Rational stable homotopy type of equivariant projective spaces and Grassmannians
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912805127454720 |
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| author | Basu, Samik Doem, Vanny Nahak, Chandal |
| author_facet | Basu, Samik Doem, Vanny Nahak, Chandal |
| contents | 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 $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02940 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rational stable homotopy type of equivariant projective spaces and Grassmannians Basu, Samik Doem, Vanny Nahak, Chandal Algebraic Topology Primary 55N91, 55P62, Secondary 55P91, 57S15 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 $V$. |
| title | Rational stable homotopy type of equivariant projective spaces and Grassmannians |
| topic | Algebraic Topology Primary 55N91, 55P62, Secondary 55P91, 57S15 |
| url | https://arxiv.org/abs/2601.02940 |