Whitehead torsion III (Ex)

Let $f \colon X \to Y$$<wikitex>; Let $f \colon X \to Y$ be a homotopy equivalence of finite CW-complexes. Let $i \colon X \to \text{cyl}(f)$ and $j \colon Y \to \text{cyl}(f)$ be the canonical inclusions and $p \colon \text{cyl}(f) \to Y$ be the canonical projection. Show that the [[Wikipedia:Whitehead torsion|Whitehead torsion]] of the maps $j, i$ and $f$ satifies $\tau(j) = 0$ and $p_*(\tau(i)) = \tau(f)$. This is {{citeD|Kreck&Lück2005|Ex 6.3}}. </wikitex> [[Category:Exercises]] [[Category:Exercises with solution]]f \colon X \to Y$ be a homotopy equivalence of finite CW-complexes. Let $i \colon X \to \text{cyl}(f)$$i \colon X \to \text{cyl}(f)$ and $j \colon Y \to \text{cyl}(f)$$j \colon Y \to \text{cyl}(f)$ be the canonical inclusions and $p \colon \text{cyl}(f) \to Y$$p \colon \text{cyl}(f) \to Y$ be the canonical projection. Show that the Whitehead torsion of the maps $j, i$$j, i$ and $f$$f$ satifies $\tau(j) = 0$$\tau(j) = 0$ and $p_*(\tau(i)) = \tau(f)$$p_*(\tau(i)) = \tau(f)$.

This is [Kreck&Lück2005, Ex 6.3].