### 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

- 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. - 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. - Show that
`$f: X^{(1)} \to Y$`

extends to`$f: X^{(2)} \to Y$`

, i.e., to a map on the $2$-skeleton. - 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 $f_*(\bar\alpha) * g_*(\alpha)$.

- Show that for an edge $e$ of $T$, the map $H$ defined as above extends to
`$e\times [0, 1]$`

. - Show that for an edge $e$
*not in*$T$, the map $H$ defined as above extends to`$e\times [0, 1]$`

(here we use`$f_* = g_*$`

). - 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`$n\geq 3$`

).