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I'm having a bit of trouble understanding why I'm wrong. Given a loop that starts on the point $(1,0)$ and does 1 loop in the clockwise direction in $S^1$ cant I construct a homotopy in the following manner: H(x,0) = the original loop H(x,1) = the point $(1,0)$. H(x,y) = a loop that starts at $(1,0)$ and then goes $(1-y)\cdot2\pi$ clockwise.

I know this isn't the most formal explanation but why is this wrong? It seems to uphold the definition of a homotopy, but it has to be wrong.

Any help would be appreciated.

John H.
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  • Nullhomotopic means homotopic to the constant curve throug a homotopy of closed curves (with endpoint and starting point coinciding). Your $H(x,y)$ does not fulfil this requirement. – Thomas Sep 13 '15 at 09:20
  • For $y\neq 0,1$, $H(x,y)$ doesn't describe a loop. – Arthur Sep 13 '15 at 09:20

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