Is there a simple construction of a Sullivan minimal model $\Lambda U \rightarrow V$ in the case that $H^1(V)\neq 0$? Do you have a reference? I envisage a degree-wise construction as in the case of $H^1(V)=0$ explaining how to deal with the new phenomenon. The existence is proven very generally in Felix, Halperin, Thomas, Rational Homotopy Theory but I find the proof very complicated and not transparent enough in this simple case (they prove that relative Sullivan models exist and that any relative Sullivan algebra is isomorphic to a product of a minimal relative Sullivan algebra and an acyclic algebra)
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1$\begingroup$ This probably isn't the answer you're looking for, but I think the most transparent construction is the one given by Sullivan himself in Infinitesimal Computations, section 5. It is not written in "formal" language, but is clear (and can be formalized). The construction is indeed degree-wise as in the simply connected case, but one might have to add generators to the same degree infinitely many times before moving on to the next degree (for example this happens if you start to model the wedge of two circles). $\endgroup$– Aleksandar MilivojevićCommented Apr 22, 2020 at 22:39
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1$\begingroup$ Thanks for the reference! I have to confess that I have never looked in the original... But I will have a look at it when I have more time and try to rewrite it here if nobody else is interested in getting the points :-) $\endgroup$– PavelCommented Apr 23, 2020 at 7:54
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Gelfand and Manin explain it very nicely in their book "Methods of Homological Algebra", last chapter.
