# induced homomorphism

I found the following in section 8.14 of Munkres Topology, a first course.

"Now suppose we have a fixed path-connected space $B$ and a fixed base $b \in B$. Let $p: E \rightarrow B$ be a covering map, where $E$ is path connected. If we choose a point $e$ in $p^{-1}(b)$ as a base point for $E$, then we have an induced homomorphism $p_*: \pi_1(E,e) \rightarrow \pi(B,b)$."

How is this homorphism induced? I can get a surjection between the first fundamental group of a base space space at a point to the fiber over that point, but since it might not be injective I can't necessarily invert it.

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Induced is not invertible. To define the homomorphism $p_*$ we need to use $p$ at some point, thus "induced" from $p$. – user27126 Apr 17 '13 at 23:07

If $\gamma : S^1\to E$ is a loop in $E$ (with base point $e$), then $p\circ\gamma : S^1\to B$ is a loop in $B$ with base point $b = p(e)$. Then we simply take the homotopy class of the projected loop: \begin{align*} p_* : \pi_1(E,e)&\to\pi_1(B,b).\\ \left[\gamma\right]&\mapsto\left[p\circ\gamma\right] \end{align*} Note that this may only be a homomorphism, not necessarily an isomorphism, so it need not be injective.