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    A mapping cylinder deformation retracts onto its target

    P-UNL871F13-PS10.2

    Area
    Topology
    Topics
    Homotopy, Quotient Spaces
    Status
    draft
    Contents

    Let \(X_f\) be the mapping cylinder of a continuous map \(f:X\to Y\), and let \[ j:Y\to X_f,\qquad j(y)=[y] \] denote the canonical inclusion. Write \(\overline Y=j(Y)\).

    1. Define \[ r:X_f\to\overline Y, \qquad r([(x,s)])=[f(x)],\quad r([y])=[y]. \] Show that \(r\) is a retraction. Show that \[ F_t([(x,s)])=[(x,t+(1-t)s)],\qquad F_t([y])=[y] \] defines a deformation retraction of \(X_f\) onto \(\overline Y\).

    2. Show that \(j\) is an embedding and conclude that \(X_f\) is homotopy equivalent to \(Y\).

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