I am interested in Hopf’s original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, but unfortunately I do not read German. Do you know a translation or a reference following the same argument? Nota Bene: I am aware […]

a 2-sphere is a normal sphere. A 3-sphere is $$ x^2 + y^2 + z^2 + w^2 = 1 $$ My first question is, why isn’t the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and animate the w parameter, as w goes from .1 to .9: Isn’t […]

What is $\pi_{31}(S^2)$ – high homotopy group of the 2-sphere ? This question has a physics motivation: 1) There are relations between (2nd and 3rd) Hopf fibrations and (2- and 3-) qbits (quantum bits) entanglement, see this reference : http://arxiv.org/pdf/0904.4925v1.pdf 2) Maybe there are relations between classification of qbits entanglements and sphere homotopy groups ? […]

Intereting Posts

What is special about the numbers 9801, 998001, 99980001 ..?
How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$
max and min versus sup and inf
How to prove l'Hospital's rule for $\infty/\infty$
how to find integer solutions for $axy +bx + cy =d$?
Proving that for reals $a,b,c$, $(a + b + c)^2 \leq 3(a^2+b^2+c^2)$
Where does the constant increase by 2 of differences between integer square values come from?
Difference between proof of negation and proof by contradiction
Do we really need constant symbols in first-order theories?
Not divisible by $2,3$ or $5$ but divisible by $7$
Do Symmetric Games with Nash Equilibria always have a symmetric Equilbrium?
Where can I learn more about the “else” operation / “else monoid”?
How would you prove $\sum_{i=1}^{n} (3/4^i) < 1$ by induction?
A first order sentence such that the finite Spectrum of that sentence is the prime numbers
Notation for repeated composition of functions