2025/12/11 by Tsuyoshi Kato, Kato, Tsuyoshi, Daisuke Kishimoto +3
Mathematics · #55M20 #57R19 #58C30 #Algebraic Topology (math.AT) #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2512.10182
openalex publication_date 2025/12/11 · openalex created_date 2025/12/13 · openalex updated_date 2026/07/28
We develop the Lefschetz fixed-point theory for noncompact manifolds of bounded geometry and uniformly continuous maps. Specifically, we define the uniform Lefschetz class \mathscrL(f) of a uniformly continuous map f\colon M→ M of a uniform simply-connected noncompact complete Riemannian manifold of bounded geometry M satisfying d(f,1)<∞, and prove that \mathscrL(f)=0 if and only if f is uniformly homotopic to a strongly fixed-point free (without fixed-points on M and at infinity) uniformly continuous map. To achieve this, we introduce a new cohomology for metric spaces, called uniform bounded cohomology, which is a variant of bounded cohomology, and develop an obstruction theory formulated in terms of this cohomology.