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I'm trying to understand intuitively the proposition that:

Any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction.

Surely a loop round a circle starting at one point and ending at the same point is very restrictive in how it travels round the circle - it either goes round or it doesn't.

What are some examples of how a loop can go around a circle?.. I.e. can the 'loop' stop half way, go back a bit, then forward until the end?

Thanks!

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    Lift the path to $\mathbb R$. You will get a continuous function from $[0,1]$ with endpoints the integers. You want to show that this can be homotoped to an injective map. This is essentially the pulling of the string mentioned in the nice @Hagen's comment.2012-12-15

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