Does Hatcher makes an error at page $45$ afirming that $f:\alpha \mapsto \inf_{(x,z)\in X \times S^1} \det|\alpha(x,z)|$ is continuous? I didn't find any reference of continuity of $\inf$ of continuous functions on a compact set. It seems to me that lower semicontinuity of $f$ is enough to affirm that $f^{-1}(0,+\infty) = \operatorname{Aut}(E\times S^1)$ is open, what am I missing?
Any help or hint would be appreciated.