Rational stable homotopy type of equivariant projective spaces and Grassmannians

TL;DR

This paper provides explicit rational stable decompositions of equivariant complex projective spaces and Grassmannians, revealing their structure as wedges of Thom spaces over irreducible representations, generalizing known cases.

Contribution

It introduces new rational stable splitting results for equivariant projective spaces and Grassmannians for arbitrary finite groups and representations, extending previous one-dimensional cases.

Findings

Rational stable splitting of $ ext{CP}(V)$ as a wedge of Thom spaces.

Rational stable splitting of $Gr_n(V)$ involving smash products of Thom spaces.

Generalization from one-dimensional to arbitrary irreducible representations.

Abstract

We prove explicit rational stable splittings of equivariant complex projective spaces $\mathbb{C}P(V)$ and Grassmannians $Gr_n(V)$, for complex representations $V$. When $V$ is a sum of one-dimensional representations, both $\mathbb{C}P(V)$ and $Gr_n(V)$ are rationally a wedge of representation spheres. For general finite groups $G$ and $V$ a sum of irreducible representations which are not necessarily one-dimensional, we show that $\mathbb{C}P(V)$ splits rationally as a wedge of Thom spaces over irreducible factors in $V$. For $Gr_n(V)$, the factors in the corresponding rational splitting are a smash product of Thom spaces over lower Grassmannians on irreducible factors in $V$.