I have the following question that I'm trying to reason out. Here it is:
Fix $n\in\mathbb{N}$ and let $V\subset\mathbb{R}^{2}$ be a closed polygon of $2n$ sides such that $\partial V=a_{1}\cup a_{1}'\cup\cdots\cup a_{n}\cup a_{n}'$, where $a_{j}$ and $a_{j}'$ are intervals with vertices $(v_{j,0},v_{j,1})$ and $(v_{j,0}',v_{j,1}')$ and the enumeration of $\partial V$ is clockwise. Let $T_{n}^{-}$ be the quotient of $V$ obtained as follows: for each $j\in\{1,2,\ldots,n\}$, identify $a_{j}$ with $a_{j}'$ via a linear homeomorphism $v_{j,0}\mapsto v_{j,0}'$ and $v_{j,1}\mapsto v_{j,1}'$. Compute $\pi_{1}(T_{n}^{-})$.
I know that I should be using the Seifert-van Kampen Theorem to actually compute this fundamental group. However, I am not sure at all as to what do with the linear homeomorphisms. I'm not sure how we can actually quotient the space. My professor barely did anything in the way of examples, and my TA belittles us for asking any questions. Thanks in advance for any help!