Assume that $ N $ is a path-connected smooth manifold. $ \alpha,\beta:\mathbb{S}^1\to N $ are two loops. $ x_1\in\alpha $ and $ x_2\in\beta $ are two points on the loops $ \alpha $ and $ \beta $. Assume that there is a homotopy $$ H:[0,1]\times\mathbb{S}^1\to N $$ such that $ H(0,\cdot)=\alpha(\cdot) $ and $ H(1,\cdot)=\beta(\cdot) $.
Since $ N $ is path-connected, we can find a curve $ \gamma:[0,1]\to N $ such that $ \gamma(0)=x_1 $ and $ \gamma(1)=x_2 $. I want to ask if $ \alpha $ and $ \gamma*\beta*\widetilde{\gamma} $ represent the same element in $ \pi_1(N,x_1) $? Here $ * $ denotes the composition of two curves and $ \widetilde{\gamma} $ is the reverse of $ \gamma $.
I guess these are the same but I cannot construct the homotopy function between them can you give me some hints or references?
