# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Go back to the page of Subsection 3.2.2.

Comment #1098 by Xiaofa Chen on

A stupid question: is there a naturally defined action of $\pi_1(X,x)$ on $\pi_n(X,x)$? When $x$ and $x'$ are two basepoints in the same connected components of $X$, is there a naturally defined (iso)morphism from $\pi_n(X,x)$ to $\pi_n(X,x')$?

Comment #1103 by Kerodon on

Yes, the fundamental group $\pi_1(X,x)$ always acts on $\pi_{n}(X,x)$.

Comment #1104 by Kerodon on

Yes, for $n \geq 1$ the construction $x \mapsto \pi_{n}(X,x)$ determines a functor from the fundamental groupoid of $X$ to the category of group.

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