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Let $f\colon X\to X$ be a homeomorphism and $M_f$ be the mapping torus of $f$. That is, $M_f=X\times [0,1]/(x,0)\sim (f(x),1)$. Is $M_f$ homeomorphic to $M_{f^{-1}}$? (Since $f$ is a homeomorphism, $f^{-1}$ is also a homeomorphism.)

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Yes, they are homeomorphic. The homeomorphism is induced by the function $X \times [0,1] \mapsto X \times [0,1]$ defined by $(x,t) \mapsto (x,1-t)$.

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