Fibers of maps to totally nonnegative spaces

This paper undertakes a study of the structure of the fibers of the Chevalley exponentiation maps $f_{(i_1,\dots ,i_d)}$. The fibers of these maps $f_{(i_1,\dots ,i_d)}$ encode the nonnegative real relations amongst exponentiated Chevalley generators. Our main theorems show that the fibers admit cell stratifications, that these cell stratifications have the same face posets as interior dual block complexes of subword complexes, and that these posets are contractible. We conjecture that each such fiber is a regular CW complex homeomorphic to the interior dual block complex of a subword complex. This conjecture is shown to have as a corollary a new proof of the Fomin-Shapiro Conjecture by way of general topological results regarding approximating maps by homeomorphisms.