Hopf Fibrations

The geometric and topological interpretation

Article

Abstract

This article defines the Hopf Fibration (HF), the concept of splitting the S3 sphere onto the twisted cartesian product of spheres S1 and S2. The basic HF applications are: 1) HF is a Fiber Bundle structure of Dirac Monopole; 2) HF is a map from the S3 in H to the Bloch sphere; 3) If HF is a vector field in R3 then exists a solution to compressible non-viscous Navier-Stokes equations. It was turned out that there are only 4 dimensions of fibers with Hopf invariant 1, namely S0, S1, S3, S7.

The article consists of two parts: 1) geometric visualization of projection of S3 to S2 (frontend); 2) formal topological version of HF in cubical type theory (backend). Consider this as basic intro and a summary of results or companion guide to the chapter 8.5 from HoTT.

$$\def\mapright#1{\xrightarrow{{#1}}} \def\mapdown#1{\Big\downarrow\rlap{\raise2pt{\scriptstyle{#1}}}} \def\mapdiagl#1{\vcenter{\searrow}\rlap{\raise2pt{\scriptstyle{#1}}}} \def\mapdiagr#1{\vcenter{\swarrow}\rlap{\raise2pt{\scriptstyle{#1}}}} $$

Geometry

Let's imagine $S^3$ as smooth differentiable manifold and build a projection onto the display as if demoscene is still alive.

Equations

Definition (Sphere $S^3$). Like a little baby in $\mathbb{R}^4$: $$ S^3 = \{ (x_0,x_1,x_2,x_3) \in \mathbb{R}^4 : \sum_{i=0}^3 x_i^2 = 1 \}; $$

after math classes in quaternions $\mathbb{H}$: $$ S^3 = \{ x \in \mathbb{H} : \|x\| = 1 \}. $$

Definition (Locus). The $S^3$ is realized as a disjoint union of circular fibers in Hopf coordinates $(\eta,\theta_1,\theta_2)$: $$ \begin{equation} x_0 = cos(\theta_1)sin(\eta), \\ x_1 = sin(\theta_1)sin(\eta), \\ x_2 = cos(\theta_2)cos(\eta), \\ x_3 = sin(\theta_2)cos(\eta). \end{equation} $$ Where $\eta \in [0,\frac{\pi}{2}]$ and $\theta_{1,2} \in [0,2\pi]$.

Definition (Mapping on $S^2$). A mapping of the Locus to the $S^2$ has points on the circles parametrized by $\theta_2$: $$ \begin{equation} x = sin(2\eta)cos(\theta_1), \\ y = sin(2\eta)sin(\theta_1), \\ z = cos(2\eta).\end{equation} $$

Code in three.js:

var fiber = new THREE.Curve(), color = sphericalCoords.color; fiber.getPoint = function(t) { var eta = sphericalCoords.eta, phi = sphericalCoords.phi, theta = 2 * Math.PI * t; var x1 = Math.cos(phi+theta) * Math.sin(eta/2), x2 = Math.sin(phi+theta) * Math.sin(eta/2), x3 = Math.cos(phi-theta) * Math.cos(eta/2), x4 = Math.sin(phi-theta) * Math.cos(eta/2); var m = mag([x1,x2,x3]), r = Math.sqrt((1-x4)/(1+x4)); return new THREE.Vector3(r*x1/m,r*x2/m, r*x3/m); };

Topology

Could we reason about spheres without a metric? Yes! But could we do this in a constructive way? Also yes.

Fiber

Definition (Fiber). The fiber of the map $p: E \rightarrow B$ in a point $y: B$ is all points $x: E$ such that $p(x)=y$.

fiber (E B: U) (p: E -> B) (y: B): U = (x: E) * Path B y (p x)

Fiber Bundle

Definition (Fiber Bundle). The fiber bundle $ F \rightarrow E \mapright{p} B$ on a total space $E$ with fiber layer $F$ and base $B$ is a structure $(F,E,p,B)$ where $p: E \rightarrow B$ is a surjective map with following property: for any point $y: B$ exists a neighborhood $U_b$ for which a homeomorphism $f: p^{-1}(U_b) \rightarrow U_b \times F$ making the following diagram commute.

$$ \begin{array}{ccc} p^{-1}(U_b) & \mapright{f} & U_b \times F \\ \mapdown{\mathbf{p}} & \mapdiagr{pr_1} & \\ U_b & & \\ \end{array} $$

Trivial Fiber Bundle

Definition (Trivial Fiber Bundle). When total space $E$ is cartesian product $\Sigma(B,F)$ and $p = pr_1$ then such bundle is called trivial $(F,\Sigma(B,F),pr_1,B)$.

Family (B: U): U = B -> U total (B: U) (F: Family B): U = Sigma B F trivial (B: U) (F: Family B): total B F -> B = \ (x: total B F) -> x.1

Theorem (Fiber Bundle is a Type Family). Inverse image (fiber) of fiber bundle $(F,B*F,pr_1,B)$ in point $y:B$ equals $F(y)$.

FiberPi (B: U) (F: B -> U) (y: B) : Path U (fiber (total B F) B (trivial B F) y) (F y)

Definition (Structure Group). The structure group of an $F$-fiber bundle is just Aut(F), the loop space of BAut(F).

Higher Spheres

Definition (2-points, Bool, Sphere $S^0$).

data bool = false | true

Definition (Suspension).

data susp (A: U) = north | south | merid (a: A) <i> [ (i = 0) -> north, (i = 1) -> south ]

Definition (Sphere $S^1$). Direct definition.

data S1 = base | loop <i> [ (i = 0) -> base, (i = 1) -> base ]

Definition (Sphere $S^n$). Recursive definition.

S: nat -> U = split zero -> susp bool succ x -> susp x S2: U = susp S1 S3: U = susp S2 S4: U = susp S3

Hopf Fibrations

Theorem (Hopf Fibrations). There are fiber bundles: $$ (S^0,S^1,p,S^1), \\ (S^1,S^3,p,S^2), \\ (S^3,S^7,p,S^4), \\ (S^7,S^{15},p,S^8). $$

Definition (H-space). H-space over a carrier $A$ is a tuple $(A,e,\mu:A\rightarrow A\rightarrow A,(a:A)\rightarrow(\mu(e,a)=a,\mu(a,e)=a))$.

Example ($S^1$ Hopf Fiber). Guillaume Brunerie.

rot: (x : S1) -> Path S1 x x = split base -> loop1 loop @ i -> constSquare S1 base loop1 @ i mu : S1 -> equiv S1 S1 = split base -> idEquiv S1 loop @ i -> equivPath S1 S1 (idEquiv S1) (idEquiv S1) ( \(x : S1) -> rot x @ j) @ i H : S2 -> U = split north -> S1 south -> S1 merid x @ i -> ua S1 S1 (mu x) @ i total : U = (c : S2) * H c

Definition (Hopf Invariant). Let $\phi: S^{2n-1} \rightarrow S^{n}$ a continuous map. Then homotopy pushout (cofiber) of $\phi$ is $cofib(\phi) = S^{n} \bigcup_\phi \mathbb{D}^{2n}$ has ordinary cohomology $$ H^{k}(cofib(\phi),\mathbb{Z})= \begin{cases} \mathbb{Z}\ for\ k=n,2n \\[2ex] 0\ otherwise \end{cases} $$.

Hence for $\alpha,\beta$ generators of the cohomology groups in degree $n$ and $2n$, respectively, there exists an integer $h(\phi)$ which expresses the cup product square of $\alpha$ as a multiple of $\beta$ — $\alpha\sqcup\alpha=h(\phi)\cdot\beta$. This integer $h(\phi)$ is called Hopf invariant of $\phi$.

Theorem (Adams, Atiyah). Hopf Fibrations are only maps that have Hopf invariant $1$.