Fundamental group - space of copies of circle $S_1$

Question

For $n>1$ an integer, let $W_n$ be the space formed by taking $n$ copies of the circle $S_1$ and identifying all the $n$ base points to form a new base point, called $w_0$. What is $\pi_{1}$($W_n , w_0 $ )?

The fundamental group is the set of all homotopy classes of loops with base point $w_0$ forms the fundamental group of $W_n$ at the point $w_0$

I know the fundamental group of the circle is $\mathbb{Z}$, the set of integers

So would it be $n\mathbb{Z}$?

Answer 1

You maybe want to look at Van Kampen Theorem, and for example, Hatcher Chapter 1, Example 1.21, in which it is shown that $\pi_1(S^1\vee S^1)=\mathbb{Z}*\mathbb{Z}$, where $*$ denotes free products, and $A\vee B$ denotes $A\amalg B /{\sim}$, with $\sim$ defined by the identification of $a_0, b_0$ for pointed spaces $A,B$, with points $a_0$, $b_0$.

Try to generalise the above to the wedge sum of $n$ circles.

Answer 2

You can understand $W_n$ as a CW-complex, so you have one $0$-cell $e^0_1$ and $n$ copies of $1$-cells $e^1_i$ ($i=1,\dots, n$) and any $m$-cell with $m>1$, so it means that if you want the fundamental group, you have $n$ diferent generators (one for each $1$-cell) and you don't have any restriction, so it means that you obtain that the fundamental group is $\pi_1(W_n)=\ast_{i=0}^n\mathbb{Z}e^1_i\cong \ast_{i=0}^n\mathbb{Z} \cong F[n]$ (where $F[n]$ is the amalgamated group of $n$ generators). In conclusion, $\pi_1(W_n)\cong \langle a_1, a_2, \dots, a_n \rangle$