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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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    $\begingroup$ 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 $\endgroup$
    – Tyler Lawson
    Commented Dec 21, 2009 at 5:09
  • $\begingroup$ 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. $\endgroup$
    – algori
    Commented Dec 21, 2009 at 16:07

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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.

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  • $\begingroup$ Thanks, Joey! Is there a reference for that? $\endgroup$
    – algori
    Commented Dec 21, 2009 at 14:06
  • $\begingroup$ Not that I know of. I heard about it from some friends who were at a talk of his in Oberwolfach.. $\endgroup$
    – Joey Hirsh
    Commented Dec 21, 2009 at 16:04
  • $\begingroup$ When was this talk then? Now they publish extended abstracts to Oberwolfach talks. $\endgroup$
    – algori
    Commented Dec 21, 2009 at 16:10
  • $\begingroup$ He gave an unscheduled talk at the "Strings, Fields and Topology" workshop there in June '09. $\endgroup$
    – Joey Hirsh
    Commented Dec 27, 2009 at 23:05
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    $\begingroup$ I took notes at this talk. I have added a link at ncatlab; they are at math.northwestern.edu/~gabriel/notes/sum09/ow_june_11.pdf $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Jan 6, 2011 at 20:11

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