The Holonomy Groupoids of Singularly Foliated Bundles
We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211306 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | The Holonomy Groupoids of Singularly Foliated Bundles. Lachlan Ewen MacDonald. SIGMA 17 (2021), 043, 34 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862738854030606336 |
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| author | MacDonald, Lachlan Ewen |
| author_facet | MacDonald, Lachlan Ewen |
| citation_txt | The Holonomy Groupoids of Singularly Foliated Bundles. Lachlan Ewen MacDonald. SIGMA 17 (2021), 043, 34 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in the sense of Kamber-Tondeur and singular foliations. We define hierarchies of diffeological holonomy groupoids associated with such bundles, which arise from the parallel transport of jet/germinal conservation laws. We show that the groupoids associated in this manner to trivial singularly foliated bundles are quotients of Androulidakis-Skandalis holonomy groupoids, which coincide with Androulidakis-Skandalis holonomy groupoids in the regular case. Finally, we prove the functoriality of all our constructions under appropriate morphisms.
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| first_indexed | 2026-04-17T17:16:43Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211306 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T17:16:43Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | MacDonald, Lachlan Ewen 2025-12-29T11:07:06Z 2021 The Holonomy Groupoids of Singularly Foliated Bundles. Lachlan Ewen MacDonald. SIGMA 17 (2021), 043, 34 pages 1815-0659 2020 Mathematics Subject Classification: 53C05; 53C12; 53C29 arXiv:2006.14271 https://nasplib.isofts.kiev.ua/handle/123456789/211306 https://doi.org/10.3842/SIGMA.2021.043 We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are in plentiful supply, arising naturally for any Lie groupoid-equivariant bundle, and simultaneously generalising regularly foliated bundles in the sense of Kamber-Tondeur and singular foliations. We define hierarchies of diffeological holonomy groupoids associated with such bundles, which arise from the parallel transport of jet/germinal conservation laws. We show that the groupoids associated in this manner to trivial singularly foliated bundles are quotients of Androulidakis-Skandalis holonomy groupoids, which coincide with Androulidakis-Skandalis holonomy groupoids in the regular case. Finally, we prove the functoriality of all our constructions under appropriate morphisms. This research was supported by the Australian Research Council through the Discovery Project grant DP200100729. I extend special thanks to V. Mathai for taking me on as a postdoc at the University of Adelaide, and to I. Androulidakis, for enlightening discussions and correspondence in late 2018 and for encouraging me to think about singular foliations. I thank B. McMillan, for helpful discussions concerning conservation laws. Finally, I extend deep thanks to the anonymous referees, whose careful consideration and critique of the paper have greatly improved its exposition. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Holonomy Groupoids of Singularly Foliated Bundles Article published earlier |
| spellingShingle | The Holonomy Groupoids of Singularly Foliated Bundles MacDonald, Lachlan Ewen |
| title | The Holonomy Groupoids of Singularly Foliated Bundles |
| title_full | The Holonomy Groupoids of Singularly Foliated Bundles |
| title_fullStr | The Holonomy Groupoids of Singularly Foliated Bundles |
| title_full_unstemmed | The Holonomy Groupoids of Singularly Foliated Bundles |
| title_short | The Holonomy Groupoids of Singularly Foliated Bundles |
| title_sort | holonomy groupoids of singularly foliated bundles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211306 |
| work_keys_str_mv | AT macdonaldlachlanewen theholonomygroupoidsofsingularlyfoliatedbundles AT macdonaldlachlanewen holonomygroupoidsofsingularlyfoliatedbundles |