Discrete degree of symmetry of manifolds

2021/12/10 by Ignasi Mundet i Riera, Riera, Ignasi Mundet i · 3 citations
Mathematics · #54H15 #57S17 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2112.05599

openalex publication_date 2021/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define the discrete degree of symmetry disc-sym(X) of a closed n-manifold X as the biggest m≥ 0 such that X supports an effective action of (\mathbf Z/r)^m for arbitrarily big values of r. We prove that if X is connected then disc-sym(X)≤ 3n/2. We propose the question of whether for every closed connected n-manifold X the inequality disc-sym(X)≤ n holds true, and whether the only closed connected n-manifold X for which disc-sym(X)=n is the torus Tⁿ. We prove partial results providing evidence for an affirmative answer to this question.

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• Finite abelian group actions on weakly Lefschetz cohomologically symplectic manifolds
• Iterated finite group actions on closed connected aspherical manifolds
• Large and iterated finite group actions on manifolds admitting non-zero degree maps to nilmanifolds

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