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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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    Yes, a loop can indeed stop half way, go back a bit, then forward until the end. A loop is just a continuous map with the same values at 0 and 1, so there's no problem with it hitting the same point(s) more than once.2012-12-15
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    Yes, it can. Nail one end of a string into a circular column, run around with the string, back and forth, whatever, five times aorund the column forward, dance around, three times backwards, until you stand at the nail agin. Now pull the string tight. Finally it will either wrap around the column one or more times clockwise or one or more times conterclockwise - or not at all. At least it won't change direction.2012-12-15
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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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