1
$\begingroup$

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!

  • 0
    @anon: Ha :-O! Best explanation ever! Now it seems blatantly obvious, thank you!2019-03-23

1 Answers 1

3

It might be that your geometric intuition is being led astray by the two-dimensional picture. It looks as if there's a "closed contour" to the right of the loop, and all you need to do is slide the loop along that. But that's deceptive -- consider what happens to the entire loop when you do that. As two points of the loop move along what's shown as the "contour" in the two-dimensional picture, the rest of the loop must keep connecting those two points. But as the points move past the hole, the rest of the loop that's connecting them would have to move through the hole.

  • 0
    Yeah, see anon's most excellent comment. Now I'm not sure how I could not see it o_O2011-07-25