Consider the following question.
Let $X$ be a topological space and $x_0$ be any point in $X$. Let $\gamma\in \pi_1(X,x_0)$ be a non-constant loop about the base point $x_0$.
Then can it happen that $\gamma*\gamma=e_{x_0}$ (here $e_{x_0}$ denotes the constant loop)?
I don't see how this is possible.
For if $\gamma(t_0)\neq x_0$ for some $t_0>0$, then $\gamma*\gamma(t_0/2)=\gamma(t_0)\neq x_0$.
Hence $\gamma* \gamma$ is not the constant loop.
Why this confuses me is because I know for a fact that the funcdamental goup of $\mathbf RP^2$ (the real projective plane) is $\mathbf Z/2\mathbf Z$.
So there must be a non-constant loop in $\pi_1(\mathbf RP^2,x_0)$ such that $\gamma* \gamma=e_{x_0}$, in contradiction to my previous "inference".
I must be making a stupid mistake somewhere.
Can somebody please point it out.
Thanks.
The concatenation of two nonconstant loops can be homotopic to a constant loop (i.e. nulhomotopic), but if one of the loops is nonconstant then the concatenation itself won't be constant.