Arnold-Kuiper-Massey theorem in nLab
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Context
Topology
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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fiber space, space attachment
Extra stuff, structure, properties
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Kolmogorov space, Hausdorff space, regular space, normal space
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sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
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open subspaces of compact Hausdorff spaces are locally compact
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compact spaces equivalently have converging subnet of every net
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continuous metric space valued function on compact metric space is uniformly continuous
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paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
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injective proper maps to locally compact spaces are equivalently the closed embeddings
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locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Complex geometry
Contents
Statements
(Arnold 71, Massey 73, Kuiper 74, Arnold 88)
In fact, this is is the beginning of a small pattern indexed by the real normed division algebras:
(Arnold 99, Atiyah-Witten 01, Sec. 5.5)
References
AKM-theorem for the complex projective plane
The original proof that the 4-sphere is a quotient of the complex projective plane by an action of Z/2:
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Vladimir Arnold, On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.
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William Massey, The quotient space of the complex projective space under conjugation is a 4-sphere, Geometriae Didactica 1973 (pdf)
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Nicolaas Kuiper, The quotient space of by complex conjugation is the 4-sphere, Mathematische Annalen, 1974 (doi:10.1007/BF01432386)
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Vladimir Arnold, Ramified covering , hyperbolicity and projective topology, Siberian Math. Journal 1988, V. 29, N 5, P.36-47
See also
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José Seade, Section V.5 in: On the Topology of Isolated Singularities in Analytic Spaces, Progress in Mathematics, Birkhauser 2006 (ISBN:978-3-7643-7395-5)
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J. A. Hillman, An explicit formula for a branched covering from to (arXiv:1705.05038)
The SO(3)-equivariant enhancement:
- Le, José Seade, Alberto Verjovsky, Quadrics, orthogonal actions and involutions in complex projective space, L’Enseignement Mathématique, t. 49 (2003) (e-periodica:001:2003:49::488)
Generalization to the quaternionic projective plane
The generalization to the 7-sphere being a U(1)-quotient of the quaternionic projective plane is due to
- Vladimir Arnold, Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67; English translation: Proceedings of the Steklov Institute of Mathematics, 1999, 224, 46–56 (mathnet:tm691)
and independently due to
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Michael Atiyah, Edward Witten, Section 5.5 of: -Theory dynamics on a manifold of -holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177, doi:10.4310/ATMP.2002.v6.n1.a1)
(in the context of M-theory on G₂-manifolds)
Generalization to the octonionic projective plane
Another proof of these cases and further generalization to the 13-sphere being an Sp(1)-quotient of the octonionic projective plane:
- Michael Atiyah, Jürgen Berndt, Projective planes, Severi varieties and spheres, in: Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 (arXiv:math/0206135, doi:10.4310/SDG.2003.v8.n1.a1)
Last revised on July 18, 2024 at 10:45:05. See the history of this page for a list of all contributions to it.