Hopf maps as static solutions of the complex eikonal equation

We demonstrate that a class of torus-shaped Hopf maps with arbitrary linking number obeys the static complex eikonal equation. Further, we explore the geometric structure behind these solutions, explaining thereby the reason for their existence. As this equation shows up as an integrability condition in certain non-linear field theories, the existence of such solutions is of some interest.Comment: 13 pages, slight changes in presentation, one paragraph on the symmetries of the eikonal equation added. Version accepted for publication in JM

Moduli Spaces and Formal Operads

Let overline{M}_{g,n} be the moduli space of stable algebraic curves of genus g with n marked points. With the operations which relate the different moduli spaces identifying marked points, the family (overline{M}_{g,n})_{g,n} is a modular operad of projective smooth Deligne-Mumford stacks, overline{M}. In this paper we prove that the modular operad of singular chains C_*(overline{M};Q) is formal; so it is weakly equivalent to the modular operad of its homology H_*(overline{M};Q). As a consequence, the "up to homotopy" algebras of these two operads are the same. To obtain this result we prove a formality theorem for operads analogous to Deligne-Griffiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field.Comment: 36 pages (v3: some typographical corrections

A Cartan-Eilenberg approach to Homotopical Algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and functor categories with a triple, in the last case we find examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications, we prove the existence of filtered minimal models for \emph{cdg} algebras over a zero-characteristic field and we formulate an acyclic models theorem for non additive functors