Representations up to homotopy and Bott's spectral sequence for Lie groupoids

By C. A. Abad, M. Crainic. 2009

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Abstract In this paper we study the notion of representation up to homotopy of a Lie groupoid and the resulting derived category, and show that the adjoint representation is well defined as a representation up to homotopy. As an application, we extend Bottʼs spectral sequence converging to the cohomology of classifying spaces of Lie groups to the case of Lie groupoids. We explain the relation of this construction with the models of Cartan and Getzler for equivariant cohomology. Our work is closely related to and inspired by Behrendʼs [3] , Bottʼs [4] , and Getzlerʼs [9] .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1c55da466a21eecc5545063945e5dac2a9dfec2d

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Citations in the corpus

VB-groupoids and representation theory of Lie groupoids (2010, background): “Recall [1, 8] that the adjoint representation of a Lie groupoid G ⇒ M is a representation up to homotopy on the 2-term complex A ρ → TM , where A is the Lie algebroid of G and ρ is the anchor map.”
Vector bundles over Lie groupoids and algebroids (2014, background): “choosing a splitting of the core exact sequence of (3.1), one can associate to every arrow in G⇒Ma linear map between the ﬁbers of the complex C−→tΓ E, and this yields a representation up to homotopy [2]. It is proven in [14] that there is a one-to-one correspondence between isomorphism classes of V”
Riemannian metrics on Lie groupoids (2014, background): “[1]), however a groupoid may not have a multiplicative connection.”
Multiplicative forms and Spencer operators (2012, background): “3 Cartan connections on groupoids The notion of Cartan connections on a Lie groupoid G arises when looking at the adjoint representation of G [1].”
Geometry of orbit spaces of proper Lie groupoids (2010, methodology): “ there is a way to lift the representations on the normal bundles to a non-associative action GEOMETRY OF ORBIT SPACES OF PROPER LIE GROUPOIDS 13 on TG0, part of a representation up to homotopy as in [AbCr], called the adjoint representation. This construction depends on a choice of an Ehresmann con”