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Questions tagged [hopf-fibration]

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0answers
41 views

Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
2
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1answer
193 views

Does the Hopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I ...
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0answers
74 views

What is known about the Hopf map for quadratic field extensions?

This question is related to my previous post: Is this generalization of the Hopf map for quadratic field extensions surjective? I still would like to know more and, while that post got several votes,...
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0answers
189 views

When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?

Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...
6
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1answer
276 views

Is this generalization of the Hopf map for quadratic field extensions surjective?

Let $k$ be a field, and let $L$ be a quadratic extension of $k$. Denote by $\sigma$ the non-trivial element of $\operatorname{Gal}(L/k)$. Let $M_2(L)$ be the vector space over $L$ of two-by-two ...
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0answers
58 views

Largest elementary neighbourhood

The real projective space $\mathbb{R}P^n$ can be defined as the quotient space of $\mathbb{S}^n$by the equivalence relation that identifies antipodal points. The largest open set of $\mathbb{S}^n$ ...
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0answers
162 views

Two ways to view the three-sphere

Think of the three-sphere as given by $\lbrace|z|^2+|w|^2=1, \;z,w\in \mathbb{C}^2\rbrace$. We can regard it in terms of Hopf coordinates \begin{align*} z&= \cos(\theta/2)e^{i(\phi+\psi)}\\ w&=...
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2answers
749 views

Are there nonlinear projective spaces?

This is actually a series of questions posed by Guram Berishvili about the structure he calls marao. Everything I am going to write here I took (and messed up) from his home page which is all in ...
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2answers
252 views

Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
7
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2answers
658 views

Generalising the Penrose Twistor Fibration

As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" ...
2
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0answers
172 views

A (different) foliation arising from Hopf fibration

In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...
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2answers
334 views

Complementary integrable vector fields

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric ...
2
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1answer
283 views

Splitting the Hopf map in two

Given the Hopf map $h:S^3\to S^2$ and an inclusion $i:S^2\hookrightarrow S^3$, the map $h\circ i:S^2\to S^2$ has mapping degree zero. Therefore, it is homotopic to the constant map and the image of ...
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0answers
74 views

Example of n-parameter family of real-analytic diffeomorphisms acting on $S^3$, constant on the Hopf fibres

I am trying to construct an n-parameter family of measure preserving real analytic diffeomorphisms on $S^3$ which preserves the $S^1$ fibres of the Hopf fibration but acts transitively on the image $S^...
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3answers
706 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in $\pi_3(S^2)...
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0answers
554 views

How to see the quaternionic hopf map generates the stable 3-stem?

I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example: third-stable-...
13
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0answers
498 views

“Homogeneity” of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]

My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
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2answers
474 views

Hopf Tori in $S^3$

By means of the Hopf fibration $\pi: S^3 \rightarrow S^2$, U. Pinkall showed that every compact Riemannian surface of genus one can be conformally embedded in $S^3$. More precisely: Let $p$ be a ...
6
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1answer
356 views

The ring of integers looks like the 3-dimensional sphere viewed as the Hopf fibration

This question is based on the following phrase: "In a sense, $\textrm{Spec} \ \mathbf{Z}$ looks topologically like a 3-dimensional sphere viewed as the Hopf fibration over $\mathbf{S}^2$." See page ...
7
votes
1answer
640 views

Hopf fibration inside the retraction of R^4 minus line -> S^2?

This was inspired by this question. Let $Y = {\mathbb R}^4 \setminus$a coordinate line, which retracts to ${\mathbb R}^3 \setminus$a point, which retracts to $S^2$. What is an explicit immersion $S^...
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5answers
1k views

Simplicial Model of Hopf Map?

The Hopf fibration is a famous map S3 --> S2 with fiber S1, which is the generator in pi_3(S2). We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces ...