Proof of Whitehead's theorem

due by Thursday, Apr 2, 2020

In this set of exercises, we complete some details of the proof of the theorem of Whitehead that was sketched as in the worksheet.

Part 1 - Existence of maps
  1. For the map $f: X^{(1)} \to Y$ defined on the $1$-skeleton, show that $f_* = \varphi\circ i_*$ where `$i: X^{(1)} \to X$ is the inclusion of the $1$-skeleton.
  2. Conclude that of $\theta: S^1 \to X^{(1)}$ is the attaching map of a $2$-cell in $X$, then $f\circ\theta: S^1 \to Y$ is homtopically trivial.
  3. Show that $f: X^{(1)} \to Y$ extends to $f: X^{(2)} \to Y$, i.e., to a map on the $2$-skeleton.
  4. Finally show that there exists $f: (X, x_0)\to (Y, y_0)$ such that $f_*=\varphi$.
Part 2 - Homotopies of maps

Recall that we have a fixed maximal tree $T$ containing the base point $x_0$, and we define the homotopy $H$ on $X^{(0)} \times [0, 1]$ as follows: given a vertex $v$, there is a unique reduced path $\alpha$ in the tree $T$ from $x_0$ to $v$; map $v \times [0, 1]$ to $f_*(\bar\alpha) * g_*(\alpha)$.

  1. Show that for an edge $e$ of $T$, the map $H$ defined as above extends to $e\times [0, 1]$.
  2. Show that for an edge $e$ not in $T$, the map $H$ defined as above extends to $e\times [0, 1]$ (here we use $f_* = g_*$).
  3. Show that $H$ extends to a homotopy from $f$ to $g$ on $X\times [0, 1]$ (note that the this step involves only $n$-cells fof $n\geq 3$).