Let $a$ be a point on the unit sphere $S=\{(x,y,z)|x^2+y^2+z^2=1\}$.

  1. How do I show that $S\backslash\{a\}$ is contractible?
  2. How do I show that a non-surjective loop $\phi\in P(S,s)$ with base point $s$ is path-homotopic to the constant loop with image $s$?

What I know:

  1. Contractible means that there should be a $s\in S\backslash\{a\}$ such that the constant map $S\backslash\{a\}\rightarrow S\backslash\{a\}$ with image $s$ is homotopic to the identity on $S\backslash\{a\}$.
    So I want a continuous map (homotopy) $f:[0,1]\times S\backslash\{a\}\rightarrow S\backslash\{a\}$ such that $f(0,x)=s$ and $f(1,x)=x$.
    What am I still missing in my proof?

  2. A loop is a closed path: a continuous map $\gamma:[0,1]\rightarrow S$ with $\gamma(0)=\gamma(1)=s$. So the constant loop $\gamma'$ has $\gamma'(t)=s$ for all $t\in[0,1]$.
    Path-homotopic (between $\phi$ and $\gamma'$) means that there is a continuous map $g:[0,1]\times[0,1]\rightarrow S$ such that $$g(0,a)=\phi(a)\text{, }g(1,a)=\gamma'(a)=s\text{, }g(b,0)=s\text{, }g(b,1)=s$$ for all $a,b\in [0,1]$.

So I got the hint that I should use the contracting homotopy from 1., but I wouldn't know what to do with it, and which homotopy is meant.
Also if I prove that $\mathbb{R}^2$ is contractible, have I then proven 1. because of the homeomorphism with $S\backslash\{a\}$?

  • $\begingroup$ 1. $\mathbb{S}^2 \setminus \left\{pt\right\}$ is homeomorphic to $\mathbb{R}^2$. 2. Use the contracting homotopy you find in 1. $\endgroup$ – cjackal Apr 27 '16 at 12:43
  • $\begingroup$ @cjackal What is $t$? $\endgroup$ – user335050 Apr 27 '16 at 12:45
  • $\begingroup$ @T.Eskin How can I use that? $\endgroup$ – user335050 Apr 27 '16 at 12:45
  • $\begingroup$ I think "pt" is just short for the word "point" there. @cjackal means the sphere with a point removed. $\endgroup$ – Akiva Weinberger Apr 28 '16 at 0:04
  • $\begingroup$ @AkivaWeinberger Could you help with the other questions? $\endgroup$ – user335050 Apr 28 '16 at 7:45

HINT: for the first question you can use the stereographic projection to see that a sphere minus a point is homeomorphic to a plane, in particular all homotopy groups are the same.

For the second question: use the Van-Kampen theorem to see that $\pi_1(S)$ is trivial. The conclusion follows.

  • $\begingroup$ Is there another way for the second question without using Van-Kampen? $\endgroup$ – user335050 Apr 30 '16 at 8:42

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