Let $K$ be a simplicial complex in $\mathbb{R}^{N}$, which we may also consider as lying in $\mathbb{R}^{N+1}$ using the first $N$ coordinates. Write $c=(0,0, \ldots, 0,1) \in \mathbb{R}^{N+1}$. Show that if $\left\langle v_{0}, v_{1}, \ldots, v_{n}\right\rangle$ is a simplex of $K$ then $\left\langle v_{0}, v_{1}, \ldots, v_{n}, c\right\rangle$ is a simplex in $\mathbb{R}^{N+1}$.

Let $L \leqslant K$ be a subcomplex and let $\bar{K}$ be the collection

$$K \cup\left\{\left\langle v_{0}, v_{1}, \ldots, v_{n}, c\right\rangle \mid\left\langle v_{0}, v_{1}, \ldots, v_{n}\right\rangle \in L\right\} \cup\{\langle c\rangle\}$$

of simplices in $\mathbb{R}^{N+1}$. Show that $\bar{K}$ is a simplicial complex.

If $|K|$ is a Möbius band, and $|L|$ is its boundary, show that

$$H_{i}(\bar{K}) \cong \begin{cases}\mathbb{Z} & \text { if } i=0 \\ \mathbb{Z} / 2 & \text { if } i=1 \\ 0 & \text { if } i \geqslant 2\end{cases}$$