论文标题
美元
$\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no $\mathbf{RP}^n$-structure
论文作者
论文摘要
歧管$ m $具有$ \ mathbf {rp}^n $ - 结构，如果它的图表由图表映射到$ \ mathbf {s}^n $，则过渡地图位于$ \ mathrm {sl} _ \ pm pm pm pm pm（n+1，\ 1，\ mathbf {r}）中。在这种情况下，我们提供简洁的证明，证明$ \ Mathbf {rp}^n \＃\ Mathbf {rp}^n $，而其他一些歧管则没有$ \ mathbf {rp}^n $结构$ n \ egeq3 $。值得注意的是，我们的证明比Cooper-Goldman提供的$ n = 3 $和Coban的证明要短，而GEQ 4 $。为此，我们谴责$ \ mathbf {rp}^n $ -manifolds与贝诺斯特（Benoist）的无限循环自律组的分类，原因很小。如Benoist和Smillie所介绍的那样，我们将利用$ \ mathbf {rp}^n $ -manifolds的八雄性概念。
A manifold $M$ possesses an $\mathbf{RP}^n$-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 an $\mathbf{RP}^n$-structure when $n\geq3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and Coban for $n\geq 4$. To do this, we reprove the classification of closed $\mathbf{RP}^n$-manifolds with infinite cyclic holonomy groups by Benoist due to a small error. We will leverage the concept of octantizability of $\mathbf{RP}^n$-manifolds with nilpotent holonomy groups, as introduced by Benoist and Smillie, which serves as a powerful tool.
