June  2022, 14(2): 377-380. doi: 10.3934/jgm.2022012

Kirill Mackenzie, Bangor and holonomy

Mathematics Division, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, UK

Received  September 2021 Published  June 2022 Early access  June 2022

Citation: Ronald Brown. Kirill Mackenzie, Bangor and holonomy. Journal of Geometric Mechanics, 2022, 14 (2) : 377-380. doi: 10.3934/jgm.2022012
References:
[1]

M. E.-S. A.-F. Aof and R. Brown, The holonomy groupoid of a locally topological groupoid, Top. App., 47 (1992), 97-113.  doi: 10.1016/0166-8641(92)90065-8.

[2]

I. Androulidakis and M. Zambon, Holonomy transformations for singular foliations, Advances in Mathematics, 256 (2014), 348-397.  doi: 10.1016/j.aim.2014.02.003.

[3]

R. Brown, Ten topologies for $X \times Y$, Quart. J. Math., 14 (1963), 303-319.  doi: 10.1093/qmath/14.1.303.

[4]

R. Brown, Groupoids and crossed objects in algebraic topology, Homology, Homotopy and Applications, 1 (1999), 1-78.  doi: 10.4310/HHA.1999.v1.n1.a1.

[5]

R. Brown, Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions, Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland), 76 (2007), 51-63.  doi: 10.4064/bc76-0-3.

[6]

R. Brown, Modelling and Computing Homotopy Types: I, Indagationes Math. (Special issue in honor of L.E.J. Brouwer), 29 (2018), 459–482. doi: 10.1016/j.indag.2017.01.009.

[7]

R. Brown, Not just an idle game: The story of the search for higher dimensional versions of the Poincaré fundamental group, Math Intell., (2021), Open Access https://link.springer.com/content/pdf/10.1007/s00283-021-10110-9.pdf. doi: 10.1007/s00283-021-10110-9.

[8]

R. Brown and J. F. Glazebrook, Connections, local subgroupoids, and a holonomy Lie groupoid of a line bundle gerbe, arXiv: math/0210322.

[9]

R. Brown and P. J. Higgins, The algebra of cubes, Colimit theorems for relative homotopy groups, J. Pure Applied Algebra, 21 (1981), 233-260.  doi: 10.1016/0022-4049(81)90080-3.

[10]

R. Brown, P. J. Higgins and R. Sivera, Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids, Tracts in Mathematics, 15, European Mathematical Society, 2011. doi: 10.4171/083.

[11]

R. Brown and I. Icen, Towards a 2-dimensional notion of holonomy, Advances in Mathematics, 178 (2003), 141-175.  doi: 10.1016/S0001-8708(02)00074-9.

[12]

R. Brown and I. Icen, Lie local subgroupoids and their holonomy and monodromy Lie groupoids, Top. Appl., 115 (2001), 125-138.  doi: 10.1016/S0166-8641(00)00062-6.

[13]

R. BrownI. Icen and O. Mucuk, Examples and coherence properties of local subgroupoids, Top. Appl., 127 (2003), 393-408.  doi: 10.1016/S0166-8641(02)00101-3.

[14]

R. BrownG. Janelidze and G. Peschke, Van Kampen's theorem for locally sectionable maps, Theory and Applications of Categories, 36 (2021), 48-64. 

[15]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.

[16]

R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid, Cah. Top. Géom. Diff. Cat., 36 (1995), 345-369. 

[17]

R. Brown and O. Mucuk, Foliations, locally Lie groupoids, and holonomy, Cah. Top. Géom. Diff. Cat., 37 (1996), 61-71. 

[18]

R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cah. Top. Géom. Diff., 17 (1976), 343-362. 

[19]

A. Grothendieck, Esquisse d'un Programme, 1984, 54 pp.

[20]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc, Lecture Note Series, 124, Cambridge, 1987. doi: 10.1017/CBO9781107325883.

[21]

J. Martins and R. F. Picken, Surface holonomy for non-abelian 2-bundles via double groupoids, Advances in Mathematics, 226 (2011), 3309-3366.  doi: 10.1016/j.aim.2010.10.017.

[22]

J. Martins and R. F. Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module, Differential Geometry and its Applications, 29 (2011), 179-206.  doi: 10.1016/j.difgeo.2010.10.002.

[23]

J. Pradines, Théorie de Lie groupoïdes différentiables. Relations entres propriétés locales et globales, C. R. Acad. Sci. Paris., 261 (1966), 907-910. 

[24]

J. Pradines, In Ehresmann's footsteps: From group geometries to groupoid geometries, preprint, arXiv: 0711.1608v1. doi: 10.4064/bc76-0-5.

[25]

J. Virsik, On the holonomy of higher order connections, Cah. Géom. Diff., (1971), 197–212.

show all references

References:
[1]

M. E.-S. A.-F. Aof and R. Brown, The holonomy groupoid of a locally topological groupoid, Top. App., 47 (1992), 97-113.  doi: 10.1016/0166-8641(92)90065-8.

[2]

I. Androulidakis and M. Zambon, Holonomy transformations for singular foliations, Advances in Mathematics, 256 (2014), 348-397.  doi: 10.1016/j.aim.2014.02.003.

[3]

R. Brown, Ten topologies for $X \times Y$, Quart. J. Math., 14 (1963), 303-319.  doi: 10.1093/qmath/14.1.303.

[4]

R. Brown, Groupoids and crossed objects in algebraic topology, Homology, Homotopy and Applications, 1 (1999), 1-78.  doi: 10.4310/HHA.1999.v1.n1.a1.

[5]

R. Brown, Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions, Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland), 76 (2007), 51-63.  doi: 10.4064/bc76-0-3.

[6]

R. Brown, Modelling and Computing Homotopy Types: I, Indagationes Math. (Special issue in honor of L.E.J. Brouwer), 29 (2018), 459–482. doi: 10.1016/j.indag.2017.01.009.

[7]

R. Brown, Not just an idle game: The story of the search for higher dimensional versions of the Poincaré fundamental group, Math Intell., (2021), Open Access https://link.springer.com/content/pdf/10.1007/s00283-021-10110-9.pdf. doi: 10.1007/s00283-021-10110-9.

[8]

R. Brown and J. F. Glazebrook, Connections, local subgroupoids, and a holonomy Lie groupoid of a line bundle gerbe, arXiv: math/0210322.

[9]

R. Brown and P. J. Higgins, The algebra of cubes, Colimit theorems for relative homotopy groups, J. Pure Applied Algebra, 21 (1981), 233-260.  doi: 10.1016/0022-4049(81)90080-3.

[10]

R. Brown, P. J. Higgins and R. Sivera, Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids, Tracts in Mathematics, 15, European Mathematical Society, 2011. doi: 10.4171/083.

[11]

R. Brown and I. Icen, Towards a 2-dimensional notion of holonomy, Advances in Mathematics, 178 (2003), 141-175.  doi: 10.1016/S0001-8708(02)00074-9.

[12]

R. Brown and I. Icen, Lie local subgroupoids and their holonomy and monodromy Lie groupoids, Top. Appl., 115 (2001), 125-138.  doi: 10.1016/S0166-8641(00)00062-6.

[13]

R. BrownI. Icen and O. Mucuk, Examples and coherence properties of local subgroupoids, Top. Appl., 127 (2003), 393-408.  doi: 10.1016/S0166-8641(02)00101-3.

[14]

R. BrownG. Janelidze and G. Peschke, Van Kampen's theorem for locally sectionable maps, Theory and Applications of Categories, 36 (2021), 48-64. 

[15]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.

[16]

R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid, Cah. Top. Géom. Diff. Cat., 36 (1995), 345-369. 

[17]

R. Brown and O. Mucuk, Foliations, locally Lie groupoids, and holonomy, Cah. Top. Géom. Diff. Cat., 37 (1996), 61-71. 

[18]

R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cah. Top. Géom. Diff., 17 (1976), 343-362. 

[19]

A. Grothendieck, Esquisse d'un Programme, 1984, 54 pp.

[20]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc, Lecture Note Series, 124, Cambridge, 1987. doi: 10.1017/CBO9781107325883.

[21]

J. Martins and R. F. Picken, Surface holonomy for non-abelian 2-bundles via double groupoids, Advances in Mathematics, 226 (2011), 3309-3366.  doi: 10.1016/j.aim.2010.10.017.

[22]

J. Martins and R. F. Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module, Differential Geometry and its Applications, 29 (2011), 179-206.  doi: 10.1016/j.difgeo.2010.10.002.

[23]

J. Pradines, Théorie de Lie groupoïdes différentiables. Relations entres propriétés locales et globales, C. R. Acad. Sci. Paris., 261 (1966), 907-910. 

[24]

J. Pradines, In Ehresmann's footsteps: From group geometries to groupoid geometries, preprint, arXiv: 0711.1608v1. doi: 10.4064/bc76-0-5.

[25]

J. Virsik, On the holonomy of higher order connections, Cah. Géom. Diff., (1971), 197–212.

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