We exhibit the first examples of compact, orientable, hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions n≥4. The core of the argument is the construction of a compact, oriented, hyperbolic 4–manifold M that contains a surface S of genus 3 with self-intersection 1. The 4–manifold M has an odd intersection form and is hence not spin. It is built by carefully assembling some right-angled 120–cells along a pattern inspired by the minimum trisection of CP2. The manifold M is also the first example of a compact, orientable, hyperbolic 4–manifold satisfying either of these conditions: H2(M,Z) is not generated by geodesically immersed surfaces. There is a covering ˜M that is a nontrivial bundle over a compact surface.
Compact hyperbolic manifolds without spin structures
Bruno Martelli;Stefano Riolo;Leone Slavich
2020-01-01
Abstract
We exhibit the first examples of compact, orientable, hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions n≥4. The core of the argument is the construction of a compact, oriented, hyperbolic 4–manifold M that contains a surface S of genus 3 with self-intersection 1. The 4–manifold M has an odd intersection form and is hence not spin. It is built by carefully assembling some right-angled 120–cells along a pattern inspired by the minimum trisection of CP2. The manifold M is also the first example of a compact, orientable, hyperbolic 4–manifold satisfying either of these conditions: H2(M,Z) is not generated by geodesically immersed surfaces. There is a covering ˜M that is a nontrivial bundle over a compact surface.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
