# zbMATH — the first resource for mathematics

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].

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