@@ -840,7 +839,7 @@ A map $r \colon Y \to X$ such that $r \circ j = \id_X$ is called \CmMark{element

But this is a very simple CW-complex with differential $\id\colon\Z\pi\to\Z\pi$ (for $\pi=\pi_1(Y)$) from degree $n$ to degree $n-1$ and thus $\tau(C_{*}(\tilde Y,\tilde X))=0$.

The converse is more complicated.

\item Let $A \in\GL_n(\Z\pi)$ ($n \geq3$) and let $X' = X \vee\bigvee_{j =1}^nS^{n-1}$, where we denote the $j$-th inclusion of $S^{n-1}$ into the wedge by $b_j$.

\item Let $A \in\GL_n(\Z\pi)$ ($n \geq3$) be an element representing $a \in\Wh(\pi(X))$and let $X' = X \vee\bigvee_{j =1}^nS^{n-1}$, where we denote the $j$-th inclusion of $S^{n-1}$ into the wedge by $b_j$.

We attach $n$$n$-cells to $X'$ via attaching maps $f_j \colon S^{n-1}\to X'$ (relevant: $[f_j]\in\i_{n-1}(X')$ which yields $Y$.

\begin{align*}

\begin{pmatrix}

...

...

@@ -853,10 +852,14 @@ A map $r \colon Y \to X$ such that $r \circ j = \id_X$ is called \CmMark{element

\ni\bigoplus_{j = 1}^n \pi_{n-1}(X')

\text{ for } b_j \in\pi_{n-1}(X') \curvearrowleft\pi_1(X') = \pi_1(X) = \pi.

\end{align*}

One sees by closer inspection that

One sees by closer inspection that the relative chain complex $C_{*}^{\text{CW}}(\tilde Y, \tilde X)$ has only two non-trivial chain groups in two consecutive dimensions, with

boundary map realized by the $\Z\pi$-isomorpism $A$. Thus, $C_{*}^{\text{CW}}(\tilde Y, \tilde X)$ is acyclic, and since $\pi_1(\tilde Y,\tilde X)=0$,

we get from the {\itshape relative Hurewicz theorem} that $\pi_k(\tilde Y,\tilde X)=0$ for all $k \in\mathbb N$. Since $n \geq3$, we have $\pi_1(Y,X)=0$, so we can conclude

that $\pi_k(X)\cong\pi_k(Y)$ for all $k$, where the isomorphism is realised by the map $\pi_k(X \hookrightarrow Y)$. By {\itshape Whitehead's theorem}, $X \hookrightarrow Y$ is

a homotopy equivalence, so we can indeed compute its Whitehead torsion. We get:

\begin{align*}

\tau(\tilde X \hookrightarrow\tilde Y) = \tau(C_{*}^{\text{CW}}(\tilde Y, \tilde X))