##### 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 $n\geq 3$`).