(see also Chern-Weil theory, parameterized homotopy theory)
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
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Under construction
The octonionic Hopf fibration is the fibration
of the 15-sphere over the 8-sphere with fiber the 7-sphere. This may be derived by the Hopf construction on the 7-sphere $S^7$ with its Moufang loop structure.
Alternatively, we may construct a fibration by first decomposing $\mathbb{O}^2$ into the octonionic lines,
$l_m \coloneqq \{(x, m x)|x \in \mathbb{O}\}$ and $l_{\infty} := \{(0, y)|y \in \mathbb{O}\}$.
In this way the fibration $\mathbb{O}^2 \setminus (0, 0) \to S^8 = \{m \in \mathbb{O}\} \union \{\infty\}$ is obtained, with fibers $\mathbb{O} \setminus 0$, and the intersection with the unit sphere $S^{15} \subset \mathbb{O}^2$ provides the octonionic Hopf fibration (see OPPV 12, p. 7).
This second construction yields the standard parameterization of the octonionic Hopf fibration via $(x,y) \mapsto x y^{-1}$ (in one chart) and $(x,y ) \mapsto x^{-1} y$ (in the other), while the Hopf construction gives $(x,y) \mapsto x y$. The latter yields the generator $-1$ of $\pi_{15}(S^8) \cong \mathbb{Z}$, while the former yields $+1$.
The automorphism group G2 of the octonions, as a normed algebra, manifestly acts on the octonionic Hopf fibration, such that the latter is equivariant.
(see also Cook-Crabb 93)
But the octonionic Hopf fibration is equivariant even with respect to the Spin(9)-action, the one on $S^8 = S(\mathbb{R}^9)$ induced from the canonical action of $Spin(9)$ on $\mathbb{R}^9$, and on $S^{15} = S(\mathbb{R}^{16})$ induced from the canonical inclusion $Spin(9) \hookrightarrow Spin(16)$.
(Gluck-Warner-Ziller 86, Prop. 7.1)
This equivariance is made fully manifest by realizing the octonionic Hopf fibration as a map of coset spaces as follows (Ornea-Parton-Piccinni-Vuletescu 12, p. 7):
The complex and quaternionic Hopf fibrations are not subfibrations of the octonionic one (Parton & Piccinni 18, p. 4).
The octonionic Hopf fibration does not admit any $S^1$ subfibration? (Parton & Piccinni 18, p. 13).
Herman Gluck, Frank Warner, Wolfgang Ziller, The geometry of the Hopf fibrations, L’Enseignement Mathématique, t.32 (1986), p. 173-198
Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, Transformation Groups (2013) 18: 845 (arXiv:1208.0899, doi:10.1007/s00031-013-9233-x)
Maurizio Parton, Paolo Piccinni, The role of $Spin(9)$ in octonionic geometry, (arXiv:1810.06288).
Reiko Miyaoka, The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. (Euclid)
Discussion in parameterized homotopy theory is in
A. L. Cook, M.C. Crabb, Fiberwise Hopf structures on sphere bundles, J. London Math. Soc. (2) 48 (1993) 365-384 (pdf)
Kouyemon Iriye, Equivariant Hopf structures on a sphere, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 (Euclid)
Last revised on December 10, 2020 at 20:52:24. See the history of this page for a list of all contributions to it.