A universal Euler characteristic of non-orbifold groupoids and Riemannian structures on Lie groupoids Article Swipe
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2310.12073
· OA: W4387801520
We introduce the universal Euler characteristic of an orbit space definable groupoid, a class of groupoids containing cocompact proper Lie groupoids as well as translation groupoids for semialgebraic group actions. Generalizing results of Gusein-Zade, Luengo, and Melle-Hernández, we show that every additive and multiplicative invariant of orbit space definable groupoids with an additional local triviality hypothesis arises as a ring homomorphism applied to the universal Euler characteristic. This in particular includes the $Γ$-orbifold Euler characteristic introduced by the first and third authors when $Γ$ is a finitely presented group. For Lie groupoids with Riemannian structures in the sense of del Hoyo-Fernandes, and for Cartan-Lie groupoids with Riemannian structures in the sense of Kotov-Strobl, we study the Riemannian structures induced on the suborbifolds of the inertia space given by images of isotropy groups. We realize the $\mathbb{Z}$-orbifold Euler characteristic as the integral of a differential form defined on the arrow space or the space of composable pairs of arrows of the original groupoid.
