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The fields of cohomological dimension 1 are exactly the fields with projective absolute Galois group. The analogue of the weak section conjecture holds for geometrically connected curves over such fields if and only if the field is PAC, see Theorem 238. At first glance, it seems strange to examine the assertions of the section conjecture for varieties over such fields. Among those fields are the finite fields, which were discussed at length in Chap 15. In Theorem 243, we will construct an infinite algebraic extension of \({\mathbb{F}}_{p}\) such that the profinite Kummer map for every smooth projective subvariety of an abelian variety is injective with dense image with respect to the topology from Chap. 4, but never surjective.Another source of fields with cohomological dimension 1 is given by the maximal cyclotomic extensions of algebraic number fields. A thorough understanding of the assertions of the section conjecture in this case might actually provide valuable insights, see Theorem 247, and may ultimately justify this chapter.

1. Brief Historical Sketch. Together with the theory of continua, dimension theory is the oldest branch of general topology. The first concepts and facts predate Hausdorff’s definition in 1914 of general Hausdorff topological spaces and, so, involved only subsets of Euclidean spaces. In its infancy, dimension theory was nurtured by the work of three outstanding mathematicians: Poincare, Brouwer, and Lebesgue. Peano’s construction in 1890 of a continuous map of a segment onto a square gave rise to the problem of whether the dimension of Euclidean space was a topological invariant. This problem was solved by Brouwer in 1911 (see the article [B1]) using the concept he introduced of the degree of a map. In the same paper Brouwer proved that, for e < 1/2, the Euclidean cube I n could not be mapped by an e-shift to a nowhere dense subset A of I n (we now know that such a set A has dimension less than n).