PreprintCC BY 4.0arXiv
Complex Projective Spaces in Rational SO(2)-Equivariant Stable Homotopy Theory
Abstract
We show that for almost every finite dimensional complex representation W of SO(2) the associated complex projective space ℂP(W)+ does not have the rational stable homotopy type of a specific wedge of representation spheres constructed from a chosen filtration of ℂP(W)+. This contrasts with the case for representations of finite Abelian groups where such an equivalence does hold. In the finite group case, the splitting result provides us with a classification result for projective spaces up to rational stable equivalence. In light of the non-splitting as a particular wedge of spheres in the case of projective spaces of SO(2)-representations, we also prove a separate classification theorem.
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BibTeX
@misc{lush2026complex,
title = {{Complex Projective Spaces in Rational $SO(2)$-Equivariant Stable Homotopy Theory}},
author = {Kian Lush},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28039},
eprint = {2610.28039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28039},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complex Projective Spaces in Rational SO(2)-Equivariant Stable Homotopy Theory
AU - Kian Lush
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28039
UR - https://arxiv.org/abs/2610.28039
AB - We show that for almost every finite dimensional complex representation W of SO(2) the associated complex projective space ℂP(W)_+ does not have the rational stable homotopy type of a specific wedge of representation spheres constructed from a chosen filtration of ℂP(W)_+. This contrasts with the case for representations of finite Abelian groups where such an equivalence does hold. In the finite group case, the splitting result provides us with a classification result for projective spaces up to rational stable equivalence. In light of the non-splitting as a particular wedge of spheres in the case of projective spaces of SO(2)-representations, we also prove a separate classification theorem.
ER -