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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isn'tan 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