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Cohomological approach to class field theory in arithmetic topology. (English) Zbl 1492.57012

Arithmetic topology is, roughly speaking, a dictionary between 3-dimensional manifolds and fields of algebraic numbers. For example, knots in \(\mathbb{R}^3\) correspond to prime ideals \(p\mathbb{Z}\) in the ring \(\mathbb{Z}\) and links are interpreted as the ideals \(m\mathbb{Z}\). The number theory part of the dictionary is well developed, while the topological counterpart is falling behind. It is an interesting and important problem to understand how the concepts of modern number theory translate into the language of topology.
Class field theory studies abelian extensions of the number fields. The main result of such a theory says that the Galois group of the extension is isomorphic to the class group of the ground field. The idèle picture of class field theory (due to Chevalley) deals with the absolute Galois groups and the idèle class groups to grasp the global behavior of the extensions. The paper under review lays the foundations for class field theory in the framework of arithmetic topology. Namely, an extension of the number field corresponds to the covering of a 3-manifold and an absolute Galois group gives a pro-finite completion of such coverings. This leads to new topological concepts such as the so-called stably generic links, and many others. The paper is well organized and thoroughly written, and can be used as a springboard for further research in the area.

MSC:

57K31 Invariants of 3-manifolds (including skein modules, character varieties)
57K10 Knot theory
57M12 Low-dimensional topology of special (e.g., branched) coverings
11Z05 Miscellaneous applications of number theory
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
11R37 Class field theory
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
57P05 Local properties of generalized manifolds
11S31 Class field theory; \(p\)-adic formal groups
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