Proposition 6

The mapping $\sigma_f:$ Cyl($f$)$\times \mathbf{I}\rightarrow$ Cyl($f$) is a strong contraction of Cyl($f$) onto its base. Its term is the canonical retraction of Cyl($f$) onto its base.

The mapping $\sigma_f$ is a homotopy. The relations $\sigma_f(\alpha_f(x, s),0) =$ $\alpha_f(x, s)$ and $\sigma_f(\beta_f(y),0) =\beta_f(y)$ imply that the origin of $\sigma_f$ is the identity mapping of Cyl($f$). Let $r_f$ denote the term of $\sigma_f$. The relations

and $\sigma_f(\beta_f(y),1) =\beta_f(y) = (\beta_f\circ \gamma_f)(\beta_f(y))$ imply that $r_f$ is the canonical retraction $\beta_f\circ \gamma_f$ of Cyl($f$) onto its base.

For $(x, s)\in X\times \mathbf{I}$ and $t\in \mathbf{I}$, we have

For $y\in Y$ and $t\in \mathbf{I}$, we have $r_f(\sigma_f(\beta_f(y), t)) =r_f(\beta_f(y))$. Consequently, $\sigma_f$ is a strong contraction of Cyl($f$) onto its base.

The mapping $\sigma_f$ is called the canonical contraction of the cylinder of $f$ onto its base.