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Let $f : S 1 → S 1$ be a map of degree $grad(f) = n$.

Consider the space $M(n) = S 1 ∪_f D^ 2$ obtained by attaching a $2$-disc to a circle along its boundary using the map $f$ — i.e.,

the quotient $(S 1 $ $\bigcup\ D^ 2 )/ ∼$ where $∼$ is the equivalence relation generated by the relations $x ∼ f(x)$ for $x ∈ ∂D^ 2 = S^ 1$.

(1) Make a sketch of this identification.

(2) Show that $π_1(M(n))$ is isomorphic to $Z/nZ$.

Does anyone have any idea how this sketch should look like? For $(2)$ I was trying to use van-Kampen Theorem but then I got stuck..

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    Just sketch a regular $n$-gon and identify all the edges. Yes, Van Kampen is good: one set will be the neighborhood of the boundary and another the interior: then you see that $n$ times the generator is trivial. – 2017-02-05
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    Thanks, but how do I get that this sketch is a regular n-gon? For Van Kampen I was thinking to take a point $x_0$ in the interior and then for $U$ take $M$ \ $x_0$ and for V a small ball around $x_0$, but again I get in trouble when computing the fundamental group of $U$ intersected $V$ – 2017-02-05
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    Fundamental group of $V$ is trivial; fundamental group of $U$ is $\Bbb Z$, generated by one edge (note that $U$ deformation retracts to the boundary, which is topologically just one edge of the $n$-gon). The intersection $U\cap V$ has fundamental group also $\Bbb Z$ the generator is a "full circle" and its image in $U$ is $n$ times the generator of $U$. – 2017-02-05

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