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

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SUMMARY

The fundamental group of a pointed space is dependent on the choice of the base point when the space is path connected. Specifically, if there exists a path \(\gamma\) from one base point \(x_0\) to another base point \(x_1\), the induced map between the homotopy classes of loops in \(\pi_1(X, x_0)\) and \(\pi_1(X, x_1)\) is an isomorphism. However, different paths can yield different isomorphisms, indicating that while the groups are isomorphic, they cannot be identified as the same. Thus, the fundamental group itself varies with the base point, although the isomorphism class remains constant.

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  • Understanding of fundamental groups in algebraic topology
  • Familiarity with path connected spaces
  • Knowledge of homotopy theory
  • Basic concepts of isomorphism in group theory
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quasar987
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Simple question: is the fundamental group of a pointed space independent of the base point?
 
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If the space is path connected. Then if \gamma is a path from x1 to x0, the map sending the homotopy class of a loop \alpha in \pi_1(X,x_0) to the homotopy class of the loop \gamma \alpha \gamma^{-1} in \pi_1(X,x_1)is easily shown to be an isomorphism.
 
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.
 

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