Formality of surfaces The de Rham dg algebra $\Omega(F)$
 of a closed orientable surface $F$
 is formal
 (that is, weakly equivalent to its cohomology algebra).
 This is a special case of the fact of formality of Kähler manifolds.
Can one prove formality of $\Omega(F)$ without using complex analysis?
 Say, by explicitly constructing its Sullivan model
 (which has to be infinitely generated, I guess).
 A: Semen:   
In Felix, Halperin, Thomas: Rational Homotopy Theory II, World Scientific 2015, it is shown (see Ch. 8, section 5) that orientable surfaces are even better than formal. They are intrinsically formal. That is, any commutative cochain algebra with the same cohomology $H_g$ as a surface (of genus $g$ here) has the same minimal model as that of $(H_g,d=0)$. They don't use Deligne-Griffiths-Morgan-Sullivan, but they do base their arguments on earlier material, so be prepared for a bit of a slog. As one of the previous commentors said, any path connected space has a minimal model, but if the fundamental group and/or its action on higher homotopy groups is bad (i.e. non-nilpotent say), then the construction is via a transfinite induction which makes the resulting model very hard to understand and work with. So the bulk of RHT II is determining a class of spaces called Sullivan spaces where techniques akin to those in the authors' book RHT (Springer Grad. texts) can be used. The definition of Sullivan space builds in the exact amount of control needed for the fundamental group and its action to be amenable to rational homotopy algebraic analysis. One unsatisfying part of the definition however is that the rational cohomology of the universal cover must be finite-dimensional in each degree and so simple examples are not Sullivan spaces.
