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2
votes
0answers
102 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 ...
2
votes
2answers
92 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
votes
1answer
217 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 ...
1
vote
0answers
66 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 ...
16
votes
3answers
609 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 ...
11
votes
0answers
305 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: ...
8
votes
0answers
347 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 ...
1
vote
2answers
361 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 ...
4
votes
1answer
312 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
597 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 ...
10
votes
4answers
596 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 ...