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The Orbit Space and Basic Forms of a Proper Lie Groupoid

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Current Trends in Analysis, its Applications and Computation

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is generalized to the case of a proper Lie groupoid, in which the orbit space is equipped with the quotient diffeological structure. As an application of this, we obtain a de Rham theorem for the de Rham complex on the orbit space.

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Notes

  1. 1.

    The statement of this lemma was communicated to the author by Eugene Lerman; however, the proof is the author’s.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Central Michigan University, Mount Pleasant, MI, USA

    Jordan Watts

Authors
  1. Jordan Watts
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Correspondence to Jordan Watts .

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Editors and Affiliations

  1. Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Paula Cerejeiras

  2. Institut of Applied Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Michael Reissig

  3. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Irene Sabadini

  4. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Joachim Toft

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Watts, J. (2022). The Orbit Space and Basic Forms of a Proper Lie Groupoid. In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-87502-2_52

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