Why fundamental groups are defined only for loops- better explained than in Munkres’ Topology.

by ayushkhaitan3437

I want to point out why fundamental groups are defined for loops, and not path homotopy classes. This is something that Munkres’ Topology does not do a good job of explaining.

Munkres says that we cannot define groups on path homotopy classes because for some pair [f] and [g], [f*g] may not be defined. This is because it is possible that f(1)\neq g(0).

This seems like a bit of an arbitrary requirement to me. Although one can clearly see the benefits of defining [f]*[g] only when f(1)=g(0), this can be worked around. For starters, if the path homotopy class between f(1) and g(0) is [h], one could define [f]*[g] as [f*h*g]. Obviously this would require the axiom of choice if there are multiple path homotopy classes between f(1) and g(0), but I still think this can be worked around.

The reason why fundamental groups are defined for loops is that there is a unique identity and inverse for every element (path homotopy class). For example, if f is a path from x_1 to x_2, then [e_{x_1}]*[f]=[f]*[e_{x_2}]=[f]. Clearly [e_{x_1}]\neq[e_{x_2}]. However for a loop, the identity is unique- namely [e_{x_0}], where x_0 is the point where the oop starts and ends.

The same argument works for inverses.