Fundamental Group Notation

Question

I am currently working on a problem in Spanier's Algebraic Topology text. Specifically, I'm working on Problem B4 in Chapter 2, which asks one to prove that $\pi(S^1 \vee S^1)$ is nonabelian.

Now, earlier in the text, the author states that $\pi(X,x_0)$ is the fundamental group of $X$ based at $x_0$. Thus, my question is, does the notation $\pi(S^1 \vee S^1)$ here simply denote the fundamental group of $S^1 \vee S^1$ without a choice of basepoint ? In general, is the notation $\pi(X)$ used often to simply mean the fundamental group of $X$ ? This would make sense to me, since the fundamental group of $X$ is independent of the choice of the basepoint.

Lastly, if $\pi(X)$ really is something different from the fundamental group of $X$, can one accomplish showing that $\pi(X)$ is nonabelian by showing that the fundamental group of $X$ is nonabelian ?

Thank you !

Answer 1

Yes, it is common to omit the basepoint when referring to the fundamental group. This is an abuse of notation, but it is often harmless since, as you mention, the fundamental group of a path-connected space is independent of the choice of basepoint up to isomorphism. (Though, one should be careful with this, since the isomorphism is non-canonical when the fundamental group is nonabelian!)

I would mention, though, that the notation $\pi(X,x_0)$ (or $\pi(X)$) is not commonly used for the fundamental group. Instead, it is usually called $\pi_1(X,x_0)$ (or $\pi_1(X)$ when the basepoint is suppressed).

Answer 2

You don't need a basepoint point, because $S^1\vee S^1$ is path connected. All the fundamental groups at the different possible base points would be isomorphic.

And $\pi_1(X)$ is usually the fundamental group of $X$, ($\pi_n(X)$ being the $n$-th homotopy group of $X$, where you consider homotopy classes of maps from $S^n$ into $X$.)