I am working on my undergrad thesis, which is about the Hopf-fibration of $S^3$. One of the key properties I want to show is that it can't be trivial. Usually people argue this using the fundamental group, and I understand these proofs.
However, given the space constraints and the other things I want to include, I feel it would not be best to introduce the whole fundamental group machinery just to prove this point. The fundamental group wasn't introduced in any of my lectures, but I do have "access" to homotopy theory.
EDIT: In particular, the sources I've found define simple connectedness in terms of the fundamental group. The definition I would like to use is that closed curves are null-homotopic/any two paths from $a$ to $b$ are homotopic.
In particular, it seems to be entirely sufficient to find some counterexample to show that $S^1$ can't be simply connected. $S^3$ is easily shown to be simply connected, so this would imply $S^3\not\sim S^2\times S^1$, which is exactly what is needed.
Looking at this question, it seems possible to show that $S^1$ can't be simply connected using differential topology tools. However, this question is very specific to this textbook, which I don't have, so I can't really follow the reasoning.
Alternatively, is it perhaps possible to formalize the rubber-band analogy for $S^1$ not being simply connected? One counterexample is all that is needed...
I am not really versed in the history of topology, but I believe that in the old days simple connectedness was defined as getting disconnected after a suitable connected codimension 1 subvariety is removed (or some more precise definition of this idea).
For instance the sphere $S^2$ is simply connected because removing in fact any closed curve on it disconnects it.
From this point of view $S^1$ is not simply connected because removing a point does not discnnect it.