International audienceThe aim of this paper is to study sub-algebras of the Z/2-equivariant Steenrod algebra (for cohomology with coefficients in the constant Mackey functor F 2) which come from quotient Hopf algebroids of the Z/2-equivariant dual Steenrod algebra. In particular, we study the equivariant counterpart of profile functions, exhibit the equivariant analogues of the classical A(n) and E(n) and show that the Steenrod algebra is free as a module over these

Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical examples of complex projective spaces under complex conjugation. Using tools from stable equivariant homotopy theory, we provide a characterization of conjugation spaces in terms of purity. This conceptual viewpoint, compared to the more computational original definition, allows us to recover all known structural properties of conjugation spaces.

Abstract. We show that the Z/2-equivariant n th integral Morava Ktheory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrum KO as a corollary. The study of Z/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of RO(Z/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.Conventions: In this paper, F denotes the field with two elements. When considering the Steenrod algebra and the chromatic tower, the prime number is assumed to be p = 2. The category of abelian groups is denoted Ab. For E an object in the category of spectra (resp. Z/2-equivariant spectra), we denote E * (resp. E ⋆ ) the cohomology theory represented by E, and E * (resp. E ⋆ ) the homology theory represented by E. The homotopy of E is denoted E * (resp. E ⋆ ). Equivariant cohomology theories are graded over the orthogonal representation ring, thus ⋆ is an orthogonal representation of Z/2. We denote 1 the trivial one dimensional representation, and α the sign representation.

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.