Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.3.4.8. Let $p$ and $q$ be nonnegative integers. Then the function $H_{p,q}: | \Delta ^{p} | \star | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$ of Example 4.3.4.7 is a homeomorphism of topological spaces.

Proof. Since $| \Delta ^ p | \star | \Delta ^ q |$ is compact and $| \Delta ^{p+1+q} |$ is Hausdorff, the continuous function $H_{p,q}$ is automatically closed. To complete the proof, it will suffice to show that $H_{p,q}$ is bijective. Fix a point $x$ of $| \Delta ^{p+1+q} |$, given by a sequence of nonnegative real numbers $( \overline{u}_0, \ldots , \overline{u}_ m, \overline{v}_0, \overline{v}_1, \ldots , \overline{v_ n} )$ satisfying

\[ \overline{u}_0 + \cdots + \overline{u}_ m + \overline{v}_0 + \cdots + \overline{v}_ n = 0. \]

Set $t = \overline{v}_0 + \cdots + \overline{v_ n}$. If $t = 0$, the set $H_{p,q}^{-1} \{ x\} $ consists of a single point of $| \Delta ^{p} |$ (regarded as a subset of $| \Delta ^{p} | \star | \Delta ^{q} |$), given by the sequence $( \overline{u}_0, \ldots , \overline{u}_ m)$. If $t = 1$, the set $H_{p,q}^{-1} \{ x \} $ consists of a single point of $| \Delta ^ q |$ (regarded as a subset of $| \Delta ^{p} | \star | \Delta ^{q} |$), given by the sequence $( \overline{v}_0, \ldots , \overline{v}_ m)$. In the case $0 < t < 1$, the set $H_{p,q}^{-1} \{ x\} $ consists of a single point of $| \Delta ^{p} | \star | \Delta ^{q} |$, given as the image of the triple

\[ ( \frac{\overline{u}_0}{1-t}, \ldots , \frac{ \overline{u}_ m}{1-t} ), t, ( \frac{ \overline{v}_0}{t}, \ldots , \frac{ \overline{v}_ n}{t} ) ) \in | \Delta ^{p} | \times [0,1] \times | \Delta ^{n} |. \]
$\square$