is there a notation to designate the induced homomorphism including base point?

Question

Let $X,Y$ be a topological spaces.

Let $f:X\rightarrow Y$ be a continuous function.

Fix $x_0\in X$.

Define $f_*([r])=[f\circ r]$ for every loop $r$ at $x_0$.

Then, $f_*:\pi_1(X,x_0) \rightarrow \pi_1(X,f(x_0))$ is called the induced homomorphism.

As you can see the notation $f_*$ itself shows no information about which base point is chosen.

Is there a typical notation for resolve this problem?

For example, when one wants to emphasize on where a limit is taken, one usually denotes like $\lim_{x\to p, x\in A}$.

I want to know whether there is a corresponding notation for $f_*$.

Answer 1

To my knowledge there is no common notation for this. This is perhaps because the conjugacy class of a fundamental group of a connected manifold is actually independent of the base point:

Given two base points $x_0, y_0 \in X$ and any path $\alpha: [0, 1] \to X$ such that $\alpha(0) = x_0, \alpha(1) = y_0$, we get a map $$\Phi: \pi_1(X, x_0) \to \pi_1(X, y_0)$$ defined by $$\Phi([\gamma]) = [\alpha \cdot \gamma \cdot \alpha^{-1}].$$ Here, $\cdot$ denotes path concatenation and $\alpha^{-1}$ is the curve $\alpha$ run backward, and checking shows that it is well-defined.

So, while the fundamental group itself depends on a choice of base point, the conjugacy class of the fundamental group does not, i.e., it is an invariant of the (connected) manifold. In this sense, the base point of a fundamental group is inessential in a way that, e.g., the "base point" of a limit is not.

Answer 2

There is another viewpoint different from the answer of @Travis, in which there is no ambiguity up to conjugacy.

Namely, it is common to put the base point into the notation for the map $f$ itself, like this: $$f : (X,x_0) \to (Y,y_0) $$ In this notation, one regards $f$ not just a morphism in the category of topological spaces and continuous maps, but as a morphism in the category of base-pointed topological spaces and base-point preserving continuous maps. In this manner the base points for induced homomorphism $f_* : \pi_1(X,x_0) \to \pi_1(Y,y_0)$ are indeed unambiguous.