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تسجيل مستخدم جديدThe $A_infty$-structure of the index map
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Let $F$ be a local field with residue field $k$. The classifying space of $GL_n(F)$ comes canonically equipped with a map to the delooping of the $K$-theory space of $k$. Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of A-infinity-spaces $GL_n(F)to K_k$. Using a generalized Waldhausen construction, we construct an explicit model built for the $A_infty$-structure of this map, built from nested systems of lattices in $F^n$. More generally, we construct this model in the framework of Tate objects in exact categories, with finite dimensional vector spaces over local fields as a motivating example.
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Suppose k is a field of characteristic 2, and $n,mgeq 4$ powers of 2. Then the $A_infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $(H^*(C_m,k)$ are well known. We give results characterizing an $A_infty$-structure on $H^*(C_ntimes
C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.
Following [44], we introduce the notion of families of projective operators on fibrations equipped with an Azumaya bundle $mathcal{A}$. We define and compute the index of such families using the cohomological index formula from [7]. More precisely, a
family of projective operators $A$ can be pulled back in a family $tilde{A}$ of $SU(N)$-transversally elliptic operators on the $PU(N)$-principal bundle of trivialisations of $mathcal{A}$. Through the distributional index of $tilde{A}$, we can define an index for the family $A$ of projective operators and using the cohomological index formula from [7], we obtain an explicit cohomological index formula. Let $1 to Gamma to tilde{G} to G to 1$ be a central extension by an abelian finite group. As a preliminary result, we compute the index of families of $tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$.
We show that an old conjecture of A.A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree $n$ to Milnor K-theory in degree $n$ admits an interpretation in terms of unstable ${mathbb A}^1$-homotopy sheaves of the general
linear group. Using this identification, we establish Suslins conjecture in degree $5$ for infinite fields having characteristic unequal to $2$ or $3$. We do this by linking the relevant unstable ${mathbb A}^1$-homotopy sheaf of the general linear group to the stable ${mathbb A}^1$-homotopy of motivic spheres.
We define and study the index map for families of $G$-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic propert
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We define the Chern character of the index class of a $G$-invariant family of $G$-transversally elliptic operators, see [6]. Next we study the Berline-Vergne formula for families in the elliptic and transversally elliptic case.
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