(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
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,
$l_m := \{(x, m x)|x \in \mathbb{O}\}$ 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, 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.
(see also Cook-Crabb 93)
The subgroup $Spin(9) \subset SO(16)$ acts transitively on the octonionic Hopf fibration, which can be consider as a homogeneous fibration
Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, (arXiv:1208.0899)
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
Last revised on May 12, 2017 at 05:49:03. See the history of this page for a list of all contributions to it.