# Do chains and cochains know the same thing about the manifold?

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:

1. 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.)

2. 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?

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I believe that the second part of Scott Wilson's thesis is roughly about these questions (added as a comment because I believe he shows that the chains themselves have an $E_\infty$-structure up to quasi-isomorphism): qcpages.qc.cuny.edu/~swilson –  Tyler Lawson Dec 21 '09 at 5:09
Thanks, Tyler! Indeed, Wilson constructs a quasi-isomorphism of partial $E_\infty$ algebras (in the sense of Kriz and May) which goes from the chains of a manifold to some (true) $E_\infty$ algebra. What is not clear to me at the moment is whether the "cap with the fundamental cycle" map can be completed to a map of partial $E_\infty$-algebras. –  algori Dec 21 '09 at 16:07

You may also want to look up David Chataur's work on the subject. I've heard that he proves that for any Poincare Duality space, each "Poincare Duality isomorphism" from cohomology to homology gives rise to a "unique" $E_{\infty}$ quasi-isomorphism from the $E_{\infty}$ structure on cochains to Wilson's $E_{\infty}$ structure on chains.