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.

  • 1
    $\begingroup$ 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 $X$. $\endgroup$ – Jeffrey Giansiracusa Jun 3 '11 at 9:42
  • $\begingroup$ Jeffrey, you're right, I'll edit the question. Thanks. $\endgroup$ – Mark Grant Jun 3 '11 at 10:13

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.