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Just a silly question here - what's the difference between free homotopy classes and homotopy classes? Pre-thanks for your response!

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    I assume that "homotopy classes" means "...while fixing endpoints." You're talking about continuous paths, right?2012-05-31
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    yes indeed. Ah so "free" here means that we don't fix endpoints?2012-05-31
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    Yep. The endpoints are free.2012-05-31
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    Super! Thanks Yuan!2012-05-31
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    @Sean: Perhaps you would consider answering your own question and accepting it? Now that you know?2012-05-31
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    this is in particular used for closed curves or manifolds without boundary. Free homotopy classes are allowed to homotop freely around, for the other ones keeps on point fixed (or even the image of a set $A$ is required to be mappped to a set $B$.2012-05-31

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