Definition 8.3.
The mapping cylinder of a map \(e \colon Y \to Z\) is
\begin{equation*}
M_e := \bigl(Y \times I \,\sqcup\, Z\bigr) \,\big/\, (y, 1) \sim e(y).
\end{equation*}
Section 8.3 The Mapping Cylinder
The mapping cylinder allows us to apply this description of connectivity to an arbitrary map \(e \colon Y \to Z\text{:}\) it replaces \(e\) by an inclusion followed by a homotopy equivalence.
There is an inclusion \(i \colon Y \hookrightarrow M_e\text{,}\) \(y \mapsto [(y, 0)]\text{,}\) and a retraction \(r \colon M_e \to Z\) sending \([(y, s)] \mapsto e(y)\) and \([z] \mapsto z\text{.}\) Linear interpolation \(\rho_t[(y, s)] = [(y, s(1-t) + t)]\text{,}\) \(\rho_t[z] = [z]\text{,}\) is a deformation retraction of \(M_e\) onto \(Z \subseteq M_e\text{,}\) constant on \(Z\text{.}\) Hence \(e = r \circ i\) factors as
with \(r\) a homotopy equivalence and \(i\) the inclusion of a closed subspace.
Lemma 8.5.
A map \(e \colon Y \to Z\) with \(Y\neq\varnothing\) is an \(n\)-equivalence if and only if the pair \((M_e, Y)\) is \(n\)-connected.
Proof.
Since \(r\) is a homotopy equivalence, \(e_* = r_* \circ i_*\) is a bijection (respectively surjection) on \(\pi_q\) exactly when \(i_*\) is. So \(e\) is an \(n\)-equivalence if and only if \(i \colon Y \hookrightarrow M_e\) is, which is equivalent to \((M_e, Y)\) being \(n\)-connected by the long exact sequence of the pair.
