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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? –  Qiaochu Yuan May 31 '12 at 14:55
    
yes indeed. Ah so "free" here means that we don't fix endpoints? –  Sean May 31 '12 at 15:22
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Yep. The endpoints are free. –  Qiaochu Yuan May 31 '12 at 15:26
    
Super! Thanks Yuan! –  Sean May 31 '12 at 15:28
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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 $B$. –  user20266 May 31 '12 at 18:14
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