# the proof of lemma 55.3 in Munkres's book Topology (second edition)

What does the fact that $$[p_0]$$ generates the fundamental group of $$S^1$$ mean? I don't know where this fact was used.

A part of Lemma 55.3 in the book proves the following:

Let $$h:S^1\to X$$ be a continuous function. Assume that $$h_*$$ is the trivial homomorphism of fundamental groups. Then the map $$h:S^1\to X$$ is nullhomotopic (The converse is also proven in the lemma).

The lemma is proved as follows in the book:

Let $$p:\mathbf R\to S^1$$ be the standard covering map $$x\mapsto (\cos 2\pi x, \sin 2\pi x)$$ and let $$p_0=p|_I$$, where $$I=[0, 1]$$. Then $$[p_0]$$ is a loop in $$S^1$$ based at $$b_0:=(1, 0)$$ and it generates $$\pi_1(S^1, b_0)$$.

Let $$x_0=h(b_0)$$. Since $$h_*$$ is trivial, the loop $$f:=h\circ p_0$$ can be path-homotoped to the constant loop based at $$x_0$$. So let $$F$$ be a path homotopy in $$X$$ between the loops $$f$$ and the constant loop $$e_{x_0}$$. Now the map $$p_0\times id:I\times I\to S^1\times I$$ is a closed continuous sujective map and is thus a quotient map. Also, $$(0, t)$$ and $$(1, t)$$ are mapped to $$(b_0, t)$$ under this map for each $$t\in I$$. The path homotopy $$F$$ maps $$0\times I$$, $$1\times I$$ and $$I\times 1$$ to $$x_0$$ of $$X$$, and so it induces a continuous map $$H:S^1\times I\to X$$ that is a homotopy between $$h$$ and a constant map.

I thought about it for a day. it's a guess. I think Munkres said it just to easily prove that the map $$H$$ is induced by the path homotopy $$F$$ (Theorem 22.2)
the fact that $$[p_0]$$ is the generator implies that the map $$p_0\times id$$ rotates $$I\times I$$ just once. So $$p_0\times id$$ has a injectivity except for $$0\times t$$,$$1\times t$$.Then ,for each $$(s,t)\in S^1\times I$$, $$s\not= b_0$$, $$(p_0\times id)^{-1}(s,t)$$ is a one point set. Of course, on this one point set, F is constant.