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I'm trying to self-study some algebraic topology, reading Hatcher. His questions seem much less straightforwardly worded than Munkres - with Munkres it was always clear that you weren't expected to know much coming in, whereas Hatcher seems to assume a lot of knowledge. Often simple seeming statements of his just seem vague and unclear to me, even though I also think they ought to be basic.

Right now I'm stuck deciphering exercise 1.2:

Show that the change-of-basepoint homomorphism $\beta_h$ depends only on the homotopy class of $h$.

At a guess, this means that given any two homotopic paths $h_1$ and $h_2$, for all loops $f$, there exists a homotopy between $\beta_{h_1}(f)$ and $\beta_{h_2}(f)$ ? I'm uncertain, as though I'm guessing at what Hatcher means rather than actually understanding him. Is this the correct interpretation, or am I off? And in general, anything I could do to more easily understand Hatcher? Thanks!

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    $\begingroup$ since $\beta_h$ is a map of groups, you can't speak of homotopy. Instead, you have to prove that if $h_1\sim h_2$, then $\beta_{h_1}=\beta_{h_2}$ $\endgroup$
    – Riccardo
    Jun 18, 2016 at 14:43
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    $\begingroup$ To me, that statements seems quite unambiguous. If $h_1$ and $h_2$ are two paths from $x_0$ to $x_1$ (as in Prop. 1.5), then we can define two functions $\beta_{h_1}$ and $\beta_{h_2}$. And the statement is that if $h_1$ and $h_2$ are in the same homotopy class, then $\beta_{h_1}=\beta_{h_2}$. $\endgroup$ Jun 18, 2016 at 14:43

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What I think is meant by this question is this. Given a topological space $X$ and a point $p \in X$ we define the fundamental group of $X$ with basepoint $p$ to be $\pi_1(X, p) = \{[\gamma] \mid \gamma \text{ is a closed loop around }p\}$. Here $[\gamma]$ denotes the homotopy class of $\gamma$.

Given $p, q \in X$ and a path $h$ in $X$ connecting $p$ to $q$ we can define a function $$\beta_h:\pi_1(X, p) \to \pi_1(X, q)$$ by $$\beta_h([\gamma]) = [\overline h * \gamma * h]$$ where $\overline h$ denotes the reverse of $h$.

As an exercise you can show that this mapping is well defined, and furthermore it is a group homomorphism. I believe that this is what Hatcher refers to as the change of basepoint homomorphism.

Now the question asks you to prove that if $h, h'$ are homotopic then $\beta_h = \beta_{h'}$.

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The natural environment for "change of base point" is that of groupoid and the fundamental groupoid $\pi_1 X$ of a space $X$. In a groupoid $G$, we have "object groups" $G(x)$for each object $x$ of $G$, and if $a: x \to y$ in $G$, then $a$ determines an isomorphism, "conjugation by $a$", $a_\#: G(x) \to G(y)$. This is dealt with in Chapter 6 of Topology and Groupoids, and in Philip Higgins' downloadable 1971 book Categories and Groupoids.

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