Modèle minimal équivariant et formalité

Abstract:

We study the rational equivariant homotopy type of a topological space $X$ equipped with an action of the group of integers modulo $n$. For $n= {p^k}$ ($p$ prime, $k$ a positive integer), we build an algebraic model which gives the rational equivariant homotopy type of $X$. The homotopical fixed-point set appears in the construction of a model of the fixed-points set. In general, this model is different from ${\text {G}}$. Triantafillou’s model $[{\text {T1}}]$. For $n= p$ ($p$ prime), we then give a notion of equivariant formality. We prove that this notion is equivalent to the formalizability of the inclusion of fixed-points set $i:{X^{{\mathbb {Z}_p}}} \to X$. Examples and counterexamples of ${\mathbb {Z}_p}$-formal spaces are given.