Abstract:
Half a century ago, Barry Mazur launched the industry known today as ‘arithmetic topology’ by observing that the ‘homotopy theory’ of a number ring O_F closely resembles that of a 3-manifold. Indeed, class field theory implies a kind of 3-dimensional Poincaré Duality for the étale cohomology of O_F. Unfortunately, this analogy has always suffered from deficiencies with real places and with 2-local coefficients. We propose a different, homotopical compactification, which enjoys a 3-dimensional duality theorem with no fudging around real places and the prime 2. The compactification is closely linked to the geometrisation of Galois and Weil groups, which we will also discuss.