(see also Chern-Weil theory, parameterized homotopy theory)
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cobordism theory, Introduction
The complex Hopf fibration (named after Heinz Hopf) is a canonical nontrivial circle principal bundle over the 2-sphere whose total space is the 3-sphere.
Its canonically associated complex line bundle is the basic line bundle on the 2-sphere.
This we discuss below in
More generally, there are four Hopf fibrations, on the 1-sphere, the 3-sphere, the 7-sphere and the 15-sphere, respectively. This we discuss in
The Eilenberg-MacLane space $K(\mathbb{Z},2) \simeq B S^1$ is the classifying space for circle group principal bundles. By its very nature, it has a single nontrivial homotopy group, the second, and this is isomorphic to the group of integers
This means that there is, up to homotopy, a canonical (up to sign), continuous map from the 2-sphere
such that $[\phi] \in \pi_2(K(\mathbb{Z},2)) = \pm 1 \in \mathbb{Z}$.
As any map into $K(\mathbb{Z},2)$ this classifies a circle group principal bundle over its domain. This is the Hopf fibration, fitting into the long fiber sequence
In other words, the Hopf fibration is the $U(1)$-bundle with unit first Chern class on $S^2$.
An explicit topological space presenting the Hopf fibration may be obtained as follows. Identify
and
Then the continuous function $S^3 \to S^2$ defined by
gives the Hopf fibration. (Thus, the Hopf fibration is a circle bundle naturally associated with the canonical line bundle.) Alternatively, if we use
and identify this presentation of the 2-sphere with the complex projective line via stereographic projection, the Hopf fibration is identified with the map $S^3 \to S^2$ given by sending
Alternatively, we may regard $S^3 \simeq S(\mathbb{H})$ as the unit sphere in the quaternions and $S^2 \simeq S\left( \mathbb{H}_{\mathrm{im}}\right)$ as the unit sphere in the imaginary quaternions. Under this identification, the complex Hopf fibration is equivalently represented by
where $\mathbf{i} \in S\left( \mathbb{H}_{\mathrm{im}}\right)$ is any unit imaginary quaternion.
Regard $S^1 = U(1)$ as equipped with its circle group structure. This makes $S^1$ in particular an H-space. The Hopf fibration $S^1 \to S^3 \to S^2$ is the Hopf construction applied to this H-space.
Consider
the Spin(3)-action on the 2-sphere $S^2$ which is induced by the defining action on $\mathbb{R}^3$ under the identification $S^2 \simeq S(\mathbb{R}^3)$;
the Spin(3)-action on the 3-sphere $S^3$ which is induced under the exceptional isomorphism $Spin(3) \simeq Sp(1) = U(1,\mathbb{H})$ by the canonical left action of $U(1,\mathbb{H})$ on $\mathbb{H}$ via $S^3 \simeq S(\mathbb{H})$.
Then the complex Hopf fibration $S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2$ is equivariant with respect to these actions.
A way to make the $Spin(3)$-equivariance of the complex Hopf fibration fully explicit is to observe that it is equivalently the following map of coset spaces:
For each of the normed division algebras over $\mathbb{R}$, the real numbers, complex numbers, quaternions, octonions
there is a corresponding Hopf fibration of Hopf invariant one.
The total space of the fibration is the space of pairs $(\alpha, \beta) \in A^2$ of unit norm: ${|\alpha|}^2 + {|\beta|}^2 = 1$. This gives spheres of dimension 1, 3, 7, and 15 respectively. The base space of the fibration is projective 1-space $\mathbb{P}^1(A)$, giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is the map
($n = 1, 2, 3, 4$) which sends the pair $(\alpha, \beta)$ to $\alpha/\beta$.
When $X$ is a sphere that is an $H$-space, namely, one of the groups $S^0 = 1$ the trivial group, $S^1 = \mathbb{Z}/2$ the group of order 2, the 3-sphere special unitary group $S^3 = SU(2)$; or the 7-sphere $S^7$ with its Moufang loop structure, then the Hopf construction produces the above four Hopf fibrations:
Let
$H_{\mathbb{C}} \in \pi_3(S^2)$,
$H_{\mathbb{H}} \in \pi_7(S^4)$
$H_{\mathbb{O}} \in \pi_{15}(S^8)$
be the homotopy class of the complex Hopf fibration, the quaternionic Hopf fibration and the octonionic Hopf fibration, respectively. Then their suspensions are the generators of the corresponding stable homotopy groups of spheres:
see this MO comment
When line bundles are regarded as models for the topological structure underlying the electromagnetic field the Hopf fibration is often called “the magnetic monopole”. We may think of the $S^2$ homotopically as being the 3-dimensional Cartesian space with origin removed $\mathbb{R}^3 - \{0\}$ and think of this as being 3-dimensional physical space with a unit point magnetic charge at the origin removed. The corresponding electromagnetic field away from the origin is given by a connection on the corresponding Hopf fibration bundle.
In complex K-theory, the Hopf fibration represents a class $H$ which generates the cohomology ring $K_U(S^2)$, and satisfying the relation $H^2 = 2 \cdot H - 1$, or $(H-1)^2 = 0$. (So in particular $H$ has an inverse $H^{-1} = 2- H$, see at Bott generator.)
A succinct formulation of Bott periodicity for complex K-theory is that for a space $X$ whose homotopy type is that of a CW-complex, we have
(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for $K(S^2)$ to the topos of sheaves over $X$.)
The Hopf fibrations over other normed division algebras also figure in the more complicated case of real K-theory $K_O$: they can be used to provide generators for the non-zero homotopy groups $\pi_n(B O)$ for the classifying space of the stable orthogonal group, which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra $\mathbb{O}$). [To be followed up on.]
Reviews include
For more discussion in homotopy type theory see also at
Relation to skyrmions:
Discussion of supersymmetric Hopf fibrations:
A. P. Balachandran, G. Marmo, B.-S. Skagerstam and A. Stern, section 9.3 of Gauge Symmetries and Fibre Bundles, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 (arXiv:1702.08910)
Simon Davis, section 3 of Supersymmetry and the Hopf fibration (doi:10.4995/agt.2012.1623)
Last revised on April 28, 2020 at 03:10:25. See the history of this page for a list of all contributions to it.