A universal Euler characteristic of non-orbifold groupoids and Riemannian structures on Lie groupoids
| Autor: | Farsi, Carla, Proctor, Emily, Seaton, Christopher |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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-Hern\'andez, 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 $\Gamma$-orbifold Euler characteristic introduced by the first and third authors when $\Gamma$ 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. Comment: 23 pages |
| Databáze: | arXiv |
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