## Departmental PhD Thesis Exam – Joshua Lackman

Tuesday, April 12, 2022 at 11:00 a.m. (sharp)

PhD Candidate: Joshua Lackman
Supervisor: Marco Gualtieri
Thesis title: The van Est Map on Geometric Stacks

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We generalize the van Est map and isomorphism theorem in three ways. First, we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the foliated cohomology of a stack $\mathcal{H}\to\mathcal{G}$ mapping into it. At the level of Lie grouoids, this amounts to describing the van Est map as a map from Lie groupoid cohomology to the cohomology of a particular LA-groupoid. We do this by, essentially, associating to any
(nice enough) homomorphism of Lie groupoids $f:H\to G$ a natural foliation
of the stack $[H^0/H]\,.$ In the case of a wide subgroupoid $H\xhookrightarrow{}G\,,$ this foliation can be thought of as equipping
the normal bundle of $H$ with the structure of an LA-groupoid. This generalization allows us to derive results that couldn’t be obtained with the usual van Est map for Lie groupoids. In particular, we recover classical results, including van Est’s isomorphism theorem about the maximal compact subgroup, which we generalize to proper subgroupoids, as well as the Poincar\'{e} lemma. Secondly, we generalize the functions that we can take cohomology of in the context of the van Est map; instead of using functions valued in representations, we can use functions valued in modules — for example, we can use $S^1$-valued functions and $\mathbb{Z}$-valued functions. This allows us to obtain classical results about linearizing group actions, as well as results about lifting group actions to gerbes. Finally, everything we do works in the holomorphic category in addition to the smooth category.

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The draft of the thesis can be found here: Thesis Draft March 21