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$\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure

Suhyoung Choi

TL;DR

The paper proves that $\mathbb{RP}^n \# \mathbb{RP}^n$ (and a class of related manifolds) admit no real projective structure for $n\ge 3$, by refining Benoist’s classification of manifolds with infinite-cyclic holonomy and leveraging octantizability. The authors develop a streamlined approach that combines deformation-space analysis, diagonalizable holonomy perturbations, and Kuiper completion to derive topological obstructions, culminating in a constructive brick decomposition into Benoist- or Hopf-type pieces. This reduces the problem to ruling out compatible global actions on the developing image, and the method extends to generalizations beyond the $n=3$ Cooper–Goldman/Çoban cases. The work thus clarifies the landscape of real projective structures on high-dimensional manifolds and provides a robust toolkit for excluding such structures via holonomy-end geometry and deformation theory, with a focus on diagonalizable and nilpotent holonomy cases.

Abstract

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.

$\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure

TL;DR

The paper proves that (and a class of related manifolds) admit no real projective structure for , by refining Benoist’s classification of manifolds with infinite-cyclic holonomy and leveraging octantizability. The authors develop a streamlined approach that combines deformation-space analysis, diagonalizable holonomy perturbations, and Kuiper completion to derive topological obstructions, culminating in a constructive brick decomposition into Benoist- or Hopf-type pieces. This reduces the problem to ruling out compatible global actions on the developing image, and the method extends to generalizations beyond the Cooper–Goldman/Çoban cases. The work thus clarifies the landscape of real projective structures on high-dimensional manifolds and provides a robust toolkit for excluding such structures via holonomy-end geometry and deformation theory, with a focus on diagonalizable and nilpotent holonomy cases.

Abstract

A manifold possesses a real projective structure if it has an atlas consisting of charts mapping to , where the transition maps lie in . In this context, we present a concise proof demonstrating that and a few other manifolds do not possess a real projective structure when . Notably, our proof is shorter than those provided by Cooper-Goldman for and Çoban for . 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.
Paper Structure (19 sections, 19 theorems, 19 equations)

This paper contains 19 sections, 19 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Comments on the results of Benoist
  3. Outline
  4. Geometric structures and Benoist manifolds
  5. Geometric structures and the deformation spaces
  6. Projective automorphism groups
  7. Benoist manifolds or $A_g$ or $B_g$
  8. Splitting of real projective manifolds
  9. Topology of Hopf and Benoist manifolds
  10. Reproving Proposition 4.9 of Benoist
  11. $\Delta$-orbits
  12. Simplices
  13. Ambient pieces
  14. Decomposition
  15. General cases
  16. ...and 4 more sections

Key Result

Corollary 1.3

Let $M$ be a closed real projective manifold, $n \geq 3$, where $\pi_1(M)$ acts trivially on $\pi_i(M) \otimes {\mathbb Z}_2$ for $i \geq 1$. Then the holonomy group and the fundamental group of cannot be a nontrivially twisted finite extension of ${\mathbb Z}$.

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3: Lemma 2.1 of Benoist94
  • Proposition 2.4
  • proof
  • Proposition 2.5: from Proposition 1 of Benoist Benoist94
  • ...and 34 more