This question was inspired by Poincaré quasi-isomorphism

Let $M$ be a closed oriented $n$-manifold. The cap product with the fundamental class of $M$ induces an isomorphism $H^i(M,\mathbf{Z})\to H_{n-i}(M,\mathbf{Z})$. Both the source and the target of this are rings. (For the definition of the homology intersection product see e.g. McClure http://arxiv.org/abs/math/0410450 or M. Goresky and R. MacPherson's first paper on the intersection homology.) It is not too difficult to show that the Poincar\'e isomorphism respects the ring structure.

The question is: to which extent is this true on the chain level?

More precisely, Goresky and MacPherson's PL chains of a manifold form a partial commutative dga (see McClure's paper mentioned above). Singular cochains form a non-commmutative dga that can be completed to an $E_{\infty}$-algebra, which is a different kind of structure. So one way to make the above question precise would be as follows:

Is there a natural way to turn the PL-chains on a PL-manifold into an $E_\infty$ algebra? (In the above-mentioned paper McClure promises to do this in another paper, but I don't know if the details are available.)

If the answer to 1. is positive, then can one complete the chain level cap product with the fundamental cycle into an $E_{\infty}$ morphism?