(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
The real Hopf fibration is the fibration
of the 1-sphere over itself with fiber the 0-sphere, which is induced via the Hopf construction from the product operation
on the real numbers.
This may also be understood as the Spin(2)-double cover of SO(2).
Here are different but equivalent ways of realizing this explicitly:
If the domain $S^1$ is regarded as the unit sphere $\{(x,y) | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\}$ in $\mathbb{R}\times \mathbb{R}$ and the codomain $S^1$ is regarded as the real projective space, then $p$ is given simply by
One can view the real Hopf fibration as the boundary of a Möbius strip, which is the non-trivial double cover of the circle.
As an element in the first (stable) homotopy group of spheres $\pi_1(S^1) \simeq \mathbb{Z}$, the real Hopf fibration represents $p_{\mathbb{R}} = 2$.
We spell out the real Hopf fibration realized as the Hopf construction
(as defined there) on
regarded with its cyclic group-structure
Here the equivalence relation for the suspension $\Sigma X = (X \times [0,1])_{/\sim}$ on the right is
while the equivalence relation for the join $X\ast X = (X \times [0,1] \times X)_{/\sim}$ on the left is
So on the left we have a circle $S^1$ realized by gluing four copies of the interval $[0,1]$ labeled in $\mathbb{Z}/2 \times \mathbb{Z}/2$ by identifying them pairwise at $0 \in [0,1]$ and pairwise the other way at $1 \in [0,1]$. while on the right we have a circle $S^1$ realized by two copies, labeled by $\mathbb{Z}/2$.
A full path around the circle on the left is given, in terms of the above coordinates in $[0,1] \times (\mathbb{Z}_2 \times \mathbb{Z}_2)$, by
As we map this path over to the other $S^1$ by adding up the two coordinate labels in $\mathbb{Z}/2$, we trace out the following path on the right, with coordinates in $[0,1] \times \mathbb{Z}/2$:
That’s twice around the circle on the right, for once on the left, manifestly showing that the real Hopf fibration is the non-trivial double cover of the circle by itself:
Last revised on February 18, 2019 at 10:09:05. See the history of this page for a list of all contributions to it.