I have a problem with understanding homotopy in loop curves. So I have these two curves σ and υ. It is defined an operation * such that we add the second curve at the first one creating another curve. I need to know if this new curve is homotopic to the original curves (or other curves from a given set). I know that a curve is homotopic with another curve if I can "bend" them to obtain the same line. But what happens when curves are loop?
is the curve generated from σ and υ homotopic to this one? and why?
I first state a theorem, whenever $\alpha_1\in L(X,x_0)$ homotopic to $\alpha_2\in L(X,x_0)$ and $\beta_1\in L(X,x_0)$ homotopic to $\beta_2\in L(X,x_0)$ we have $\alpha_1*\beta_1\in L(X,x_0)$ homotopic to $\alpha_2*\beta_2\in L(X,x_0)$ where $L(X,x_0)$ denote set of all loops in $X$ based at $x_0\in X$.
Now if possible let $\gamma_1*\delta_1$ is homotopic to $\gamma_2*\delta_2$ then we have $[\gamma_1*\delta_1]=[\gamma_2*\delta_2]$ also $[\gamma_1]=[\gamma_2]$ therefore we have $[\gamma_1]^{-1}[\gamma_1*\delta_1]=[\gamma_2]^{-1}[\gamma_1*\delta_1]$ i.e. $[\delta_1]=[\delta_2]$.But according to the second picture we have $[\delta_1]=[\delta_2]^2$ that gives $ \delta_2$ is homotopic to constant loop based at $x_0$ , which is impossible as inside of $\delta_2$ contains a hole.
So the conclusion is $\gamma_1*\delta_1$ is not homotopic to $\gamma_2*\delta_2$ .