Given maps \(f_i: X_i \to X_{i+1}\) for integers \(i < 0\), show that the “reverse mapping telescope” obtained by gluing together the mapping cylinders of the \(f_i\)’s in the obvious way deformation retracts onto \(X_0\). Similarly, if maps \(f_i: X_i \to X_{i+1}\) are given for all \(i \in \mathbb{Z}\), show that the resulting “double mapping telescope” deformation retracts onto any of the ordinary mapping telescopes contained in it, the union of the mapping cylinders of the \(f_i\)’s for \(i\) greater than a given number \(n\).
Solution
For \(i<0\), let \(M_i\) be the mapping cylinder of \[ f_i:X_i\to X_{i+1}. \] The reverse telescope is \[ T^- =\cdots\cup M_{-2}\cup M_{-1}, \] with the copy of \(X_{i+1}\) at the top of \(M_i\) identified with the copy at the bottom of \(M_{i+1}\).
Each mapping cylinder \(M_i\) deformation retracts onto its top \(X_{i+1}\) by sliding points along the interval direction. These deformations are compatible with the gluing and can be performed successively on the strips \[ M_{-1},\ M_{-2},\ldots \] with the time intervals chosen, for example, as \[ [0,1/2],\ [1/2,3/4],\ [3/4,7/8],\ldots. \] At every time before \(1\), only finitely many cylinders have been collapsed, and at time \(1\) every point has been carried into \(X_0\). On each compact subset only finitely many cylinders are involved, so the resulting homotopy is continuous. It fixes \(X_0\) throughout. Hence \[ T^-\simeq X_0 \] by deformation retraction.
Now let \[ T=\bigcup_{i\in\mathbb Z}M_i \] be the double telescope, and for a fixed integer \(n\) let \[ T_{\ge n}=\bigcup_{i\ge n}M_i \] be the ordinary right-hand telescope starting at \(X_n\). The complementary left-hand part \[ \cdots\cup M_{n-2}\cup M_{n-1} \] is a reverse telescope ending at \(X_n\), so the preceding deformation retraction collapses it onto \(X_n\) while fixing \(X_n\). Extending this deformation by the identity on \(T_{\ge n}\) gives a deformation retraction \[ T\searrow T_{\ge n}. \] Thus the double telescope deformation retracts onto every ordinary tail telescope.