Endomorphisms of the Cohomology Algebra of the Even Orthogonal Grassmannian
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918356984004608 |
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| author | Goswami, Arnab Sarkar, Swagata |
| author_facet | Goswami, Arnab Sarkar, Swagata |
| contents | Let $M_{n,k}$ denote the even orthogonal Grassmanian, $SO(2n) / (U(k) \times SO(2n-2k) )$. We study endomorphisms of the rational cohomology algebra of $M_{n,k}$. We prove that an endomorphism of the rational cohomology algebra of $M_{n,k}$, which maps all the Chern classes of the canonical $k$-plane bundle over $M_{n,k}$ to zero, or maps all the Pontrjagin classes of the canonical, real, oriented $(2n-2k)$-plane bundle over $M_{n,k}$ to zero, is the zero endomorphism. Additionally, we prove that if an endomorphism of the rational cohomology algebra of $M_{n,k}$ vanishes on $H^{2}(M_{n,k}; \mathbb{Q})$, and admits a splitting, then the splitting equals zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Endomorphisms of the Cohomology Algebra of the Even Orthogonal Grassmannian Goswami, Arnab Sarkar, Swagata Algebraic Topology Primary: 57T15, 14M17, 55M99, Secondary: 14M15, 55S99 Let $M_{n,k}$ denote the even orthogonal Grassmanian, $SO(2n) / (U(k) \times SO(2n-2k) )$. We study endomorphisms of the rational cohomology algebra of $M_{n,k}$. We prove that an endomorphism of the rational cohomology algebra of $M_{n,k}$, which maps all the Chern classes of the canonical $k$-plane bundle over $M_{n,k}$ to zero, or maps all the Pontrjagin classes of the canonical, real, oriented $(2n-2k)$-plane bundle over $M_{n,k}$ to zero, is the zero endomorphism. Additionally, we prove that if an endomorphism of the rational cohomology algebra of $M_{n,k}$ vanishes on $H^{2}(M_{n,k}; \mathbb{Q})$, and admits a splitting, then the splitting equals zero. |
| title | Endomorphisms of the Cohomology Algebra of the Even Orthogonal Grassmannian |
| topic | Algebraic Topology Primary: 57T15, 14M17, 55M99, Secondary: 14M15, 55S99 |
| url | https://arxiv.org/abs/2509.09363 |