I am beginning study in algebraic topology and came across this problem, which is rather upsetting to me. My understanding is that elements of the fundamental group are loops, so how is this dashed path defined? If it is meant to represent the loop from the initial point that then travels across the dashed path and then along the sides, then why is it not the identity? The square itself is just a polygon and I thought its fundamental group would be trivial because every loop at any base point is homotopic to the trivial loop.
Best Answer
The space is not the square $I \times I$; it is a quotient of $I \times I$. The edges are identified as indicated by the arrows. Think of it as gluing each edge to the others in such a way that the arrows go in the same direction. In particular, all corners of the original square are the same point in the quotient space, so that the dashed line is indeed a loop.