(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 quaternionic Hopf fibration is the Hopf fibration induced by the quaternions, hence it is the fibration
of the 7-sphere over the 4-sphere with fiber the 3-sphere, which is induced via the Hopf construction from the product operation
on the quaternions, or else from
to match standard conventions.
This means that if $S^7$ is regarded as the unit sphere $\{(x,y) | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\}$ in $\mathbb{H}\times \mathbb{H}$ and $S^4$ is regarded as the quaternionic projective space, then $p$ is given (on points $(x,y)$ with $y \neq 0$) simply by
Since the automorphism group of the quaternions, as an $\mathbb{R}$-algebra, is the special orthogonal group $SO(3)$
acting by rotation of the imaginary quaternions, via the Hopf construction it follows that the 7-sphere and 4-sphere inherit $SO(3)$-actions under which the quaternionic Hopf map is equivariant.
Notice that this means that $SO(3)$ acts on $S^7$ here diagonally on the two copies of the imaginary octonions in $S^7 \hookrightarrow \mathbb{H} \oplus \mathbb{H}$ (as opposed to, say, via any one of the embeddings $SO(3) \hookrightarrow SO(8)$ and the following canonical action of $SO(8)$ on $S^7 \hookrightarrow \mathbb{R}^8$).
(see also Cook-Crabb 93)
The quaternionic Hopf fibration gives an element in the 7th homotopy group of the 4-sphere
and in fact it is a generator of the non-torsion factor in this group.
Stably, i.e. as a generator for the stable homotopy groups of spheres in degree $7-4 = 3$, the quaternionic Hopf map becomes a torsion generator
Fix a finite subgroup $G \hookrightarrow SO(3)$ which does not come from $SO(2) \hookrightarrow SO(3)$ – i.e. not a cyclic group, but one of the dihedral groups or else the tetrahedral group or octahedral group or icosahedral group (by the ADE classification).
Regard both $S^7$ and $S^4$ as pointed topological G-spaces via the $SO(3)$-action induced via automorphisms of the quaternions, as above. Write
for the corresponding equivariant suspension spectra.
Notice that if we took trivial $G$, then in the stable homotopy category
by the above. In contrast:^{1}
In $G$-equivariant homotopy theory this becomes a non-torsion group, i.e.
with the quaternionic Hopf fibration, regarded as a $G$-equivariant map, representing a non-torsion element.
First use the Greenlees-May decomposition which says that for any two $G$-equivariant spectra $X,Y$ and writing $\pi_\bullet(X), \pi_\bullet(Y)$ for their equivariant homotopy groups, organized as Mackey functors $H \mapsto \pi_n^H(X)$ for all subgroups $H \subset G$, then the canonical map
is rationally an isomorphism.
With this we are reduced to showing that there exists $n \in \mathbb{Z}$ and a morphism of Mackey functors of equivariant homotopy groups $\pi_n(\Sigma^\infty_G S^7) \to \pi_n(\Sigma^\infty_G S^4)$ which is not a torsion element in the abelian hom-group of Mackey functors.
To analyse this, we use the tom Dieck splitting which says that the equivariant homotopy groups of equivariant suspension spectra $\Sigma^\infty_G X$ contain a direct summand which is simply the ordinary stable homotopy groups of the naive fixed point space $X^H$:
Now observe that the fixed points of the $SO(3)$-action on the quaternionic Hopf fibration that we are considering is just the real Hopf fibration:
since $SO(3)$ acts transitively on the quaternionic quaternions and fixes the real quaternions. By our assumption that $G \subset SO(3)$ does not come through $SO(2) \hookrightarrow SO(3)$ it follows that this statment is still true for $G$:
But the real Hopf fibration defines a non-torsion element in $\pi_0^S \simeq \mathbb{Z}$.
In conclusion then, at $n = 1$ and $H = G$ we find that the $G$-equivariant quaternionic Hopf fibration contributes a non-torsion element in
which appears as a non-torsion element in
and hence in $[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G$.
See also at equivariant stable cohomotopy
Discussion in parameterized homotopy theory includes
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)
Discussion in homotopy type theory is in
The proof of prop. profited from Charles Rezk, who suggested here that the reduction to fixed points will make the real Hopf fibration give a non-torsion contribution, and from David Barnes who amplified the use of the Greenless-May splitting theorem. ↩
Last revised on August 14, 2018 at 10:07:43. See the history of this page for a list of all contributions to it.