a ij ) 2 = (a ij ) and (c ij ) 2 = ( ũ p 2 ũ-1 ) 2 = ũ p 2 2 ũ-1 = ũ p 2 ũ-1 = (c ij ).
(57) It remains to verify that p ∈ M n (Λ µ ), i.e. j 1 * (a ij ) = j 2 * (c ij ). This follows from (40) combined with (54)

j 1 * (p 1 ) = u j 2 * (p 2 ) u -1 = j 2 * (ũ p 2 ũ-1 ) = j 2 * (p 2 ). ( 58
)
This ends the proof of Lemma 35.

Lemma 36 Let p 1 = (a ij ) ∈ Idemp n (Λ 1,µ ) and p 2 = (b ij ) ∈ Idemp n (Λ 2,µ ) be idempotents.
Suppose the idempotents j 1 * (p 1 ), j 2 * (p 2 ) are conjugated through an inner automorphism defined by u

∈ GL n (Λ ′ µ ), i.e. j 1 * (p 1 ) = u j 2 * (p 2 ) u -1 . (59)
Then -i) j 1 * (p 1 ⊕ 0 n ) and j 2 * (p 2 ⊕ 0 n ) are conjugated by U := u ⊕ u -1 ∈ GL 2n (Λ ′ µ ), i.e. j 1 * (p 1 )⊕0 n = j 1 * (p 1 ⊕0 n ) = U j 2 * (p 2 ⊕0 n ) U -1 = ( u j 2 * (p 2 ) u -1 )⊕0 n .
(60) -ii) Supposing that j 2 is surjective, the invertible matrix U lifts to an invertible matrix Ũ ∈ M 2n (Λ 2,µ ). Let

p2 := Ũ (p 2 ⊕ 0 n ) Ũ-1 . (61)
Then

j 1, * (p 1 ⊕ 0 n ) = (u j 2, * (p 2 ) u -1 ) ⊕ 0 n = j 2, * (p 2 ); (62)
i.e. the matrices p 1 ⊕ 0 n , p2 form a double matrix idempotent in Idemp 2n (Λ µ ), denoted p := (p 1 , p 2 , u) ∈ Idemp 2n (Λ µ ) and

(i 1 * , i 2 * )p = (p 1 , p2 ) (63)
-iii) The pair of idempotents

p 1 ∈ Idemp n (Λ 1,µ ), p 2 ∈ Idemp n (Λ 2,µ
) is stably equivalent to the pair of idempotents p 1 ⊕ 0 n , p 2 ⊕ 0 n and p 2 ⊕ 0 n ∼ l p2 . In other words

([p 1 ], [p 2 ]) = ([p 1 ], [p 2 ]) ∈ T 0 (Λ 1,µ ) ⊕ T 0 (Λ 2 , µ). ( 64
)
Note that in this lemma, in comparison with the preceding Lemma 35, the size of the desired idempotent doubles; otherwise, the important modifications still occur onto matrices associated with Λ 2,µ .

Proof of Lemma 36 . Part -i) is clear.

The proof of -ii) uses O 2n (u), where u ∈ GL n (A µ ). Recall that O 2n (u) may be written as a product of elementary matrices and a scalar matrix (27)

U := O 2n (u) = u 0 0 u -1 = 1 u 0 1 1 0 -u -1 1 1 u 0 1 0 -1 1 0 ∈ M 2n (Λ 2,µ ).
(65) Having in mind the localisation of the ring A, it is important to note that the entries of the formula (65) depend on u and u -1 only.

The proof will be complete after we will have shown that the invertible matrix U has an invertible lifting Ũ ∈ Λ 2,µ . This follows from the properties of elementary matrices (valid for matrices and block matrices) to be discussed next.

Definition 37 Elementary Matrices.

A matrix E ij (a) ∈ GL n (A µ ) having all entries equal to zero, except for the diagonal entries equal to 1 and just one (i, j)-entry a ∈ A µ , 0 ≤ i = j ≤ n is called elementary matrix with entry in A µ .

The space of elementary matrices with entries in

A µ is by defi- nition E n (A µ ) := {E ij (a) | 1 ≤ i = j ≤ n, a ∈ A µ }. ( 66
)
Let E n (A) the sub-group generated by all elementary matrices and

E(A) := inj lim n∈N E n (A). ( 67
)
Lemma 38 Begin The elementary matrices satisfy

E i,j (a).E i,j (b) = E i,j (a + b) (68)
E i,j (a) -1 = E i,j (-a), (69)
therefore E i,j (a) ∈ GL(A µ ). Any elementary matrix is a commutator

E ij (A) = [E ik (A), E kj (1)], f or any i, j, k distinct indices. ( 70
)
We come back to the proof of Lemma 36 -ii). As j 2 is surjective, each of the entries of factors of the RHS of (65) has a lifting in M 2n (Λ 2µ ); each of the elementary matrix factor lifts as an invertible elementary matrix. The last factor lifts as it is. Therefore, U has an invertible lifting Ũ ∈ GL 2n (Λ 2,µ ). Lemma 35 completes the proof of Lemma 36 -ii).

Lemma 36 -iii) follows from the definition of T loc 0 (Λ). This completes the proof of Lemma 21.2.

Theorem 33 follows from Lemma 34 combined with Lemma 35.

Theorem 33 refers to the construction and description of idempotents over Λ µ . We need to extend Theorem 33 to elements of T loc 0 (A), i.e. to formal differences of local idempotents.

Lemma 39 (compare [START_REF] Milnor | Introduction to Algebraic K-Theory[END_REF] Lemma 1.1) Let p 1 , p 2 q 1 , q 2 ∈ Idemp n (A µ ) be idempotents and let [ ] denote their T 0 (A µ ) class. Suppose

[p 1 ] -[p 2 ] = [q 1 ] -[q 2 ] ∈ T 0 (A µ ). ( 71
)
Then p 1 +q 2 and p 2 +q 1 are locally, stably isomorphic, p 1 +q 2 ∼ ls p 2 + q 1 .

Proof. The stabilisation and Grothendieck completion imply that there exists an idempotent s ∈ Idemp m (A µ ) such that the idempotents p 1 + q 2 + s, p 2 + q 1 + s are locally, stably isomorphic. We assume that the idempotent s is already sufficiently stabilised. This means there exists an invertible matrix u ∈ GL 2n+m (A µ ) such that

p 1 + q 2 + s, = u (p 2 + q 1 + s) u -1 .
We add to both sides of this equality the idempotent 1 m -s and we extend u to be the identity on the last summand. We get

p 1 + q 2 + s + (1 -s), = u (p 2 + q 1 + s + (1 -s)) u -1 .
From this we get further

p 1 + q 2 + 1 2m , = u (p 2 + q 1 + 1 2m ) u -1 ,
that is, the idempotents p 1 + q 2 , p 2 + q 1 are locally, stably isomorphic

p 1 + q 2 ∼ sl p 2 + q 1 .
Lemma 40 Let p, q ∈ Idemp n (A µ ) be idempotents. Suppose

[p] -[1 n ] = [q] -[1 n ] ∈ T 0 (A µ ). ( 72
)
Then -i) p and q are locally, stably isomorphic, p ∼ ls q, -ii) there exists an N ∈ N and an u ∈ GL n+N (A µ ) such that

p + 1 N = u q u -1 + 1 N = u (q + 1 N ) u -1 . (73)
Proof. -i) Lemma 39 says that the idempotents p + 1 n , q + 1 n are locally, stably isomorphic. This means that the idempotents p and q are locally, stably isomorphic. Part -ii) tells precisely this.

Theorem 41 Let p ij be idempotents

[p 1 ] = [p 11 ] -[p 12 ] ∈ T 0 (Λ 1,µ ) [p 2 ] = [p 21 ] -[p 22 ] ∈ T 0 (Λ 2,µ )
with the property that

j 1 * [p 1 ] = j 2 * [p 2 ] ∈ T 0 (Λ ′ µ ). Then there exists [p] = [p 01 ] -[p 02 ] ∈ T 0 (Λ µ ) with the property that i 1 * [p] = [p 1 ] and i 2 * [p] = [p 2 ].
Proof. We may describe the two T -theory classes differently

[p 1 ] = [p 11 ]-[p 12 ] = [p 11 +(1-p 12 )]-[p 12 +(1-p 12 )] = [p ′ 12 ]-[1 n ] ∈ T 0 (Λ 1,µ ) and [p 2 ] = [p 21 ]-[p 22 ] = [p 21 +(1-p 22 )]-[p 22 +(1-p 22 )] = [p ′ 22 ]-[1 n ] ∈ T 0 (Λ 2,µ ). Then j 1 * [p 1 ] = j 1 * ([p ′ 12 ] -[1 n ]) = (j 1 * [p ′ 12 ]) -[1 n ] and j 2 * [p 2 ] = j 2 * ([p ′ 22 ] -[1 n ]) = (j 2 * [p ′ 22 ]) -[1 n ].
The hypothesis says that

(j 1 * [p ′ 12 ]) -[1 n ] = (j 2 * [p ′ 22 ]) -[1 n ].
Lemma 39 says that the idempotents j 1 * [p ), u). The desired idempotents are

p 10 = p = (j 1 * (p ′ 12 ), (j 2 * (p ′ 2 ), u) ∈ Idemp(Λ µ ) p 20 = 1 N ∈ Idemp(Λ µ ).
18.2 Constructing invertibles over Λ µ . Theorem 42 Let s 1 ∈ GL n (Λ 1,µ ), s 2 ∈ GL n (Λ 2,µ ), be invertible matrices with entries in Λ 1,µ , resp. Λ 2,µ , such that

j 1 * (u 1 ) = u j 2 * (u 2 ) u -1 , (75)
where u ∈ GL n (Λ ′ µ ). -i) Then there exists an invertible matrix s ∈ GL 2n (Λ µ ) such that

i 1 * (s) = s 1 ⊕ 1 n and i 2 * (s) = s2 ( 76
)
where the invertible matrix s2 ∈ GL 2n (Λ 2,µ ) is conjugated to s 2 ⊕ 1 n through an inner auto-morphism defined by the invertible matrix

Ũ ∈ GL 2n (Λ 2,µ ), that is s2 = Ũ (s 2 ⊕ 1 n ) Ũ-1 (77) j 2, * s2 = (u j 2, * (s 2 ) u -1 ) ⊕ 1 n = j 1, * (s 1 ⊕ 1 n ) ( 78
)
-ii) The corresponding invertible double matrix s is denoted s = (s 1 , s 2 , Ũ ).

Proof. The proof of Theorem 42 goes along the same lines as the proof of Theorem 33.The proof of Theorem 33 is based on the following facts: -a) operations with double matrices respect Remark 19 -ii), lifting of the invertible element

U = O 2n (u) := u ⊕ u -1 ∈ GL 2n (Λ 2,µ ) (u ∈ GL n (Λ ′ 2,µ
)) by means of the factorisation of U by elementary matrices, see [START_REF] Teleman | Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces[END_REF], and -c) the fact that any inner automorphism keeps invariant any zero vector sub-space.

To prove Theorem 42 we use the same arguments -a), -b), -c) with the following changes: idempotents are replaced by invertible elements and for -c) we use the fact that the inner auto-morphisms transform the mapping 1 n into itself. This ends the proof of the theorem.

The next theorem is the analogue of Theorem 41 in the T 1 (A µ ) case.

Theorem 43 Suppose j 1 and j 2 are epi-morphisms.

Let

[s 1 ] ∈ T loc 1 (Λ 1,µ ) and [s 2 ] ∈ T loc 1 (Λ 2,µ ) be such that j 1 * [s 1 ] = j 2 * [s 2 ] ∈ T loc 1 (Λ ′ µ ). ( 79
)
Then there exists

[s] ∈ T loc 1 (Λ µ ) such that i 1 * [s] = [s 1 ] ∈ T loc 1 (Λ 1,µ ) and i 2 * [s] = [s 2 ] ∈ T loc 1 (Λ 2,µ ). ( 80
)
Proof. The definition of T loc 1 (Λ ′ µ ) involves an ambiguity belonging to the sub-module O(Λ ′ µ ). We assume the elements s 1 and s 1 are sufficiently stabilised. Equality (79) tells there exist two elements ξ 1 , ξ 2 ∈ O 2n (Λ ′ µ ) such that the invertible matrices j 1 * (s 1 ) + ξ 1 , j 2 * (s 2 )+ξ 2 are locally conjugated by means of a matrix u ∈ GL 2n (Λ ′ )

j 1 * (s 1 ) + ξ 1 = u (j 2 * (s 2 ) + ξ 2 ) u -1 . (81)
The problem of finding the element s will be split in two separate problems

1. find ξ1 ∈ O(Λ 1,µ ), resp. ξ2 ∈ O(Λ 2,µ ), lifts of the elements ξ 1 , resp. ξ 2 ,
2. apply Theorem 43 with s 1 , resp. s 2 , replaced by s 1 + ξ1 , resp. s 2 + ξ2 , and the invertible element u.

1. The first lift is obtained in two steps: -i) we find invertible lifts ξ1 ∈ Λ 1,µ , resp. ξ2 ∈ Λ 2,µ , of the elements ξ 1 , resp. ξ 2 , and we verify that -ii) such lifts belong to O 2n (Λ i,µ ), i = 1, 2.

As the lifts of the elements ξ i belong to O 2n (Λ i,µ ), the lifted elements will not change the corresponding T -completion classes.

Lifting of O(Λ

′ µ )
We illustrate the procedure on the element ξ 1 ; for ξ 2 we use the same procedure.

The element ξ 1 has the form

ξ 1 = α 1 0 0 α -1 1 = α 1 ⊕ α -1 1 ∈ O 2n (Λ ′ µ ). ( 82
)
We use the factorisation [START_REF] Teleman | Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces[END_REF] to decompose of ξ 1

ξ 1 = α 1 0 0 α -1 1 = 1 α 1 0 1 1 0 -α -1 1 1 1 α 1 0 1 0 -1 1 0
.

(83) The homo-morphism j 1 being an epi-morphism, there exists α1 , resp. β1

∈ Λ 1,µ such that j 1 ( α1 ) = α 1 , resp. j 1 ( β1 ) = α -1 1 . The lifted element is ξ1 = 1 α1 0 1 1 0 -β1 1 1 α1 0 1 0 -1 1 0 , ( 84
)
j 1 ( ξ1 ) = α 1 0 0 α -1 1 = ξ 1 .
We use formula [START_REF] Teleman | Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces[END_REF] and property (69) of elementary matrices to find the inverse of ξ1 ξ-1

1 = 0 1 -1 0 1 -α 1 0 1 1 0 β1 1 1 -α 1 0 1 . (85)
We have

j 1 ( ξ-1 1 ) = α 1 0 0 α -1 1 -1 ∼ cl ξ -1 1 because j 1 is a unital ring homo-morphism. The lifted element is ξ1 = ξ1 ⊕ ξ-1 1 ∈ M 4n (Λ 1,µ ).

Back to the proof of Theorem 43

The elements s 1 + ξ1 ∈ GL 4n (Λ 1,µ ), s 2 + ξ2 ∈ GL 4n (Λ 2,µ ) and the invertible element u satisfy the relation

j 1, * (s 1 + ξ1 ) = u (s 2 + ξ2 ) u -1 . (86)
Theorem 42 follows from Theorem 43.

19

K 1 (A) vs. T 1 (A)
The following identities are well known and used as building blocks of K-theory, see [START_REF] Whitehead | Generalised homology theories[END_REF], [START_REF] Milnor | Introduction to Algebraic K-Theory[END_REF], [START_REF] Karoubi | Homologie cyclique et K-Théorie[END_REF], [START_REF] Blackadar | K-Theory for Operator Algebras[END_REF], [START_REF] Rosenberg | Algebraic K-Theory and its Applications[END_REF], [START_REF] Rordam | An Introduction to K-Theory for C * -Algebras[END_REF], [START_REF] Weibel | K-theory[END_REF].

We summarise some basic facts from the classical algebraic K 1theory and compare them with T 1 .

Theorem 44 -i) Whitehead group K 1 (A) is K 1 (A) := inj lim n∈N GL n (A)/[GL n (A), GL n (A)] (87)
The group structure in K 1 (A) is given by matrix multiplication and direct sum addition

[A] + [B] := [ A.B ]. ( 88
)
T 1 (A) is the set of Jordan canonical forms of matrices over A modulo O(A). The sum in T 1 (A) is the direct sum.

-ii) Any commutator is stably isomorphic to a product of elements of the form O 2n (A). More specifically, for any A, B ∈ GL(A µ ) 70

ABA -1 B -1 0 0 1 = A 0 0 A -1 B 0 0 B -1 (BA) -1 0 0 BA . (89) -iii) If A ∈ GL n (A µ ), then (see (27)) O 2n (A) := A 0 0 A -1 = 1 A 0 1 1 0 -A -1 1 1 A 0 1 0 -1 1 0
(90) -iv) Any elementary matrix is a commutator

E ij (A) = [E ik (A), E kj (1)], f or any i, j, k distinct indices. (91)
-iv) For any A, B ∈ GL n (A µ ), A + B is stably equivalent to AB and BA modulo (multiplicatively) elements of the form O 2n (A)

A 0 0 B = AB 0 0 1 B -1 0 0 B = B -1 0 0 B BA 0 0 1 . (92)
-v) For any A, B ∈ GL n (A µ ) one has the identity

ABA -1 0 0 1 = A 0 0 A -1 B 0 0 1 A -1 0 0 A = A 0 0 A -1 B 0 0 1 A 0 0 A -1 -1 . ( 93
)
Theorem 45 (see [START_REF] Whitehead | Generalised homology theories[END_REF], [START_REF] Bass | Introduction to some Methods of Algebraic K-theory[END_REF], [START_REF] Swan | Vector bundles and projective modules[END_REF], [START_REF] Milnor | Introduction to Algebraic K-Theory[END_REF], [START_REF] Karoubi | Homologie cyclique et K-Théorie[END_REF], [START_REF] Rosenberg | Algebraic K-Theory and its Applications[END_REF] )

-i) [GL(A), GL n (A)] = E n (A) -ii) [ABA -1 B -1 ] = 0 ∈ K 1 (A) -iii.1) [A] + [B] = [ AB ] = [ BA ] = [B] + [A] ∈ K 1 (A). -iii.2) [A] + [B] = [B] + [A] ∈ T loc 1 (A). Therefore T loc 1 (A) is an Abelian group. -iv) [O 2n (A)] = [1 n ] = 0 in K 1 (A) and T loc 1 (A) -v) -[A] = [A -1 ] in K 1 (A) and T 1 (A) loc . -vi) [ABA -1 ] = [B] ∈ K 1 (A).
Proof. -i) Relation (70) says that any commutator is a product of matrices of type O n (A). Formula [START_REF] Teleman | Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces[END_REF] says that any matrix of type O n (A) is a product of elementary matrices and a scalar matrix 27. This proves -i).

-ii) Follows from the definition of K 1 (A).

-iii.1) and -iii.2). See definitions.

-iv) For any invertible element x 0 one has

(x 0 + (ξ + ξ -1 )) ∼ O (x 0 + (ξ + ξ -1 )) + (ξ + ξ -1 ) (94)
along with the fact that T 1 is a group.

-v) Relation for K 1 follows from A.A -1 = 1, which is the zero element in K 1 .

For T 1 the relation follows from -iv).

-vi) (ABA -1 ).B -1 is a commutator; it is the zero element in K 1 .

On the other side [B

-1 ] = [B] ∈ K 1 .
The next remark explains the specific parts of the construction of T loc 1 which make our construction different from the classical one.

Remark 46 -i) In our construction the factorisation of the elements of GL(A) through the commutator sub-group (or the subgroup generated by elementary matrices) is avoided because the number of products needed to generate these sub-groups might be unbounded. Having in mind the algebra of integral operators, or pseudodifferential operators, one realises that any product increases the size of the support; an un-bounded number of products does not allow a support control. For this reason, in the whole chapter we don't use more than three products of elements in the algebra. Products are replaced by sums and direct sums. The increase of the size of matrices replaces in our construction the need to perform multiple products. In our definition of T loc 1 (A), where products could not be avoided, the corresponding increase in the size of the supports is absorbed by the projective limit.

-ii) Our construction uses the elements O(A) additively and not multiplicatively. In the classical construction of K 1 (A), the elements of O(A) are used multiplicatively. Unfortunately, to generate the commutator sub-group it is necessary to perform an un-bounded number of products; in our construction an un-bounded number of products is not allowed.

-iii) The construction of T loc 1 (A) uses the factorisation of GL(A) through the smaller (than the commutator) sub-group of inner automorphisms. The class of an invertible element u modulo inner automorphisms, which is nothing but the abstract Jordan form J(u) of u, contains more information than the class of the invertible element modulo the commutator sub-group. Both, in the classical K 1 (A) and T 1 (A) the elements of O n (A) represent the zero element.

If A were the algebra of complex matrices and u ∈ M(C), then J(u) could be identified with the Jordan canonical form of the matrix u. The Jordan canonical form of the matrix u ⊕ u -1 is precisely J(u) ∪ J(u -1 ) modulo permutations of the Jordan blocks. It is clear that u ⊕ u -1 could never be conjugated to the the identity element unless u = 1 n . In general, u ⊕ u -1 could not be conjugated to

1 n ⊕ 1 n unless u ∼ s 1 n .
For the computation of the groups T loc i (C), where C is the algebra of complex numbers endowed with the trivial filtration, see [START_REF] Teleman | [END_REF] and §22.

The elements O(A) represent the zero element in T 1 .

For the definition of T loc 1 (A) we find it natural to consider the quotient space of GL(A) modulo the equivalence relation ∼ sl . This factorisation uses the additive sub-group O(A). This factorisation decrees that the Jordan canonical forms of the elements u and u -1 are opposite one to each other in T loc 1 . The additive group generated by elements O(A) is contained in the commutator sub-group; this property insures the fact that there exists a natural epi-morphism from T loc 1 (A) to K 1 (A). -iv) The factorisation through the sub-group O(A) does not appear to kill much information. A partial argument in support of this is the fact that for any u, v

∈ GL n (A), (u ⊕ u -1 ) ∼ l (v ⊕ v -1 ) if and only if (u ⊕ 1 n ) ∼ l (v ⊕ 1 n ).
-v) Additionally, the projective limit (Alexander-Spanier type construction, made possible by the filtration A µ of the algebra), makes the algebraic T loc i -theory richer than the classical K i -theory, i = 0, 1. Theorem 47 -i) There is a canonical epi-morphism

Π : T loc 1 (A) -→ K 1 (A) (95) Ker Π = [ GL(A), GL(A) ] / Inner(A). (96)
20 Connecting homo-morphism ∂ :

T loc 1 (Λ ′ ) -→ T loc 0 (Λ) ⊗ Z[ 1 2 ]
In this section we assume that the diagram 11 satisfies Hypotheses 1, 2. 3. In this section we define the connecting homomorphism

∂ : T loc 1 -→ T loc 0 ⊗ Z[ 1 2
]. The reader will notice that the construction of ∂ involves from the very beginning idempotents.

Definition 48 Let [u] ∈ T 1 (Λ ′ µ ). Recall that the elements of T 1 (Λ ′ µ ) are equivalence classes of invertible matrices modulo ∼ O Aµ ; we may assume that u ∈ GL n (Λ ′ µ ). Define the connecting homomophism ∂ : T 1 (Λ ′ µ ) -→ T 0 (A µ ) ⊗ Z[ 1 2 ] ∂[u] = [p(1 n , 1 n , Ũ(u)] -[Λ n (A µ )], (97)
where Ũ1 (u) is obtained through the decomposition of u ⊕ u -1 as in [START_REF] Teleman | Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces[END_REF], and lifted in Λ 1,µ , as in (84).

Proposition 49 Suppose the homo-morphisms j 1 , j 2 are epi-morphisms.

Then the connecting homomorphism is well defined.

We have to show that ∂ is compatible with the equivalence relation ∼ O Aµ . For, suppose that u 1 + ξ 1 = u 0 + ξ 0 , where u 0 , u 1 ∈ GL n (A µ ) and ξ 0 , ξ 1 ∈ O n (A µ ). We have to prove that ∂O(u 1 ) = ∂O(u 2 ) = trivial idempotent.

We proved in §18.3 that any invertible element ξ belonging to O n (Λ ′ µ ) lifts to an invertible elements ξ1 , resp. ξ2 , belonging to Λ µ,1 , resp. Λ µ,2 . In our specific case, the element 1 n (Λ 1,µ ), resp. 1 n (Λ 2,µ ), is conjugated through the lifted elements ξi , i = 1, 2.

The conjugation inside Λ µ,i preserves the

T 1 (Λ µ,i ) classes. Recall that the idempotents of Λ µ consist of pairs of idempotents (p 1 , p 2 ) ∈ Λ µ,1 ⊕ Λ µ,2 such that i 1 p 1 = i 2 p 2 . The idempotents p 1 = ξ1 1 n ξ-1 1 , p 2 = ξ2 1 n ξ-1 2
satisfy this condition. The proposition is proven.

21 Six terms exact sequence.

Theorem 50 7 The following sequences are exact -i)

T loc 0 (Λ) (i 1 ,i 2 ) -→ T loc 0 (Λ 1 ) ⊕ T loc 0 (Λ 2 ) j 1 * -j 2 * -→ T loc 0 (Λ ′ ) (98)
-ii)

T loc 1 (Λ) (i 1 * ,i 2 * ) -→ T loc 1 (Λ 1 ) ⊕ T loc 1 (Λ 2 ) j 1 * -j 2 * -→ T loc 1 (Λ ′ ) (99)
-iii)

T loc 1 (Λ 1 ) ⊕ T loc 1 (Λ 2 ) j 1 * -j 2 * -→ T loc 1 (Λ ′ ) ∂ -→ (100)
T loc 0 (Λ) ⊗ Z[ 1 2 ]
(i 1 * ,i 2 * ) -→ (T loc 0 (Λ 1 ) ⊗ Z[ 1 2 ]) ⊕ (T loc 0 (Λ 2 ) ⊗ Z[ 1 2 ]). (101)
Therefore, the following six terms sequence is exact

T loc 1 (Λ) (i 1 * ,i 2 * ) -→ T loc 1 (Λ 1 ) ⊕ T loc 1 (Λ 2 ) j 1 * -j 2 * -→ T loc 1 (Λ ′ ) ∂ -→ (102)
T loc 0 (Λ) ⊗ Z[ 1 2 ]
(i 1 * ,i 2 * ) -→ T loc 0 (Λ 1 ) ⊗ Z[ 1 2 ] ⊕ T loc 0 (Λ 2 ) ⊗ Z[ 1 2 ] j 1 * -j 2 * -→ T loc 0 (Λ ′ ) ⊗ Z[ 1 2 ]
(103)

Proof. -i) i.1.) Im(i 1 * , i 2 * ) ⊂ Ker(j 1 * -j 2 * ). Easy to verify. i.2.) Ker(j 1 * -j 1 * ) ⊂ Im(i 1 * , i 2 * )
We are going to verify it.

Lemma 51 Let p ∈ Idemp n (A µ ). Then p ⊕ (1 -p) ∼ ls 1 n . (104)
Proof. The proof is based on the following identity

p 0 0 1 -p = p 1 -p 1 -p p 1 n 0 0 0 p 1 -p 1 -p p (105)
along with the observation that

p 1 -p 1 -p p 2 = p 2 + (1 -p) 2 p(1 -p) + (1 -p)p (1 -p)p + p(1 -p) p 2 + (1 -p) 2 = 1 n 0 0 1 n (106) which shows that p 1 -p 1 -p p -1 = p 1 -p 1 -p p , (107)
which entitles us to say that the RHS of ( 107) is an inner automorphism.

For any idempotent p of size n we convine to write p := 1 n -p.

Let ([p 1 ] -[p 2 ] , [q 1 ] -[q 2 ]) ∈ T loc 0 (Λ 1,µ ) ⊕ T loc 0 (Λ 2,µ ) be such that 0 = j 1, * ([p 1 ] -[p 2 ]) -j 2, * ([q 1 ] -[q 2 ]),
where p 1 , p 2 , q 1 , q 2 are idempotents. The pair of T -theory classes may be re-written

0 = j 1, * ([p 1 + p2 ] -[p 2 + p2 ]) -j 2, * ([q 1 + q2 ] -[q 2 + q2 ]) or 0 = j 1, * ([p 1 + p2 ] -[1 r ]) -j 2, * ([q 1 + q2 ] -[1 s ]).
By adding all sides a trivial idempotent of sufficiently large size, we may assume that r = s is large. This relation may be re-written

0 = j 1, * ([p 1 + p2 ] -j 2, * ([q 1 + q2 ]).
This means that there exists an idempotent ξ ∈ Idemp N (Λ ′ µ ) such that the idempotents

(j 1, * ([p 1 + p2 ] + ξ) -(j 2, * ([q 1 + q2 ] + ξ)
are isomorphic. We add further the idempotent ξ := 1 N -ξ to get isomorphic idempotents

(j 1, * ([p 1 + p2 ] + ξ + ξ), (j 2, * ([q 1 + q2 ] + ξ + ξ) ∈ Idemp N (Λ ′ µ )
( N being the size of these idempotents) or

(j 1, * ([p 1 + p2 ] + 1 N ), (j 2, * ([q 1 + q2 ] + 1 N ).
This means there exists u ∈ GL N (Λ ′ ) which conjugates these two idempotents.

Theorem 41 says that there exits the idempotent p ∈ Idemp 2 N (Λ µ ) such that

i 1 * p = (p 1 + p2 + 1 N ) ⊕ 1 N and i 2 * p = U ((q 1 + q2 + 1 N ) ⊕ 1 N ) U -1 where U ∈ GL 2 N (Λ ′ µ ). This means the image of the T -theory class of p -1 N ∈ Λ µ through the pair of homomorphisms (i 1, * , i 2, * ) is (i 1, * , i 2, * )[p -1 N ] = = ( (p 1 + p2 + 1 N ) ⊕ 1 N -1 N , U ((q 1 + q2 + 1 N ) ⊕ 1 N ) U -1 -1 N ) = = ( (p 1 + p2 + 1 N ) ⊕ 1 N -1 N , U ((q 1 + q2 + 1 N ) ⊕ 1 N -1 N ) U -1 = = ([p 1 -p 2 ], [q 1 -q 2 ],
which completes the proof of the part i).

-ii) -ii.1.) Im(i 1 , i 2 ) * ⊂ Ker(j 1, * -j 1, * ). If u = (u 1 , u 2 ) ∈ GL(Λ µ ) belongs to Im(i 1 , i 2 ) * , then according to the definition of Λ µ , u may be chosen so that j 1 (u 1 ) = j 2 (u 2 ) and hence [u] ∈ Ker(j 1, * -j 2, * ).

-ii.2.)

Ker(j 1, * -j 2, * ) ⊂ Im(i 1 , i 2 ) * . Let u = (u 1 , u 2 ) ∈ Ker(j 1, * -j 1, * ). This means there exist two elements ξ 1 , ξ 2 ∈ O(Λ ′ µ ) so that j 1 (u 1 ) + ξ 1 = j 2 (u 2 ) + ξ 2 . The elements j 1 (u 1 ), j 2 (u 2 ) lift to u 1 ∈ Λ 1,µ , resp. u 2 ∈ Λ 2,µ . On the other side, see §18.3, the elements ξ 1 , resp. ξ 2 , lift to elements ξ1 ∈ O(Λ 1,µ ), resp. ξ2 ∈ O(Λ 2,µ ). This means u 1 + ξ1 ∈ Λ 1,µ and u 1 + ξ2 ∈ Λ 2,µ are so that j 1 (u 1 + ξ1 ) = j 2 (u 2 + ξ2 ).
Therefore,

[(u 1 + ξ1 ), (u 2 + ξ2 )] = [(u 1 , u 2 )] ∈ T 1 (Λ 1µ ) ⊕ T 1 (Λ 2µ ),
which proves the statement.

-iii) -iii.1.) Im(j 1 * -j 2 * ) ⊂ Ker∂.

It is sufficient to prove ∂ • j 1 * = 0. Consider u ∈ Λ 1,µ . We have to compute ∂ • j 1 (u). The result is independent of the lift of (j 1 (u)) ⊗ (j 1 (u)) -1 . We may use the lift u ⊗ u -1 . We get then

∂ • j 1 (u) = (u ⊗ u -1 )(1 n ⊕ 1 n )(u ⊗ u -1 ) -1 = 1 n ⊕ 1 n . ( 108
) -iii.2.) Ker∂ ⊂ Im(j 1 * -j 2 * ). Let u ∈ Λ ′ µ be such that ∂u = [(1 n , 1 n , U(u))] -[Λ(A n µ )] = 0. We have to prove that there exist v 1 ∈ Λ 1,µ , v 2 ∈ Λ 2,µ such that ][u = [j 1, * v 1 -j 2, * v 2 ].
The hypothesis says

[(1 n , 1 n , U(u))] = [Λ n µ ]
. By adding an idempotent q and its complementary to both sides of this equation, we get

(1 m+n , 1 m+n , U(u ⊕ 1 m )) ∼ c Λ m+n µ .
22 Relative T -groups: T i (A µ , J).

Definition 52 Let A be a localised ring and J be a localised bilateral ideal in A. Here we make reference to the Mayer -Vietoris Diagram 11; we take

Λ µ = J µ , Λ 1,µ = Λ 2,µ = A µ (112) Λ µ = {(λ 1 , λ 2 ) | λ 1 ∈ Λ 1,µ , λ 2 ∈ Λ 2,µ , λ 1 -λ 2 ∈ J µ .} (113)
and

Λ ′ µ = A µ /J µ ( 114
)
The above structure will be called (A, J ) localised ideal. Define, for i = 1, 2

j i : Λ µ -→ A/J , j i (λ 1 , λ 2 ) := λ i mod.J µ . ( 115
)
Definition 53

T loc 0 (A, J ) := Ker ( T loc 0 (Λ µ ) j 2, * -→ T loc 0 (A µ /J µ ) ) (116)
T loc 1 (A, J ) := Ker ( T loc 1 (Λ µ ) j 2 , * -→ T loc 1 (A µ /J µ ) ) (117)
Theorem 54 -i) One has the exact sequence

0 -→ J µ ι -→ A µ π -→ A µ /J µ -→ 0, ( 118
)
where ι is the inclusion and π is the canonical projection.

-ii) The exact sequence 7 becomes

T loc 1 (A, J ) i * -→ T loc 1 (A) π * -→ T loc 1 (A/J ) ∂ -→ (119)
T loc 0 (A, J ) ⊗ Z[ 1 2 ] i * -→ T loc 0 (A) ⊗ Z[ 1 2 ] π * -→ T loc 0 (A/J ) ⊗ Z[ 1 2
].

(120)

Proof. We leave the check to the reader.

Connecting homo-morphism -second form

In this section we define Mayer -Vietoris diagramms associated with elliptic operators between two different Hilbert spaces and we exibit the corrresponding connecting homomorphism.

With A and B one associates

S 0 = 1 -BA ∈ M n (Λ 1,µ-1 ), S 1 = 1 -AB ∈ M n (Λ 1,µ-1 ).
The matrices S 0 , S 1 satisfy

j 1 (S 0 ) = j 1 (S 1 ) = 0. ( 127
)
With these matrices one associates the invertible matrix (ref. [START_REF] Connes | Cyclic Cohomology, The Novikov Conjecture and Hyperbolic Groups[END_REF])

L = S 0 -(1 + S 0 )B A S 1 ∈ GL 2n (Λ 1, µ-2 ); ( 128
)
the inverse of the matrix L is

L -1 = S 0 (1 + S 0 )B -A S 1 ∈ GL 2n (Λ 1, µ-2 ). ( 129
)
Let e 1 , e 2 be the idempotents

e 1 = 1 0 0 0 ∈ Λ 1, µ-2 , e 2 = 0 0 0 1 ∈ Λ 2, µ-2 . (130)
The invertible matrix L is used to produce the idempotent

P := Le 1 L -1 = S 2 0 S 0 (1 + S 0 )B S 1 A 1 -S 2 1 ∈ Idemp 2n (Λ 1, µ-2 ). ( 131
)
with j 1 (P ) = 0 0 0 1 .

The idempotent P is used to construct the double matrix idempotent P U

P U := (S 2 0 , 0) (S 0 (1 + S 0 )B, 0) (S 1 A, 0) (1 -S 2 1 , 1) ∈ Idemp 2n (Λ 1,µ-2 ) ⊕ Idemp 2n (Λ 2,µ-2 ). ( 132
)
with

π (P U ) = 0 0 0 1 ∈ Idemp(Λ 2,µ ).
The idempotent P U satisfies (j 1, * -j 2, * )(P U ) = 0. Therefore, P U ∈ M 2n (Λ µ-2 ).

Definition 58 23 Connecting homomorphism -second form. (for K-theory see [START_REF] Connes | Cyclic Cohomology, The Novikov Conjecture and Hyperbolic Groups[END_REF]).

For any [U] ∈ T loc 1 (Λ ′ ) we define the connecting homomorphism

∂ II : T loc 1 (Λ ′ ) -→ T loc 0 (Λ) by ∂ II [U] := [P U ] -[(e 2 , e 2 )] ∈ T loc 0 (Λ). (133)
In this case it is easy to check that ∂ is compatible with the equivalence relation ∼ O . Indeed, any element ξ = u ⊕ u -1 ∈ Λ ′ µ has an invertible lifting in Λ 1, µ , see §18.3. For this reason, both A and B may be chosen to be inverse one to each other. Threfore, S 0 = 0 and S 1 = 0 and hence

P = 0 0 0 1 , ∂[U] = 0.

Connecting homomorphism and stabilisation

In this sub-section we show how ∂ depends on stabilisations. In the proof of the exactness, Theorem 7 .-iii), we will need to know how the connecting homomorphism, Definition 23, behaves with respect to stabilisations. For this purpose we consider a more general situation than that considered in the previous section §23.

Let U ∈ GL m+n (Λ ′ ) and

e (0,n) = 0 0 0 1 n ∈ M m+n (Λ ′ ) (134) be such that the diagram Λ ′ m+n U ---→ Λ ′ m+n   e (0,n)   e (0,n) Λ ′ m+n U ---→ Λ ′ m+n (135)
is commutative. This condition will be needed to show that P U ∈ Idemp m+n (Λ), formula (). The case discussed in §23 corresponds in this subsection to m = 0 .

We proceed as in §23. Let A, B ∈ GL m+n (Λ 1,µ ) be liftings of U, resp. U -1 , in M m+n,µ (Λ 1 ). Such liftings exist because we assume j 1 is surjective.

With A and B one associates S 0 = 1 -BA ∈ GL m+n (Λ 1,µ ) and S 1 = 1 -AB ∈ GL m+n (Λ 1,µ ). The matrices S 0 , S 1 satisfy j 1, * (S 0 ) = j 1, * (S 1 ) = 0.

(

) 136
With these matrices one associates the invertible matrix

L = S 0 -(1 + S 0 )B A S 1 ∈ GL 2(m+n) (Λ 1,µ-2 ); ( 137
)
the inverse of the matrix L is

L -1 = S 0 (1 + S 0 )B -A S 1 ∈ GL 2(m+n) (Λ 1,µ-2 ). ( 138
)
Let e 1 be the idempotent

e 1 = e (0,n) 0 0 0 ∈ GL 2(m+n) (Λ 1,µ ). ( 139
)
and

e 2 = 0 0 0 e (0,n) ∈ GL 2(m+n) (Λ 1,µ ). (140)
The invertible matrix L is used to produce the idempotent

P U := Le 1 L -1 = S 0 e (0,n) S 0 S 0 e (0,n) (1 + S 0 )B A e (0,n) S 0 A e (0,n) (1 + S 0 )B ∈ Idemp 2(m+n) (Λ 1,µ-2 ).
(141)

Definition 59 R(U) := P U -e 2 . (142)
The idempotent P U is used to construct the double-matrix idempotent

P U := (S 0 e (0,n) S 0 , 0) (S 0 e (0,n) (1 + S 0 )B, 0) (A e (0,n) S 0 , 0) (A e (0,n) (1 -S 0 )B, e (0,n) ) (143) ∈ Idemp 2(m+n) (Λ 1,µ-2 ⊕ Λ 2,µ-2 ). ( 144
)
The matrix P U is an idempotent in M 2(m+n) (Λ µ-2 ). We may verify directly that (j 1, * -j 2, * )P U = 0. Indeed, j 1 * is a ring homomorphism and j 1 * (S 0 ) = j 1 * (S 1 ) = 0; finally, the hypothesis (135) gives 23.1j 1 (A e (0,n) (1 -S 0 )B = Ue (0,n) U -1 = e (0,n) = j 2 (e (0,n) ). ( 145) Therefore P U ∈ M 2n (Λ µ-2 ). The fact that the matrix P U is an idempotent in M 2(m+n) (Λ µ-2 ) follows from the fact that P and e 2 are idempotents along with the discussion above.

Definition 60 Connecting homomorphism -third form.

We suppose the assumptions and constructions of §23 above are in place. For any

[U] ∈ T loc 1 (Λ ′ ) one defines the connecting homo- morphism ∂ : T loc 1 (Λ ′ ) -→ T loc 0 (Λ) by ∂[U] := [P U ] -[(e 2 , e 2 )] = (146) (0, 0) (0, 0) (0, 0) (Ae (0,n) B, e (0,n) ) -[(e 2 , e 2 )] ∈ T loc 0 (Λ). ( 147
)
It remains to follow up how P U depends of the choice of the lifts A and B of U and U -1 . A different choice of A and B has the effect of modifying the matrix L. We will show that if A ′ and B ′ are two such different lifts and L ′ is the corresponding matrix, then

L ′ = L L, with L ∈ GL 2(m+n) (Λ) (148)
and hence the corresponding idempotents P U := Le 1 L -1 , P ′ U := L ′ e 1 L ′-1 are conjugate.

To better organise the computation, we change the liftings one at the time.

We begin with A. Let à = A + T with j 1 (T ) = 0. Let S0 = 1 -B Ã, S1 = 1 -ÃB, L, P and PU be the corresponding elements. A direct computation gives

LL -1 = 1 -BT -BT B T 1 + T B (149) or L = 1 -BT -BT B T 1 + T B L. (150)
We know that L and L are invertible matrices; therefore the RHS of (??) is an invertible matrix.

The corresponding idempotent P is

P = L.e 1 . L-1 = 1 -BT -BT B T 1 + T B L.e 1 .L -1 1 -BT -BT B T 1 + T B -1 = (151) = 1 -BT -BT B T 1 + T B P 1 -BT -BT B T 1 + T B -1 (152)
and furthermore

PU = ( 1 -BT -BT B T 1 + T B , 1) P U ( 1 -BT -BT B T 1 + T B , 1) -1 . (153)
Therefore, [ PU ] = [P U ] ∈ T loc 0 (Λ). It remains to see what happens if A remains unchanged and the lifting B is changed. Let B = B + H, with j 1 (H) = 0. Let L, P and PU be the corresponding matrices. A direct computation gives

L.L -1 = 1 + ∆ 11 ∆ 12 ∆ 21 1 + ∆ 22 ∈ M 2n (Λ µ-4 ) (154)
where

∆ 11 =HA -HAHA -BAHA ∆ 12 = -2H + HAB + HAH + BAH -BAHAB -HAHAB ∆ 21 =AHA ∆ 22 = -AH + AHAB.
The RHS of (154) is a product of invertible matrices; therefore, it is an invertible matrix. Proceeding as above we get

PU = ( 1 + ∆ 11 ∆ 12 ∆ 21 1 + ∆ 22 , 1) P U ( 1 + ∆ 11 ∆ 12 ∆ 21 1 + ∆ 22 , 1) -1 . (155)
This completes the discussion about the choice of the liftings A and B.

Part III Topological index and analytical index. Reformulation of index theory.

24 Level I: Index theory at the T loc * -theory level.

Definition 61 Let J ⊂ A be a localised bi-lateral ideal of the unital ring A. There correponds the short ring exact sequence

0 -→ J µ ι -→ A µ π -→ A µ /J µ -→ 0. ( 156
)
Definition 62 The ring A µ /J µ is the analogue of the Calkin ring. If U ∈ A µ then π(U) ∈ A/J is called the symbol of U.

Consider the 6-term exact sequence in T loc -theory (119)

119T loc 1 (A, J ) i * -→ T loc 1 (A) π * -→ T loc 1 (A/J ) ∂ -→ (157)
T loc 0 (A, J ) ⊗ Z[ 1 2 ] i * -→ T loc 0 (A) ⊗ Z[ 1 2 ] π * -→ T loc 0 (A/J ) ⊗ Z[ 1 2
].

(158)

1. Topological index of u is T op T Index (u) := δ[u] ∈ T loc 0 (A) ⊗ Z[ 1 2 ], (159)
where [u] ∈ T loc 1 (A/J ).

Analytical Index of u is

An T Index (u) := R (δ[u]) = δ II [u] ∈ T loc 0 (A) ⊗ Z[ 1 2 ] ( 160
)
where δ II is the second definition of boundary map, see Definition 23.

Case 1.

Problem 63 Define significant classes of extensions ( 156)

∆(A, J ) := T op T Index (u) -An T Index(u) (161)
can be computed.

Case 2.

Conjecture 64 -1) Let M 2l be a closed compact quasi-conformal manifold. Let Ω * (M) be the algebra of differential forms on M 2l . Let H be the Hilbert space of L 2 -forms of degree l. Let be the exact sequence associated to the short exact sequence

0 -→ L (1,∞) ι -→ ΨDif f π -→ ΨDif f /L 1,∞ -→ 0. For any u ∈ GL N (ΨDif f /L 1,∞ ) one has (T op T Index) (u) ⊗ Q = (An T Index) (u) ⊗ Q. (162)
-2) Let A be an elliptic pseudo-differential operator on M. Let u = π(A) ∈ GL N (A/J ) be the image of A in the quotient space. Then

An T Index (u) ⊗ Q = T op T Index (u) ⊗ Q = (163) = 1 (2πi) q q! (2q)! (-1) dimM Ch(u) ∩ [T * M]. (164)
Ch(u) in the formula (163) is the periodic cyclic homology of u.

-3) Let A be a pseudo-differential elliptic operator on the quasiconformal manifold M. Let u be its classical symbol. Produce the residue operator R(u) (K 1 is replaced by T loc 1 ). Let f ∈ C q AS (M) be an Alexander -Spanier co-cycle; let [f ] be its co-homology class. Let τ (T * (M)) be the Todd class of M.

Then

An T Index (u) ⊗ Q [f ] = T op T Index (u) ⊗ Q [f ] = (165) = 1 (2πi) q q! (2q)! (-1) dimM Ch(u) ∪ τ (T * (M)) ∩ [T * M]. ( 166
)
25 Level II: Index Theory in local periodic cyclic homology.

Definition 65 Let A µ be a localised ring. Consider the exact sequence 156. Let u ∈ T loc 1 (A µ /J µ ).

1. Topological index of u is

T op Ch Index (u) := Chδ[u] ∈ C loc,per,λ (A), (167)
2. Analytical Index of u is

An Ch Index (u) := Ch ev (δ [u]) = Ch ev (δ II [u]) ∈ H loc,per,λ 0 (A), (168) see Definition 23.
The topological index could be defined in a different way. One could consider the connecting homomorphism δ λ in the local cyclic periodic homology instead of the connecting homomorphism in the T loc * exact sequence. This leads to the topological index

T op Ch λ Index (u) := Chδ λ [u] ∈ H loc,per,λ ev (A), (169) Case 1
Problem 66 Define significant classes of extensions (156) for which the difference

∆ Ch (A, J ) := T op Ch Index (u) -An Ch Index (u) (170)
can be computed.

Case II.

Conjecture 67 Let M be a quasi-conformal closed manifold. Let u ∈ GL N (A µ /J µ ), where (A µ ) is the algebra of pseudo-differential operators on M and J ) = L ( 1 n ,∞) , localised by the support of operators about the diagonal. Then T op Ch Index(u) = An Ch Index(u).

(171)

26 Level III: Index theory restricted at the diagonal.

This situation applies only when the local periodic cyclic homology has a limit to the diagonal. It depends on the regularity of the structure. The classical index theorems belong to this class.

Part IV Noncommutative Topology

Abstract We intend to produce a theory which generalises topological spaces; we call it non-commutative topology. In non-commutative differential geometry the basic homology theory is the periodic cyclic homology, based on the bi-complex (b, B). In non-commutative topology this structure will be replaced by the bi-complex ( b, d); the boundary b is called modified Hochschild boundary. These ideas combine A. Connes' work [START_REF] Connes | Noncommutative Geometry[END_REF] with ideas of the articles by Teleman N. and Teleman K. [START_REF] Teleman | A geometrical definition of some Andre Weil forms which can be associated with an infinitesimal connection, (Roumanian)[END_REF], [START_REF] Teleman | Sur le charact'ere de Chern dun fibre complexe differentiable[END_REF], [START_REF] Teleman | Local 3 Index Theorem[END_REF], [START_REF] Teleman | Local Hochschild Homology of Hilbert-Schmidt Operators on Simplicial Spaces[END_REF], [START_REF] Teleman | Local Algebraic K-Theory[END_REF], [START_REF] Teleman | The Local Index Theorem[END_REF], [START_REF] Teleman | [END_REF].

Modified Hochschild homology

The Alexander -Spanier co-homology uses solely the topology of the space; it does not require any kind of analytical regularity. We use the Alexander -Spanier construction and the definition of local periodic cyclic co/homology as the departure point for non-commutative topology. Recall we extended the Alexander -Spanier co-homology to arbitrary localised rings, see §21.

As an application we compute the local modified periodic cyclic homology of the topological algebra of smooth functions, Theorem , of the Banach agebra of continuous functions, Theorem , and of the algebra of arbitrary functions, Theorem ??, on a smooth manifold. These results are significant because it is known that the Hochschild and (periodic) cyclic homology of Banach algebras are either trivial or not interesting, see Connes [START_REF] Connes | Noncommutative differential Geometry[END_REF], [START_REF] Connes | Noncommutative Geometry[END_REF], [START_REF] Connes | Cyclic Cohomology, The Novikov Conjecture and Hyperbolic Groups[END_REF]. Entire cyclic cohomology, due to Connes [START_REF] Connes | Noncommutative Geometry[END_REF] ( [START_REF] Connes | Cyclic Cohomology, The Novikov Conjecture and Hyperbolic Groups[END_REF], [START_REF] Connes | Conjectures de Novikov et fibrés presque plats[END_REF]) gives a different solution to the problem of defining the cyclic homology of Banach algebras. The chains of the entire cyclic cohomology are elements of the infinite product (b, B) which satisfy a certain bi-degree asymptotic growth condition. Connes constructed a Chern character of θsummable Fredholm modules with values in the entire cyclic cohomology, see Connes [START_REF] Connes | Entire cyclic cohomology of Banach algebras and characters of θ-summable Fredholm modules[END_REF].

Proposition 73 178For any ring A, the homology of the modified Hochschild complex contains the Hochschild homology.

Proof. As said before, the operator b contains the factor b both to the left as well as to the right. For this reason any Hochschild cycle is a bn cycle; for the same reason any b boundary is a b boundary.

29.1

Local modified periodic cyclic homology of the algebra of smooth functions.

In this subsection we consider the Fréchét topological algebra of smooth functions over a compact smooth manifold M . We know that the Hochschild boundary is well defined on germs at the diagonals and that the Hochschild homology depends only on the quotient complex, see Teleman [START_REF] Teleman | Microlocalisation de l'Homologie de Hochschild[END_REF].

The first two terms of the spectral sequence associated to the first filtration (with respect to d and then with respect to b) of the ( b, d) bicomplex are E 1 p,q = Hp(C * ,q, d) ∼ = H q-p dR (M ) and E 2 p,q = Hq(E 1 p, * , b) ∼ = H q-p dR (M ). In fact, the E 1 -term is the Alexander -Spanier co-homology of the complex of smooth chains. The Alexander -Spanier chain complex contains the sub-complex σ * C * (A). We know that this complex is co-homologous with the original Alexander -Spanier complex. For this reason, for the computation of E 2 , we may represent any element of E 1 by an element of the this sub-complex. The term E 2 = b-homology is a quotient group of a sub-group of E 1 ; from this we get dim{ b-homology of a quotient group of a sub-group of E 1 } = dim E 2 ≤ dim E 1 .

(183) The term E 1 , being the Alexander -Spanier co-homology of M , is isomorphic to the periodic b-homology of the algebra A.

From the equations (183), ( 184) and (185) we get

E 2 ∼ = H * dR (M ). ( 186
)
We have proved the following result.

Theorem 74 27 The local modified periodic cyclic homology of the algebra of smooth functions on a compact smooth manifold is isomorphic to the Z2-graded de Rham co-homology of the manifold.

30 Characteristic classes of idempotents.

Definition 75 Let A be a localised ring which contains the rational numbers. Let e ∈ MN (A) be an idempotent. The Chern character of e is the even local periodic cyclic homology class defined by Ch ( e ) := e + (-1) 1 1 ! e (de)

Remark 76 It is important to notice that the Chern character defined above does not use the trace.

Theorem 77 Let A be a localised ring. Suppose the ring K contains Q.

Let e ∈ MN (A) be an idempotent.

Then the RHS of the equation ( 187) is a cycle in the bi-complex ( b, d).

Proof. We compute b e(de) 2n . We have

2Definition 1

 1 Localised rings. Localised rings.

1 .

 1 For any idempotent e ∈ M n (A µ ) define Ch ev,Aµ (e) = T r e + ) 2p .[START_REF] Getzler | On the Chern character of a theta-summable Fredholm module[END_REF]

  -i) The equivalence relation ∼ O is compatible with the addition in S.-ii) S/ ∼ O is a group.-iii) Let [u] denote the ∼ O -equivalence class of u. Then -[u] = [I(u)]. (38) Definition 28 Define O(u) := u + I(u) f or any u ∈ S (39)

′ 12 ]

 12 j 2 * [p ′ 22 ] are locally, stably isomorphic. Now we are in the position to use Theorem 21.1. Let u ∈ GL(Λ 2 (A µ )); consider the conjugation j 1 * (p ′ 12 ) = u j 2 * (p ′ 22 ) u -1 . (74) Theorem 21.1 provides the idempotent p = (j 1 * (p ′ 12 ), (j 2 * (p ′ 2

dim E 1 =

 1 dim{ bhomology of C ∞ (M ) } (184) On the other side, the b-homology contains the b-homology, Proposition 178 dim (bhomology) ≤ dim( bhomology).

(- 1 )(- 1 )(- 1 )(- 1 )(- 1 )(- 1 )

 111111 bn(e (de)2n ) = b 2n k=1 (-1) k-1 C k 2n (db) k-1 de) 2n ) = 2n e (de) 2n-1 + b 2n k=2 k-1 C k 2n (db) k-2 (db)(e(de) 2n ) = (190) 2n e (de) 2n-1 + b 2n k=2 k-1 C k 2n (db) k-2 d(e de) 2n-1 ) = (191) 2ne(de) 2n-1 + b 2n k=2 k-1 C k 2n (db) k-2 (de) 2n ) = (192) 2ne (de) 2n-1 + b 2n k=2 (-1) k-1 C k 2n 2 k-2 (de) 2n ) = (193) 2ne (de) 2n-1 + 2n k=2 (-1) k-1 C k 2n 2 k-2 (2e -1)(de) 2n-1 k-1 C k 2n 2 k-1 )e (de) 2n-1 + 2n k=2 k C k 2n 2 k-2 (de) 2n-1 . k C k 2n 2 k-2 = n.(197)

We may transfer the conjugation onto the second term of this equation to get ∂(u) = Λ m+n µ , which proves the statement.

-iii.3.) Im ∂ ⊂ Ker (i 1 , i 2 ) * .

Let u ∈ T 1 (Λ ′ µ ) and p = ∂u ∈ T 0 (Λ µ ). We have to prove that (i 1 , i 2 ) * (p) = 0.

We have

On the other side

-iii.4.)

). Let p ∈ T 0 (Λ µ ) and (i 1 , i 2 ) * (p) = 0. We have to prove that there exists u ∈ T 1 (Λ ′ µ ) such that p = ∂u ∈ T 0 (Λ µ ).

That is, we look for an invertible matrix u ∈ M n (Λ

. We stabilise the idempotents 1 n ⊕ 0 n , Λ n µ . The conjugation transforms trivial idempotents in trivial idempotents. The stabilised idempotents 1 n+m ⊕ 0 n+m , Λ n+m µ still satisfy the condition

We choose an isomorphism ũ between the trivial idempotents 1 n+m ⊕ 0 n+m , Λ n+m µ . We reduce these idempotents, modulo the ideal J; we obtain idempotents in Λ ′ (A µ ). Let u be the restriction of ũ modulo J. The element u is the element we are looking for.

If we started with the element

To motivate the next definition, let D : H 0 -→ H 1 be an (integral or pseudo-differential) elliptic operator. Let σ(D) := D mod. compact operators be its symbol.

We consider the ring

Definition 55 Consider the following structure.

Let Λ 1 = Λ 2 be a ring. Let J ⊂ Λ i , i = 1, 2, be a bi-lateral ideal.

Introduce the rings

The ring structure in Λ is

and

If Λ i , J are localised rings, then this is a localised Mayer -Vietoris diagramm.

To this localised Mayer-Vietoris diagram there corresponds a six term exact sequence, see Theorem 7.

Next, we are going to exibit the corresponding connecting homomorphism ∂ :

). Such liftings exist because the canonical mapping j 1 is surjective.

28 The idempotent Π.

Recall the operator σ was introduced in §2.4.10 It is well defined on the whole complex C * (A)

db

where d is the non-localised Alexander -Spanier co-boundary and b is the Hochschild boundary.

The main properties of σ are 1. σ commutes both with d and b.

2. σ is a chain homomorphism both in the Alexander-Spanier and in the Hochschild complex. Hence, the range of the operator σ, and its powers, are sub-complexes both in the Alexander -Spanier and Hochschild complexes.

3. The range of the homomorphism Π consists of non-degenerate chains.

4. σ is homotopic to the identity. Therefore, the inclusions of these subcomplexes into the Alexander -Spanier, resp. Hochschild, complexes induce isomorphisms between their homologies.

The operator Π (k) is defined by the formula

The opertor Π (k) has the properties 29 Modified Hochschild homology.

On the range of the idempotent Π, i.e. { C * (A)}, one has the identity

From this formula we get Proposition 68 On the complex { C * (A)} one has the identity

Proof. The relation ( 176) is obtained from the formula

by making the substitution σ = 1 -(db + bd). For more details see [START_REF] Teleman | Modified Hochschild and Periodic Cyclic Homology[END_REF]. The expression of the operator bn contains the factor b both to the left and to the right.

Definition 72 -1) bb = 0 and hence b is a boundary operator.

-2) The complex {C * (A), b} * } is called modified Hochschild complex. It makes sense even for non-localised algebras.

The homology of the modified Hochschild complex is called modified Hochschild homology.

-3) We assume A to be a localised ring; the Alexander -Spanier and the Hochschild complex are localised. On the space of normalised chains ( b, d) introduces a bi-complex structure. The boundary b defined on the space of normalised chains is called local modified Hochschild homology.

The homology of the bi-complex ( b, d) is called local modified periodic cyclic homology; it is Z2-graded.

The definition of the local modified periodic cyclic homology is analogues to the definition of periodic cyclic homology, comp. A. Connes [START_REF] Connes | Noncommutative Geometry[END_REF], see also J. -L. Loday [START_REF] Loday | Cyclic Homology, Grundlehren in mathematischen Wissenschaften 301[END_REF] Sect. 5.1.7., pag.159.

More specifically, given the localised ring A, one has

where Cp,q(A) := ⊗ p-q A, for q ≤ p. The boundary maps are b : Cp,q -→ Cp,q-1 and d : Cp,q -→ Cp+1,q;

Proof. -1)

From Lemma 78 we get b2n ( e(de) 2n ) = n (de) 2n-1 .

(201)

On the other side

These prove the theorem.

Definition 79 Let ∆ ⊂ C * (A) be the K sub-module generated by all chains e(de) n , (

where n ∈ N and e ∈ M * (A) is an arbitrary idempotent.

∆ is called characteristic sub-complex.

Proposition 80 ∆ is a sub-complex both in the (b, B) and ( b, d) complexes.

Although not all relations below are used in this chapter, we provide them for the benefit of the reader.

On the same space the formulas hold

The following relations hold on normalised local/non-local chains. Although not all of them are necessary in what folllows, we mention them for the benefit of the reader, see J. M. Garcia -Bondia, H. Figueroa, J. C. Varilly [START_REF] Jose | Elements of Noncommutative Geometry[END_REF]

show that e(de

This means the classes e(de) 2n , 1 2 (e -1 2 )(de) 2n are co-homologous in the local and non-local complex σ( C * (A)). The non-local class 1 2 (e -1 2 )(de) 2n represents the Connes -Chern character of the idempotent e in the nonlocal entire cyclic homology, see Getzler, Senes [START_REF] Getzler | On the Chern character of a theta-summable Fredholm module[END_REF] and [START_REF] Jose | Elements of Noncommutative Geometry[END_REF], Pg. 447. 31 Rational Pontrjagin classes of topological manifolds 31.1 Existance of direct connections on topological manifolds.

We intend to construct a direct connection A on M . Recall that a direct connection, see [START_REF] Teleman | Direct Connections and Chern Character[END_REF], on M consists of a set of isomorphisms A(x, y) where, in general A(x, y) = isomorphism from a neighbourhood of y to a neighbourhood of x.

(216) with the property A(x, x) = Identity.

(217)

The connection A will be constructed by induction using a handlebody decomposition of M , see [START_REF] Kirby | Foundational Essays on Topological Manifolds, Smoothings and Triangulations[END_REF] 7.1. Pg. 319.

The second condition on direct connections assures that the inductive construction of the connection A may be performed without meeting homotopic obstructions in πi(T OP ). (219)

The characteristic class

Given that A(x, x) = Identity, the chain Ψ2n is a cycle in the b ′ -complex. Therefore,

Ψ2n is a cycle in the cyclic homology of the algebra A.