Is the Fundamental Group of a Pointed Space Dependent on the Base Point?

[quasar987]

Simple question: is the fundamental group of a pointed space independant of the base point?

 

[StatusX]

If the space is path connected. Then if [itex]\gamma[/itex] is a path from x₁ to x₀, the map sending the homotopy class of a loop [itex]\alpha[/itex] in [itex]\pi_1(X,x_0)[/itex] to the homotopy class of the loop [itex]\gamma \alpha \gamma^{-1} [/itex] in [itex]\pi_1(X,x_1)[/itex]is easily shown to be an isomorphism.

 

[mathwonk]

but a different path can yield a different isomorphism. hence the two groups are isomorphic but there is no distionguished isomorphism. so the two groups cannot be identified, so the answer is no, the group itself depends on the point, but the isomorphism class does not.

we often think iof isomorphic groups as "the same" but of course they are not, else there would be no galois theory.