I’m learning about homotopy and the fundamental groups, and I’m having a hard time visualizing the transformation. In particular, I can’t see when there is no homotopy between loops.
An example from my class is the following picture (the loops are all confined to $S^1$. I draw them that way just to make clear what paths are being travelled).
I can see how the first loop could be deformed into the second one by doing less and less of the “repeated” portion.
Why can’t we have something similar, for example as shown below, to deform a loop into another loop of the opposite direction? Is the condition of continuity violated somehow?
Apologize for the bad quality of the pics.
Edit: I’m actually thinking of the fundamental group of the circle $S^1$. My understanding is that the group is comprised of the equivalence classes of loops with a base point $x_0$ (here I take it to be $(1,0)$). In what sense is a loop goes a full circle from $(1,0)$ counterclockwise not the same as a loop goes a full circle from $(1,0)$ clockwise? I don’t see how the procedure in the second picture does not describe a homotopy between the two.