For several classes of pseudodifferential operators with operator-valued symbol analytic index formulas are found. The common feature is that usual index formulas are not valid for these operators. Applications are given to pseudodifferential operators on singular manifolds.

Sur une pseudo-variété de dimension paire à une singularité conique isolée, des triplets spectraux sont construits à partir d’une classe d’opérateurs différentiels elliptiques de type Fuchs, contenant les opérateurs de Dirac à coefficients dans des fibrés plats dans la direction radiale. Ces derniers engendrent, sous une hypothèse raisonnable, le groupe de $K$-homologie pair tensorisé par $ℂ$ de la pseudo-variété et leur caractère de Chern est calculé.

We give exhaustive list of biquadratic fields $K=\mathbb{Q}(i,\sqrt{m})$ and $K=\mathbb{Q}(\sqrt{2},\sqrt{m})$ without $2$-exotic symbol, i.e. for which the $2$-rank of the Hilbert kernel (or wild kernel) is zero. Such $K=\mathbb{Q}(i,\sqrt{m})$ are logarithmic principals [J3]. We detail an exemple of this technical numerical exploration and quote the family of theories and results we utilize. The $2$-rank of tame, regular and wild kernel of $K$-theory are connected with local and global problem of embedding in a $Z_2$-extension. Global class field theory can describe the $2$-rank of the Hilbert...

We prove that for any ring $𝐑$ of Krull dimension not greater than 1 and $n\ge 3$, the group ${\mathrm{E}}_n\left(𝐑[X,X^{-1}]\right)$ acts transitively on ${\mathrm{Um}}_n\left(𝐑[X,X^{-1}]\right)$. In particular, we obtain that for any ring $𝐑$ with Krull dimension not greater than 1, all finitely generated stably free modules over $𝐑[X,X^{-1}]$ are free. All the obtained results are proved constructively.

In this paper, we shall discuss possible theories of defining equivariant singular Bott-Chern classes and corresponding uniqueness property. By adding a natural axiomatic characterization to the usual ones of equivariant Bott-Chern secondary characteristic classes, we will see that the construction of Bismut’s equivariant Bott-Chern singular currents provides a unique way to define a theory of equivariant singular Bott-Chern classes. This generalizes J. I. Burgos Gil and R. Liţcanu’s discussion...

Nous montrons une version explicite du théorème de Beilinson pour la courbe modulaire $X_1\left(N\right)$. Ce résultat est la première étape d’un travail reliant, d’une part, la valeur en $2$ de la fonction $L$ d’une forme primitive de poids $2$, et d’autre part, la fonction dilogarithme associée à la courbe modulaire correspondante, dans l’esprit de la conjecture de Zagier pour les courbes elliptiques. Comme corollaire de notre théorème, dans le cas où $N$ est premier, nous répondons à une question de Schappacher et Scholl...

Let $\Gamma ={\Gamma }_1{*}_G{\Gamma }_2$ be the pushout of two groups ${\Gamma }_i$, i = 1,2, over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces $B{\Gamma }_1\leftarrow BG\leftarrow B{\Gamma }_2$. Denote by ξ the diagram $I\frac{p}{\leftarrow }H\frac{1}{\to }X=H$, where p is the natural map onto the unit interval. We show that the $Ni{l}^{\sim }$ groups which occur in Waldhausen’s description of $K_{*}\left(\mathbb{Z}\Gamma \right)$ coincide with the continuously controlled groups ${}_{*}^{cc}\left(\xi \right)$, defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups ${}_{*}^{cc}\left({\xi }^{+}\right)$ which are known to form a homology...