1,052 research outputs found
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
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