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First of all, I apologize for the crudeness of my question. Consider the construction of the homotopy groups. We mod out the space of "loops" at point by the equivalence relation generated by homotopy equivalence then give the new space a group structure were the operation is "concatenation" of loops. My question: Could we, instead, mod out the space of "loops" (without reference to a specific point) by the equivalence relation generated by isotopy equivalence then give this space a group structure using some kind of "surgery" on the equivalence classes?

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Homotopy equivalence is already an equivalence relation. Are you talking about ambient isotopy here? What do you mean by "some kind of surgery"? –  Qiaochu Yuan Jun 28 '12 at 2:10
    
yes, ambient isotopy, was what I was referencing. By "surgery" I meant something like what follows: let f:[0,1] --> M and g:[0, 1] --> M be two loops on the space M, then remove a disk from f and a disk from g then form a new "loop" by attaching line segments to the end points in an obvious manner (perserving orientation?). –  Mathmonkey Jun 28 '12 at 2:27
    
do you want to cut out disks that "bound" the images of $f$ and $g$? This may only be possible for trivial loops, those homotopic to the constant loop. –  Olivier Bégassat Jun 28 '12 at 2:30
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What do you mean by "remove a disk from $f$"? –  Qiaochu Yuan Jun 28 '12 at 2:34
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You cannot remove such a disc from the constant loop. Besides, the line segments you want to attach are not canonical (there are several homotopy classes of them and you have to choose one). –  user17786 Jun 28 '12 at 11:49
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