Closed hyperbolic manifolds without spin^c structures

2025/01/14 by Jacopo G. Chen, Chen, Jacopo G.
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2501.07796

openalex publication_date 2025/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In all dimensions n ≥ 5, we prove the existence of closed orientable hyperbolic manifolds that do not admit any spin^c structure, and in fact we show that there are infinitely many commensurability classes of such manifolds. These manifolds all have non-vanishing third Stiefel--Whitney class w₃ and are all arithmetic of simplest type. More generally, we show that for each k ≥ 1 and n ≥ 4k+1, there exist infinitely many commensurability classes of closed orientable hyperbolic n-manifolds M with w_4k-1(M) ≠ 0.

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