Index theorem: $\delta: T^{loc}_{1} (\mathbb{B}/ \mathcal{L}^{2}) \otimes_{\mathbb{Z}} \mathbb{Z} [1/2] \longrightarrow T^{loc}_{0}(\mathcal{L}^{2})\otimes_{\mathbb{Z}} \mathbb{Z} [1/2]$ \\ is an isomorphism, \\ $\delta [\sigma (U)] = [P_{U} - e]$.
Index Theoreme: $\delta: T^{loc}_{1} (\mathbb{B}/ \mathcal{L}^{2}) \otimes_{\mathbb{Z}} \mathbb{Z} [1/2] \longrightarrow T^{loc}_{0}(\mathcal{L}^{2})\otimes_{\mathbb{Z}} \mathbb{Z} [1/2]$ \\ est un isomorphisme, \\ $\delta [\sigma (U)] = [P_{U} - e]$.
Résumé
This article deals with our program presented in \cite{Teleman_arXiv_III}.
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{\bf Part I: $T$-Theory}.
In this part we summarise and set notation for the topics presented in \cite{Teleman_arXiv_III}.
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For any associative algebra $\mathcal{A}$ we define the \emph{commutative groups}
$T_{i}(\mathcal{A})$, $i = 0, 1$. We introduce the notion of \emph{localised algebras}, \S 4,
$\mathcal{A} = \{ \mathcal{A}_{\mu} \}$, given by a \emph{linear filtration} of the algebra $\mathcal{A}$ and we associate the \emph{commutative groups} $T_{i}^{loc} (\mathcal{A})$.
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Although we define solely $T^{loc}_{i}(\mathcal{A})$ for $i = 0, \; 1$, we expect our construction could be extended in higher degrees.
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\emph{We stress that our construction of} $T_{i}(\mathcal{A})$ and
$T^{loc}_{i}(\mathcal{A})$ \emph{uses exclusively matrices}. The projective modules are totally avoided. \emph{Equivalence relation} of projective modules, used in the construction of algebraic $K$-theory, is replaced by \emph{conjugation}, which is used in both theories.
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\emph{The commutative group} $T_{0}(\mathcal{A})$ is by definition the \emph{Grothendieck completion of the space of idempotent matrices factorised through the equivalence relations}: -i) \emph{stabilisation} $\sim_{s}$, -2) \emph{conjugation} $\sim_{c}$, and -3) for \emph{localised groups}, $T_{0}^{loc}(\mathcal{A})$, \emph{projective limit with respect to the filtration}, denoted $\sim_{p}$.
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By definition, $T_{1}(\mathcal{A})$ is the quotient space of
$\mathbf{GL}(\mathcal{A})$ modulo the equivalence relation generated by
-1) stabilisation $\sim_{s}$, -2) \emph{conjugation} $\sim_{c}$ and -3)
$\sim_{\mathbf{O}(\mathcal{A}}$, where $\mathbf{O}(\mathcal{A})$ is the \emph{sub-group generated by elements of the form} $u \oplus u^{-1} $, \emph{for any} $u \in \mathbf{GL}(\mathcal{A})$.
For any $u_{1}, u_{0} \in \mathbf{GL}(\mathcal{A}) \;/\; (\sim_{s} \cup \sim_{c} ) $, we define
$u_{1} \sim_{\mathbf{O}(\mathcal{A}}) u_{0}$ provided there exist $\xi_{1}, \xi_{0} \in \mathbf{O}(\mathcal{A})$ such that
$u_{0} + \xi_{0} = u_{1} + \xi_{1}. $
The operation
\begin{equation*}
\mathbf{GL}(\mathcal{A}) \;/\; ( \sim_{s} \cup \sim_{c} ) \longrightarrow
\mathbf{GL}(\mathcal{A}) \;/\; ( \sim_{s} \cup \sim_{c} \cup \sim_{\mathbf{O}(\mathcal{A})} )
\end{equation*}
transforms the commutative \emph{semi-group} $\mathbf{GL}(\mathcal{A}) \;/\; ( \sim_{s} \cup \sim_{c} )$
in the commutative \emph{group} $\mathbf{GL}(\mathcal{A}) \;/\; ( \sim_{s} \cup \sim_{c} \cup \sim_{\mathbf{O}(\mathcal{A})} )$. This operation is a particular case of a more \emph{general completion procedure which we call} $T$-\emph{completion}, see \S 11.
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\emph{The equivalence relation} $\sim_{\mathbf{O}(\mathcal{A})}$ \emph{is introduced to insure, forcebly, existence of opposite elements in} $T_{1}(\mathcal{A})$.
By imposing a controle of the supports one obtains $T_{1}^{loc}(\mathcal{A})$.
\par
The groups $T^{loc}_{i}(\mathcal{A})$ follow the same construction as that of $T_{i}(\mathcal{A})$,
provided the supports of the elemements belong to $\mathcal{A}_{\mu}$.
\emph{Our definition of} $T_{1}(\mathcal{A})$ and $T^{loc}_{1}(\mathcal{A})$ \emph{does not use the commutator sub-group} $[\mathbf{GL}(\mathcal{A}), \mathbf{GL}(\mathcal{A})]$
\emph{nor elementary matrices in its construction}.
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We \emph{avoid using projective modules} in the construction of the $T_{i}(\mathcal{A})$ and $T_{i}^{loc}(\mathcal{A})$. Projective modules, rather than matrices, introduce more arbitrarieness and
are more difficult to controle the supports. \emph{Equivalence} of projective modules is replaced
by \emph{inner auto-morphisms} of the algebra.
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We define short exact sequences \S 12 of \emph{ localised} algebras and we get the corresponding open six terms exact sequence (Theorem 53).
Neither integral opeators nor pseudo-differential operators are used.
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We stress that one has to take the tensor product of the expected six terms exact sequence by
$\mathbb{Z}[\frac{1}{2}]$ in order to get the open six terms exact sequence. We expect the factor
$\mathbf{Z}[\frac{1}{2}]$ to have important implications related to the existence of Pontrjagin classes,
existence of a generator of the $K$-homology fundamental class and Kirby-Siebenmann obstruction
class.
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Our work shows that the basic relations which define $T_{1}$ reside in the \emph{additive} sub-group generated by elements of the form $u \oplus u^{-1}$, $u \in \mathbf{GL}(\mathcal{A})$, rather than in the \emph{multiplicative} commutator sub-group $[\mathbf{GL}(\mathcal{A}), \mathbf{GL}(\mathcal{A})]$.
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Even into the case of trivially filtered algebras, $\mathcal{A} = \{ \mathcal{A}_{\mu} \}$, for all $\mu \in \mathbb{N}$, the groups $T^{loc}_{1}(\mathcal{A})$ provides more information than the classical group $T_{1}(\mathcal{A})$.
For the computation of the groups $T_{i}^{loc}(\mathbb{C})$ see \cite{Teleman_arXiv_V}.
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{\bf Part II: Index Theory}.
We introduce \emph{short exact sequences} of operators significant to index theory.
We prove the Conjecture 19 of \cite{Teleman_arXiv_III} which re-formulates index theory.
For any Fredholm operator $U \in \mathbb{B}(H_{\mu})$, the class of $U$ in the quotient algebra
$\mathbb{B}(H_{\mu})/ \mathcal{L}^{2}(H_{\mu})$ is called \emph{total symbol} of $U$, denoted
$\sigma (U)$. The total symbol is no longer concentrated along the diagonal.
The \emph{local topological index} of the operator $U$ is by definition the class of the total symbol
$\sigma (U)$ in $T^{loc}_{1}(\mathbb{B}(H_{\mu})/ \mathcal{L}^{2}(H_{\mu}))$. The \emph{local analytic index} of the operator $U$ is the class of the element $R(U) = P - e$ in the algebraic local group $T^{loc}_{0}(\mathcal{L}^{2} (H))$, (see Theorem 62).
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The Todd class is removed from the index theorem.
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The \emph{index theorem} is: -i) the homomorphism $\delta: T^{loc}_{1} \longrightarrow T^{loc}_{0}$ is \emph{an isomorphism} and -ii)
$\delta [\sigma (U)] = [ R (U)]$.
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The main results of this article are Theorem 53 and Theorem 61.