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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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    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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