I'm reading Lee's Introduction to topological manifolds and on page 291 he writes about the two-holed torus $X$:
"In terms of our standard generators for $\pi_1(X)$ this loop is path-homotopic to either $\alpha_1 \beta_1 \alpha_1^{-1} \beta_1^{-1}$ or $\beta_2 \alpha_2 \beta_2^{-1} \alpha_2^{-1}$ so it is not null homotopic..."
where he is talking of the path around the middle bit as for example on this picture http://inperc.com/wiki/images/b/ba/Double_torus_construction.jpg the red path in the third part of the picture.
I'm confused about this because according to my understanding this path is null homotopic because I can move it to one side of the two-holed torus and then shrink it to a point as there are no holes inside it.
Can someone explain to me why this is false? Many thanks for your help!