A detailed look on actions on Hochschild complexes especially the degree 1 coproduct and actions on loop spaces

[摘要] We explain our previous results about Hochschild actions (2007/2008) pertaining, in particular, to the coproduct, which appeared in a different form in a work by Goresky and Hingston (2009), and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in a previous article (2008), is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations given in two previous articles (2007/2008). With respect to loop spaces, we show that the coproduct is well defined modulo constant loops and going one step further that in the case of a graded Gorenstein Frobenius algebra, the coproduct is well defined on the reduced normalized Hochschild complex. We discuss several other aspects such as “time reversal” duality and several homotopies of operations induced by it. This provides a cohomology operation which is a homotopy of the anti-symmetrization of the coproduct. The obstruction again vanishes on the reduced normalized Hochschild complex if the Frobenius algebra is graded Gorenstein. Further structures such as “animation”, the BV structure, a coloring for operations on chains and cochains, and a Gerstenhaber double bracket are briefly treated.

[学科分类] 神经科学

[关键词] Hochschild chains and cochains;Goresky–Hingston coproduct;string topology;loop spaces;Frobenius algebras;coproduct