arXiv subjects

Jacopo G. Chen

Publications and source records attributed to Jacopo G. Chen.

5 recordsLinked to original sources

arXiv2026

On a twisted Singer conjecture

According to the Singer conjecture, the $L^2$-Betti numbers of a closed aspherical manifold are expected to vanish outside the middle dimension. In this paper, we study a natural analog of the Singer conjecture concerning the vanishing of twisted $L^2$-Betti numbers outside the lower middle dimension. These invariants can be thought of as $L^2$-Betti numbers of an infinite cyclic cover, depending on a first cohomology class. This is related to a previous conjecture by the author, which connects the twisted $L^2$-Euler characteristic to the Thurston norm, and which we prove for a class of closed arithmetic hyperbolic manifolds.

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arXiv2025

Computing the twisted $L^2$-Euler characteristic

We present an algorithm that computes Friedl and Lück's twisted $L^2$-Euler characteristic for a suitable regular CW complex, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonné determinant. The algorithm needs to run for an extremely long time to certify its outputs, but a truncated, human-assisted version produces very good results in many cases, such as hyperbolic link complements, closed census 3-manifolds, free-by-cyclic groups, and higher-dimensional examples, such as the fiber of the Ratcliffe-Tschantz manifold.

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arXiv2025

Closed hyperbolic manifolds without $\text{spin}^c$ structures

In all dimensions $n \ge 5$, we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text{spin}^c$ 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 $w_3$ and are all arithmetic of simplest type. More generally, we show that for each $k \ge 1$ and $n \ge 4k+1$, there exist infinitely many commensurability classes of closed orientable hyperbolic $n$-manifolds $M$ with $w_{4k-1}(M) \ne 0$.

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arXiv2025

Some closed hyperbolic 5-manifolds

We give an explicit construction of a family of closed arithmetic hyperbolic 5-manifolds, tessellated by $117 964 800 = 512 \cdot 16 \cdot 14400$ copies of a Coxeter simplicial prism. We proceed to study various properties of these manifolds, such as the volume and the first Betti number. We also describe a related family of 5-manifolds with a larger volume, but a simpler construction.

math.GT↗

arXiv2025

Non-cobordant hyperbolic manifolds

In all dimensions $n \ge 4$ not of the form $4m+3$, we show that there exists a closed hyperbolic $n$-manifold which is not the boundary of a compact $(n+1)$-manifold. The proof relies on the relationship between the cobordism class and the fixed point set of an involution on the manifold, together with a geodesic embedding of Kolpakov, Reid and Slavich. We also outline a possible approach to cover the dimensions $4m+3 \ne 2^k-1$.

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