# Tagged Questions

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### mapping homotopic to the identity map

Please give me a hand with this problem, It was on my exam, and I just couldn't solve it. Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is ...
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### Smooth torus eversion

I asked a vague question about torus eversion earlier, with no hard math, so while I'm at it, how about this one, which may involve hard math: "Everybody knows" that Stephen Smale showed us how to ...
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### Homotopy of spheres $\pi_{n+1}(S^n) \simeq \mathbb{Z}_2$

I have a problem: I have to prove that $$\pi_{n+1}(S^n) \simeq \mathbb{Z}/2\mathbb{Z}$$ when $n \ge 3$. I know the Freudenthal suspension theorem and the Hopf fibration. Is there an easy method to ...
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### Hopf fibration and homotopy of spheres

Let $$S^3 \to S^7 \to S^4$$ an the Hopf fibration. We con consider the induced sequence in homotopy $$\pi_i(S^3) \to \pi_i(S^7) \to \pi_i(S^4) \to \pi_{i-1}(S^3) \to \pi_{i-1}(S^7) \to \cdots$$ ...
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### Bott periodicity and homotopy groups of spheres

I studied Bott periodicity theorem for unitary group $U(n)$ and ortogonl group O$(n)$ using Milnor's book "Morse Theory". Is there a method, using this theorem, to calculate $\pi_{k}(S^{n})$? (For ...
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### Find a regular homotopy

Firstly, we define a regular homotopy between regular closed curves as a continuous map $F:$I x I$\rightarrow \mathbb{R}^n$ satisfying the following conditions: (i) for each fixed u $\in$ I, the map ...
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### A puzzle on knotted surfaces

Only after having learned that the somehow only notion of equivalence of knots is definitely "ambient isotopy" I stumbled over this blog entry on ambient isotopy. (Had it been earlier!) What bothers ...
It's intuitively clear what it means that two knots $K,K'$ are essentially the same, but it can be termed and defined more precisely in different ways. Are all of them equivalent? $K, K'$ are ...
Consider the circle with center in $(0,0)$ and radius 1 and the circle in $(2,0)$ and radius 1.5 in the plane. Are they homotopic (a) if we remove the origin, (b) if no point is removed? I think that ...