0
$\begingroup$

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 the answers are NO for (a) and YES for (b). Am I right?

  • 0
    @HagenvonEitzen: I believe the first case is quite trivial, for the second I believe that it would not be too difficult to consider a continuous transformation that translates the big circle near the little one while shrinking it too. Thanks however.2012-10-30

1 Answers 1

1

For the second case, you can use the family of homotheties: $f_t(x,y)=(-4,0)+(1+\tfrac{t}{2})(x+4,y)$ restricted to $x^2+y^2=1$, $0\leq t\leq 1$. For $t=0$, it is the identity and, for $t=1$, you obtain the second sphere.