A cusped hyperbolic 4-manifold without spin structures

TL;DR

The paper proves the existence of a cusped orientable arithmetic hyperbolic $4$-manifold $M$ with $w_2(M) eq 0$, i.e. lacking a spin structure. It achieves this by constructing a hyperbolic 4-manifold with corners $X$ tessellated by the right-angled polytope $P^4$, containing an orientable surface $S$ with self-intersection $S rac{S}{S} = rac{1}{1}$, and then doubling along embedded facets to yield $M$. The construction relies on the Kerckhoff–Storm family of polytopes, arranging a nested chain $P^2 rac{P^3}{P^4}$ to realize a geometrically finite substructure $N ightarrow X$, which deformation retracts onto $S$ and enforces a nontrivial $w_2$ obstruction. This work extends spin-structure obstructions to cusped hyperbolic 4-manifolds, providing explicit arithmetic examples and a blueprint for generating such manifolds via polytopal tessellations and facet-doubling.

Abstract

We build a non-compact, orientable, hyperbolic four-manifold of finite volume that does not admit any spin structure.

Figures (10)

• Figure 1: On the left, a schematic picture of the three-dimensional thickening $N = N_0 \cup N_1 \cup N_2$ of the piecewise geodesic surface $S = S_0 \cup S_1 \cup S_2$, where $S_i \subset N_i$ are totally geodesic manifolds with corners. It is not a manifold near the auxiliary surface with corners $\Sigma = N_0 \cap N_1 \cap N_2$ (represented by a black dot). On the right, the thickening $X$ of $N$: a 4-manifold with corners, neighbourhood of $S$ in $M$, tessellated by some copies of $P^4$ (represented by 10 gray pentagons)
• Figure 2: The extremal, half-height and central facets $E_i \cong P^3$, $H_i$ and $C_{ij}$ of $P^4$, where $\{ i,j,k,l \} = \{ 1,2,3,4 \}$. The ideal vertices are in white. Note the compact pentagon $E_i \cap E_j \cong P^2$.
• Figure 3: The surface $\Sigma$ with corners obtained by gluing 8 copies of the right-angled pentagon $P^2$ (four edges of the big dodecagon are glued in pairs as indicated by the black arrows). It is a holed torus, and deformation retracts onto the red theta-graph $\Theta = \gamma_0 \cup \gamma_1 \cup \gamma_2$. The three red oriented curves $\gamma_0, \gamma_1$ and $\gamma_2$ go as indicated by the gray arrows.
• Figure 4: The top of the $3$-manifold with corners $\Sigma^\mathrm{thick}$ . The green lines indicate its tessellation into 8 copies of $P^3$. As usual, the ideal vertices are in white.
• Figure 5: The 3-manifold with corners $N_0$ is built by gluing some top facets of $\Sigma^\mathrm{thick}$ as indicated by the blue letters P and Q. It has 5 top facets. The four vertical blue edges are glued making an angle of $2\pi$.

Theorems & Definitions (12)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Proposition 6
• proof
• Proposition 7
• proof