Embedding theorems for spaces of $\R$-places of rational function fields and their products

We study spaces $M(R(y))$ of $\R$-places of rational function fields $R(y)$ in one variable. For extensions $F|R$ of formally real fields, with $R$ real closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in $M(F(y))$ and prove uniqueness results. Further, we study embeddings of products of spaces of the form $M(F(y))$ in spaces of $\R$-places of rational function fields in several variables. Our results uncover rather unexpected obstacles to a positive solution of the open question whether the torus can be realized as a space of $\R$-places.