Journal of Symplectic Geometry

Volume 16 (2018)

Number 2

Obstructions to the integrability of $\mathcal{VB}$-algebroids

Pages: 439 – 483



Alejandro Cabrera (Instituto de Matemática, Universidade Federal de Rio de Janeiro, Brazil)

Olivier Brahic (Departamento de Matemática, Universidade Federal do Paraná, Curitiba, Brazil)

Cristian Ortiz (Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil)


$\mathcal{VB}$-groupoids are vector bundle objects in the category of Lie groupoids: the total and the base spaces of the vector bundle are Lie groupoids and the vector bundle structure maps are required to define Lie groupoid morphisms. The infinitesimal version of $\mathcal{VB}$-groupoids are $\mathcal{VB}$-algebroids, namely, vector bundle objects in the category of Lie algebroids. Following recent developments in the area, we show that a $\mathcal{VB}$-algebroid is integrable to a $\mathcal{VB}$-groupoid if and only if its base algebroid is integrable and the spherical periods of certain underlying cohomology classes vanish identically. We illustrate our results in concrete examples. Finally, we obtain as a corollary computable obstructions for a $2$-term representation up to homotopy of Lie algebroid to arise as the infinitesimal counterpart of a smooth such representation of a Lie groupoid.

Received 9 September 2014

Accepted 10 November 2017

Published 18 July 2018