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?
Homotopy versus Isotopy
2
$\begingroup$
algebraic-topology
-
1You 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). – 2012-06-28