Speaker: Sacha Ikonicoff (Paris Diderot)
Operadic theory of divided powers and of unstable algebras over the Steenrod algebra
Divided power operations and stable cohomological operations are the two main notions that appear in the 1954-1955 Henri Cartan seminar in the study of the homology and the cohomology of the Eilenberg-MacLane spaces.
On one hand, the more general notion of divided power algebra over an operad was introduced by Fresse in the study of the homotopy of simplicial algebras over an operad. The first aim of this talk is to characterise divided power algebras over an operad as defined by Fresse in terms of monomial operations and relations, following the classical definition of Cartan.
On the other hand, the study of cohomological operations led to the construction of unstable modules over the Steenrod algebra. Unstable modules coming from topology are equipped with an associative commutative cup-product satisfying an `unstability' condition. However, several classical examples of (non-topological) unstable modules, such as Brown-Gitler, Carlsson, and Campbell-Selick modules, come equipped with an inner product that doesn't satisfy the same properties. The second aim of this talk is to create a notion of *-unstable P-algebra, for an operad P
and an operation * in P, that englobes those classical cases. We also identify some
of the classical modules to the free unstable algebras over suitable operads.