Compute the fundamental group of a five-pointed star ( boundary plus interior ).

Knowing that I have taken only chapter 1 & 2 of " introduction to knot theory " of Richard H. Crowell and Ralph H. Fox.

Which includes the fundamental group of the circle but does not include van Kampen theorem.

Could anyone give me a hint for the solution, please?

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    $\begingroup$ Perhaps a picture would help. Isn't the region you describe contractible? $\endgroup$ – lulu Jul 9 at 14:17
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    $\begingroup$ If you choose the center of your five-pointed star as base point for your fundamental group, you can shrink any closed path to the constant path by scaling. This is because the star is star-shaped around the center. $\endgroup$ – Magma Jul 9 at 14:28

Say that $\Omega \subset \Bbb R^n$ is star-shaped if there is $p \in \Omega$ with the following property : for all $ q \in \Omega$, the segment $[p,q] \subset \Omega$.

We know that

If $\Omega$ is star-shaped, then $\Omega$ is contractible, that is homotopy equivalent to a point. In particular, $\Omega$ is connected and simply-connected.

For proof of this statement see Lemma 2.11 and Theorem 3.4 here

Now we have that:

A topological space X is simply connected if and only if X is path-connected and the fundamental group of X at each point is trivial.

Por proof of this see Theorem 2.3 here

Therefore for a star shaped region, the fundamental group is trivial.

As you asked for a hint I would try prove these claims without clicking the links first.


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