RPⁿ # RPⁿ and some others admit no real projective structure

2022/09/07 by Suhyoung Choi, Choi, Suhyoung
Mathematics · #53C15 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Primary 57M50 #Secondary 53A20

paper · pdf · doi:10.48550/arxiv.2209.02924

openalex publication_date 2022/09/07 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

A manifold M possesses a real projective structure if it has an atlas consisting of charts mapping to Sⁿ, where the transition maps lie in SL_±(n+1, R). In this context, we present a concise proof demonstrating that RPⁿ#RPⁿ and a few other manifolds do not possess a real projective structure when n≥3. Notably, our proof is shorter than those provided by Cooper-Goldman for n=3 and Çoban for n≥ 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.

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