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Let $X$ be a path-connected nilpotent space (meaning $\pi_1(X)$ is nilpotent and acts nilpotently on the higher homotopy groups). Let $\rho\colon\thinspace\pi_1(X)\to \mathrm{Gl}(V)$ be a representation, where $V$ is a rational vector space. Then $\rho$ defines a local coefficient system on $X$, and we have the cohomology with local coefficients $H^\ast(X;\rho)$.

It is well known that Sullivan's theory of minimal models works well for nilpotent spaces, and one can (more or less) easily obtain a minimal, nilpotent cdga $M_X$ with $H(M_X)\cong H^*(X;\mathbb{Q})$, whose isomorphism type determines the rational homotopy type of $X$. (Here I am assuming some finiteness hypotheses, such as are satisfied if $X$ is a manifold or finite CW complex.)

My question is, is there a (more or less) easy way to obtain a minimal (in some sense) dg-module $M_{X,\rho}$ over $M_X$ which has $H(M_{X,\rho})\cong H^\ast(X;\rho)$ as modules over $H^\ast(X;\mathbb{Q})$?

Ideally I would like to be able to calculate $H^1(X;\rho)$ when $X$ is a nilmanifold (so $\pi_1(X)$ is nilpotent with trivial higher homotopy groups) and calculate cup products $a_1\cup a_2\in H^2(X;\rho_1\otimes\rho_2)$. So far I have looked at

Gómez-Tato, Antonio Théorie de Sullivan pour la cohomologie à coefficients locaux.[Sullivan's theory for cohomology with local coefficients] Trans. Amer. Math. Soc. 330 (1992), no. 1, 235–305.

and parts of Sullivan's original paper

Sullivan, Dennis Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331.

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I think you are asking for the impossible here because the cohomology with local coefficients does not generally have a cup product ring structure. However, $H^*(X;\rho)$ is of course a module over $H^*(X)$ and so you can ask for a dg-module over a cdga model for $X$. –  Jeffrey Giansiracusa Jun 3 '11 at 9:42
Jeffrey, you're right, I'll edit the question. Thanks. –  Mark Grant Jun 3 '11 at 10:13

1 Answer 1

up vote 3 down vote accepted

I think one could argue as follows: one can start by constructing a cochain complex $A^*(X,\rho)$ that computes the twisted cohomology so that it will be a module over the Sullivan cochains $A^*(X)$ of $X$ (with constant coefficients). Then one can plug it in Hinich's machinery: see http://arxiv.org/PS_cache/q-alg/pdf/9702/9702015v1.pdf , section 3. A minimal model would then be a cofibrant replacement of $A^*(X,\rho)$ as an $A^*(X)$-module.

Let me also mention a reference that may be useful: Gomez-Tato, Halperin, Tanr\'e, Rational homotopy theory for non-simply connected spaces. Trans. Amer. Math. Soc. 352 (2000), no. 4, 1493–1525.

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