On the dimension of the space of R-places of certain rational function fields

Abstract

We prove that for every n ∈ ℕ the space M(K(x 1,..., x n) of ℝ-places of the field K(x 1,..., x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1,..., x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.