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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On two-dimensional holonomy
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by João Faria Martins and Roger Picken PDF
Trans. Amer. Math. Soc. 362 (2010), 5657-5695 Request permission


We define the thin fundamental categorical group ${\mathcal P}_2(M,*)$ of a based smooth manifold $(M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$ and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps are rank-$n$ homotopic, when the rank of the differential of the homotopy between them equals $n$. Let $\mathcal {C}(\mathcal {G})$ be a Lie categorical group coming from a Lie crossed module ${\mathcal {G}= (\partial \colon E \to G,\triangleright )}$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal P}_2(M,*) \to \mathcal {C}(\mathcal {G})$, by using a notion of categorical connections, being a pair $(\omega ,m)$, where $\omega$ is a connection 1-form on $P$, a principal $G$ bundle over $M$, and $m$ is a 2-form on $P$ with values in the Lie algebra of $E$, with the pair $(\omega ,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.
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Additional Information
  • João Faria Martins
  • Affiliation: Edifício dos Departamentos de Matemática da FCUP, Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
  • Address at time of publication: Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
  • Email:,
  • Roger Picken
  • Affiliation: Departamento de Matemática, Instituto Superior Técnico (Universidade Técnica de Lisboa), Av. Rovisco Pais, 1049-001 Lisboa, Portugal
  • Email:
  • Received by editor(s): December 4, 2007
  • Received by editor(s) in revised form: April 30, 2008
  • Published electronically: June 9, 2010
  • Additional Notes: The first author was financed by Fundação para a Ciência e Tecnologia (Portugal), post-doctoral grant number SFRH / BPD / 34138 / 2006. This work was supported by the Programa Operacional Ciência e Inovação 2010, financed by the Fundação para a Ciência e a Tecnologia (FCT) and cofinanced by the European Community fund FEDER, in part through the research project Quantum Topology POCI / MAT / 60352 / 2004
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 5657-5695
  • MSC (2010): Primary 53C29; Secondary 18D05
  • DOI:
  • MathSciNet review: 2661492