Square lemma for Paths, Homotopy

[gauss mouse]

Homework Statement

In Lee's "Topological Manifolds", there is a result on page 193 called "The Square Lemma" which states that if I denotes the unit interval in \mathbb{R}, X is a topological space, F\colon I\times I\to X is continuous, and f,g,h,k are paths defined by
f(s)=F(s,0),\ g(s)=F(1,s),\ h(s)=F(0,s),\ k(s)=F(s,1),
then f\cdot g\sim h\cdot k where \cdot denotes path concatonation and \sim denotes homotopy rel \{0,1\}.

It seems to me that this is not guaranteed to be true if X is not simply connected. Indeed if we take X=I\times I and take F=\text{id}_{I\times I}, then I don't think that we have f\cdot g\sim h\cdot k.

Am I wrong about this?

 

[Dick]

What's not simply connected about I x I? And why don't you think fg and hk are homotopic? You can deform them both to a path across the diagonal of the square.

 

[gauss mouse]

Ah. Brain malfunction on my part. For some reason I was thinking of I\times I as the boundary of the square and not the whole square. That's sorted then. Thanks!