Transferring \(A_\infty\) (strongly homotopy associative) structures. (English) Zbl 1112.18007

Čadek, Martin (ed.), The proceedings of the 25th winter school “Geometry and physics”, Srní, Czech Republic, January 15–22, 2006. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 79, 139-151 (2006).
The aim of this note is to give explicit formulas for transfers of \(A_\infty\)-structures and related maps and homotopies. The author starts from a chain map \(f:(V,\partial_V) \rightarrow (W,\partial_W)\) having a left chain-homotopy inverse \(g\) via a chain-homotopy \(h\) and an \(A_\infty\)-structure \(\mu=(\mu_2,\mu_3,\dots)\) on \((V,\partial_V)\).
The author gives two sets of formulas, an inductive one and a non-inductive one, constructing an \(A_\infty\)-structure \(\nu=(\nu_2,\nu_3,\dots)\) on \((W,\partial_W)\) and an \(A_\infty\)-map \(\phi=(\phi_1,\phi_2,\phi_3,\dots):(V,\partial_V,\mu_2,\mu_3,\dots) \rightarrow (W,\partial_W,\nu_2,\nu_3,\dots)\) having a left \(A_\infty\)-homotopy inverse \(\psi\) via an \(A_\infty\)-homotopy \(H\) such that \(\phi_1=f\), \(\psi_1=g\) and \(H_1=h\).
The existence of these transfers follows, in characteristic zero, from a general theory developed by the author in a previous paper. The easier half of the formulas was already known to Kontsevich-Soibelman and Merkulov who derived them, without explicit signs, under slightly stronger assumptions than those made in this note.
For the entire collection see [Zbl 1103.53001].


18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
55S99 Operations and obstructions in algebraic topology
Full Text: arXiv