to $\sigma_i$ is not in $K_i$. It follows that $L\subset(\partial\tau_1*\dots*\partial\tau_r)*\partial\sigma$. The latter is a combinatorial sphere, which is of codimension $r+1$ in the $(m+r-1)$-simplex $(\tau_1*\dots*\tau_r)*\sigma=\Delta^{m_1}*\dots*\Delta^{m_r}$.

The preceding construction exhibits, within the sphere $S^{m-1}$ that contains $|K|$, an embedded copy of $S^{m-2}$ that contains $|L|$. Let $r{\colon}S^{m-1}\to S^{m-1}$ be the reflection in $S^{m-2}$. Then $hr^{-1}=hr$ is orientation preserving, and so is isotopic to the identity by the Alexander trick. Hence $hj$ is equivalent to $rj=j$.

\[van Kampen\] The above construction can be easily extended (cf. [@M2 Example 3.5]) to yield, for any maximal simplex $\sigma$ of $K$, a map $f_\sigma{\colon}|K|\to S^{m-1}$ with precisely one double point, one of whose two preimages lies in the interior of $|\sigma|$. This implies that every proper subpolyhedron $P$ of $|K|$ embeds in $S^{2m-2}$. Indeed, since $P$ is compact, it is disjoint from some point in the interior of $|\sigma|$ for some maximal face $\sigma$ of $K$, and therefore from a disk $D$ in the interior of $|\sigma|$. Then $P$ is embedded in $S^{m-1}$ by $f_\sigma$ precomposed with an appropriate self-homeomorphism of $|K|$, fixed outside $|\sigma|$.

We shall next see that Theorem \[self-dual\] is not as exciting as it might appear to be.

Collapsible and cell-like 