/sci/ - Science & Math

>>10147486
>just a meme
Lrn2meme fgt pls

>>10147580
explain how to meme

halp

>>10147486
First we notice that a fibre bundle [math]S^3 \rightarrow S^2[/math] with fibres [math]S^1[/math] over [math]S^2[/math] exists. Now the Pontrjagyn invariant [math]\mu:[S^2\times S^1,S^2]\rightarrow \mathbb{Z}_2[/math] characterizes homotopy classes of each local trivialization [math]S^2 \times S^1 \rightarrow S^2[/math]; the only non-trivial homotopy class is precisely the local trivializaiton of the Hopf fibration [math]S^3 \rightarrow S^2[/math]. To see why this is significant, the trivial bundle, characterized by [math]\mu = 0[/math], has unlinked fibres, while those for the Hopf fibration are linked: for each [math]x\neq y \in S^2[/math], the disjoint union [math]S^1_x \coprod S^2_y[/math] of the fibres over them is isotopic in [math]S^3[/math] to the Hopf link.
IN essence, the Hopf fibration is basically the [math]only[/math] non-trivial way to isometrically embed a bunch of circles parameterized by [math]S^2[/math] into [math]S^3[/math], up to homotopy.

>>10148114
is this relevant to physics or just random mathematical object?

>>10148157
The Kaluza-Klein monopole in string theory is made from the Hopf fibration. String theory is not my expertise though so I'll give you another example.
Suppose you have Helium-3 in its chiral superfluid-A1 phase, then its BdG Hamiltonian is parameterized by the Bloch vectors [math]{\bf k}\in\mathbb{T}^3[/math] such that [math]H = \begin{pmatrix} \epsilon_k + k_z & k_x - ik_y \\ k_x - ik_y & \epsilon_k - k_z\end{pmatrix}[/math], where [math]\epsilon_k^2 = |{\bf k}|^2 - \mu[/math] is the dispersion. By tuning the chemical potential [math]\mu[/math], you can tune the magnitude of the 3-dimensional [math]{\bf d}[/math]-vector without interfering with its topological order, hence He3-A1 is characterized by homotopy classes of maps [math]{\bf d}[/math] into the [math]2[/math]-sphere [math]S^2[/math]. Now suppose we add a vortex (i.e. string defect) through the superconducting order parameter along [math]\hat{z}[/math] - via a magnetic field threading the superfluid He3, for instance - then the [math]SO(3)[/math] symmetry is broken down to [math]SO(2)[/math] for which [math]\hat{\bf d}[/math] is parameterized by
1. the orbital coordinate [math]\phi \in S^1[/math], and
2. the "unbroken" Bloch vectors [math]{\bf k}^\perp = k_x,k_y[/math], i.e. these crystal momenta stay good quantum numbers even in the presence of the vortex, compactified such that [math]{\bf k}^\perp \in S^2[/math] and [math]{\bf d}[/math] is defined at [math]\infty[/math],
then we see that [math]\hat{\bf d} \in [S^2\times S^1,S^2][/math]. Hence the Pontrjagyn invariant [math]\mu[/math], or equivalently the Pontrjagyn index [math]\mathbb{Z}_2[/math], characterizes the topological orders of the superfluid He3-A1 in the presence of a vortex.

>>10148181
Sorry, it should be [math]k_x + ik_y[/math] in the lower left entry of [math]H[/math].

>>10147486
>what is this a hopf fibration and is it important or just a meme?
The Hopf fibration tells you how to obtain a 4D ball from a 3D ball, kind of. We understand how the Hopf fibration makes a 2D sphere become a 3D sphere, but when you do it to the surface of the ball, weird things happen to the volume of the interior of the ball. The sphere theorem is kind of related, and it is derivative of Perelman's solution to the Poincaré conjecture.
>https://www.youtube.com/watch?v=-tj190Lcw48

>>10148218
>that wasn't easy to follow, was it?
made me kek

@10148218
Poor schizoposter

>t. listens to the joe rogan experience

>>10147486
wow, math guy pages sure look different

>>10149532
alpha jew

>>10147580
>fgt
Why the homophobia?

>>10148114
>>10148181
based yukari poster, i only come to this shithole to read your posts