  Schaefer's dichotomy theorem [Schaefer, STOC'78] states that a boolean
constraint satisfaction problem (CSP) is polynomial-time solvable if one of six
given conditions holds for every type of constraint allowed in its instances.
Otherwise, it is NP-complete. In this paper, we analyze boolean CSPs in terms
of their topological complexity, instead of their computational complexity. We
attach a natural topological space to the set of solutions of a boolean CSP and
introduce the notion of projection-universality. We prove that a boolean CSP is
projection-universal if and only if it is categorized as NP-complete by
Schaefer's dichotomy theorem, showing that the dichotomy translates exactly
from computational to topological complexity. We show a similar dichotomy for
SAT variants and homotopy-universality.
