topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The twistor fibration $\mathbb{C}P^3 \to S^4$ (Atiyah 79, Sec. III.1, see also Bryant 82, ArmstrongSalamon14, ABS 19), also called, in its coset space-version $SO(5)/U(2) \to SO(5)/SO(4)$, the Calabi-Penrose fibration (apparently starting with Lawson 85, Sec. 3, see also, e.g., Loo 89, Seade-Verjovsky 03, 3 for this usage, see Nordstrom 08, Lemma 2.31 for review of Calabi’s construction Calabi 67, Calabi 68) is a fiber bundle-structure on complex projective 3-space over the 4-sphere with 2-sphere (Riemann sphere) fibers:
If one identifies the 4-sphere as the quaternionic projective line $S^4 \simeq \mathbb{H}P^1$, then the fibration $p$ here is given by sending complex lines to the quaternionic lines which they span (Atiyah 79, III (1.1), see also Seade-Verjovsky 03, p. 198):
for any $x \in \mathbb{C}^4 \simeq_{\mathbb{R}} \mathbb{H}^2$.
It is possible to define a twistor fibration over each $S^{2n}$, where the resulting manifold is a complex manifold endowed with a holomorphic? $n$-plane field transverse to the fibers of the mapping. Namely, writing $S^{2n}=SO(2n+1)/SO(2n)$, then, using the inclusion $U(n) \subset SO(2n)$, one has the coset fibration
The manifold $Z_n$ canonically has the structure of a complex manifold and is known as the twistor space of $S^{2n}$.
There are generalizations of this picture for each of the so-called ‘inner’ symmetric spaces $G/K$ where $K$ is the fixed subgroup of an involution that is an inner automorphism of $G$. The twistor fibration is of the form $G/U \to G/K$ where $U \subset K$ is a subgroup such that $K/U$ (the typical fiber of the fibration) is an Hermitian symmetric space. There are also other kinds of twistor spaces over $G/K$ that are flag manifolds of the form $G/T$ where $T \subset K$ is a maximal torus. (For these generalizations see Bryant 85.)
Eugenio Calabi, Minimal immersions of surfaces in euclidean spheres, J. Differential Geometry (1967), 111-125 (euclid:jdg/1214427884)
Eugenio Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima, Topics in Complex Manifolds (Ed. H. Rossi), Les Presses de l’Universit ́e de Montr ́eal (1968), 59-81 (naid:10006413960)
Michael Atiyah, Section III.1 of: Geometry of Yang-Mills fields, Pisa, Italy: Sc. Norm. Sup. (1979) 98 (spire:150867, pdf)
Robert Bryant, Section 1 of: Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differential Geom. Volume 17, Number 3 (1982), 455-473 (euclid:jdg/1214437137)
Robert Bryant, Lie groups and twistor spaces, Duke Mathematical Journal 52 (1985), pp. 223–261, (euclid:dmj/1077304286)
H. Blaine Lawson, Surfaces minimales et la construction de Calabi-Penrose, Séminaire Bourbaki : volume 1983/84, exposés 615-632, Astérisque no. 121-122 (1985), Talk no. 624, p. 197-211 (numdam:SB_1983-1984__26__197_0)
Simon Salamon, p. 218-220 of: Harmonic and holomorphic maps, In: E. Vesentini (eds.) Geometry Seminar “Luigi Bianchi” II - 1984, Lecture Notes in Mathematics, vol 1164. Springer 1985 (doi:10.1007/BFb0081912)
James Eells, Simon Salamon, Section 8 of: Twistorial construction of harmonic maps of surfaces into four-manifolds, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 4, p. 589-640 (numdam:ASNSP_1985_4_12_4_589_0)
Bonaventure Loo, The space of harmonic maps of $S^2$ into $S^4$, Transactions of the American Mathematical Society Vol. 313, No. 1 (1989) (jstor:2001066)
José Seade, Alberto Verjovsky, Section 2 of: Higher dimensional complex Kleinian groups, Math Ann 322, 279–300 (2002) (doi:10.1007/s002080100247)
Le, José Seade, Alberto Verjovsky, Section 4 of: Quadrics, orthogonal actions and involutions in complex projective space, L’Enseignement Mathématique, t. 49 (2003) (e-periodica:001:2003:49::488)
Jonas Nordstrom, Calabi’s construction of Harmonic maps from $S^2$ to $S^n$, Lund University 2008 (pdf, pdf)
Angel Cano, Juan Pablo Navarrete, José Seade, Section 10.1 in: Kleinian Groups and Twistor Theory, In: Complex Kleinian Groups, Progress in Mathematics, vol 303. Birkhäuser 2013 (doi:10.1007/978-3-0348-0481-3_10)
John Armstrong, Simon Salamon, Twistor Topology of the Fermat Cubic, SIGMA 10 (2014), 061, 12 pages (arXiv:1310.7150)
Bobby Acharya, Robert Bryant, Simon Salamon, A circle quotient of a $G_2$ cone, Differential Geometry and its Applications Volume 73, December 2020, 101681 (arXiv:1910.09518, doi:10.1016/j.difgeo.2020.101681)
In higher dimensions:
Over $\mathbb{H}P^3$:
Last revised on November 21, 2020 at 14:32:09. See the history of this page for a list of all contributions to it.