# Square lemma for Paths, Homotopy

1. Feb 19, 2013

### gauss mouse

1. The problem statement, all variables and given/known data
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 $f\cdot g\sim h\cdot k$.

2. Feb 19, 2013

### 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.

3. Feb 19, 2013

### 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!