Introduction to arithmetic topology

This is an introductory talk relative to the colloquium on November 15 that will be given by Caterina Consani. “Arithmetic topology” refers to analogy between number fields and closed orientable 3-manifolds; between the rational number field and the 3-sphere; between prime numbers and knots; between the Legendre symbol of a pair of prime numbers and the linking number of a pair of knots; and so forth. My goal is to make the analogy visible by introducing the fundamental objects on each side of the correspondence, mainly cohomology theories. 

References:

Connes and Consani, Knots, Primes and the adele class space.

Mazur, Notes on Étale cohomology of number fields. Shows that number fields have a 3-dimensional cohomology theory.

Morishita, Analogies between Knots and Primes, 3-Manifolds and Number Rings.

SGA 1, Grothendieck and Raynaud. One classic source for the relation between étale fundamental groups and fundamental groups. (See Théorème d’existence de Riemann.)

Hatcher, Algebraic Topology.