$\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866918191166390272 |
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| author | Choi, Suhyoung |
| author_facet | Choi, Suhyoung |
| contents | A manifold $M$ possesses a real projective structure if it has an atlas consisting of charts mapping to $\mathbf{S}^n$, where the transition maps lie in $\mathrm{SL}_\pm(n+1, \mathbf{R})$. In this context, we present a concise proof demonstrating that $\mathbf{RP}^n\#\mathbf{RP}^n$ and a few other manifolds do not possess a real projective structure when $n\geq3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and Çoban for $n\geq 4$. To do this, we reprove the classification of closed real projective manifolds with infinite-cyclic holonomy groups by Benoist due to a small error. We will leverage the concept of the octantizability of real projective manifolds with nilpotent holonomy groups, as introduced by Benoist and Smillie, which serves as a powerful tool. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_02924 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure Choi, Suhyoung Geometric Topology Primary 57M50, Secondary 53A20, 53C15 A manifold $M$ possesses a real projective structure if it has an atlas consisting of charts mapping to $\mathbf{S}^n$, where the transition maps lie in $\mathrm{SL}_\pm(n+1, \mathbf{R})$. In this context, we present a concise proof demonstrating that $\mathbf{RP}^n\#\mathbf{RP}^n$ and a few other manifolds do not possess a real projective structure when $n\geq3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and Çoban for $n\geq 4$. To do this, we reprove the classification of closed real projective manifolds with infinite-cyclic holonomy groups by Benoist due to a small error. We will leverage the concept of the octantizability of real projective manifolds with nilpotent holonomy groups, as introduced by Benoist and Smillie, which serves as a powerful tool. |
| title | $\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure |
| topic | Geometric Topology Primary 57M50, Secondary 53A20, 53C15 |
| url | https://arxiv.org/abs/2209.02924 |