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Central extensions and reciprocity laws. (English) Zbl 0886.18003

In this article the concept of a group acting on a groupoid is developed. Such an action gives rise to a canonical extension of this group; it is proved that any of its extensions is obtained this way, and the extension splits when the action has a fixed object.
Several applications of the last property are given. The first one is a categorical proof of the quadratic reciprocity law for any global field of characteristic not equal to 2, and the other is the purely symplectic nature of Atiyah-Bott’s fixed point theorem for a holomorphic line bundle over a Kaehler manifold [M. F. Atiyah and R. Bott, “A Lefschetz fixed point formula for elliptic complexes. II: Applications”, Ann. Math., II Ser. 88, 451-491 (1968; Zbl 0167.21703)].

MSC:

18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
58H05 Pseudogroups and differentiable groupoids
11R37 Class field theory
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
20E22 Extensions, wreath products, and other compositions of groups
58J20 Index theory and related fixed-point theorems on manifolds
32L05 Holomorphic bundles and generalizations

Citations:

Zbl 0167.21703
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References:

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