There is a `long' history of transfer (up to homotopy!) of algebraic structure from a dg _ algebra A to its homology H(A) (e.g. Kadeishvili for the associative case and Heubschmann for the Lie case). Schlessinger and I computed a variety of examples of rational homotopy types using this result. There have been several uses of such transfer for theoretical results. Have there been any further computational examples?
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5$\begingroup$ I cannot answer your question, but I would be very much interested if you could provide some details of the `long history of transfer' and its uses. $\endgroup$ – Leonid Positselski Jun 22 '10 at 16:52
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$\begingroup$ @Leonid: this probably isn't an example, strictly speaking, but in homology of manifolds, the Lefschetz fixed point theorem (that the index of a generic smooth tangent vector field is the Euler Character) can be proved by taking a simplicial approximation to a finite integral flow for the field, being clever to have as few fixed simplices as necessary, and arguing that induced maps in homology have the same total trace as the maps on chains. $\endgroup$ – some guy on the street Jun 22 '10 at 18:25
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1$\begingroup$ For part of the long history from a twisted point of view, see . arXiv:0902.4396 [pdf, ps, other] Title: A twisted tale of cochains and connections Authors: Jim Stasheff Comments: 17 pages, in honor of the 60-th birthday of Tornike Kadeishvilli For more complete story, see the relevant paper(s) of Huebschmann. $\endgroup$ – Jim Stasheff Jul 4 '10 at 15:20
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1$\begingroup$ More precisely: by Huebschman arXiv:0710.2645 (to appear in the Gerstenhaber-Stasheff Festschrift) arXiv:0809.4791 (to appear in the Kadeishvili Festschrift) More references can easily be found there. $\endgroup$ – Jim Stasheff Jul 4 '10 at 17:43
