$\partial Y\to{Z}$ to a triangulation of $Y\to{Z}$.

![image](figs.1){width="9cm"}\

A simplex $\tau$ in $K$ has vertices $\tau=(b_0^0,\dots,b_{q_0}^0|...|b_0^p,\dots,b_{q_p}^p)$ with $\sigma=(a_0,\dots,a_p)$ such that $\pi^\prime(b_j^i)=a_i$. Here, we give the set of vertices of the total space, the lexicographical order. So geometrically, for an open simplex $\stackrel{\circ}{\sigma}$ in $L$, we have $$\pi^{-1}(\lvert\stackrel{\circ}{\sigma}\rvert)\approx{\lvert\stackrel{\circ}{\sigma}\rvert}\times{\bigcup_{\tau\in{\pi^{-1}(\sigma)}}}\Delta^{q_0...q_p}\times{\tau}.$$

We collect all these in the formal definition below using simplicial sets. For these we recall the notation but refer otherwise to Mac Lane [@McL], May [@May].

\[2.2\] A simplicial set $S_.=\{S_q\}$ is a sequence of sets with face operators $d_i:S_q\to{S_{q-1}}$ and degeneracy operators $s_i:S_q\to{S_{q+1}}$, $i=0,...,q$, satisfying the following identities: $$\begin{aligned}
d_id_j= \left\{ \begin{array}{r@{\quad: \quad}l}d_{j-1}d_i & i<j  \\
d_jd_{i+1} & i\geq{j}, \end{array} \right.\end{aligned}$$ $$\begin{aligned}
s_is_j= \left\{ \begin{array}{r@{\quad: \quad}l}s_{j+1} s_i & i\leq{j}  \\
s_j s_{i-1} & i>j, \end{array} \right.\end{aligned}$$ and $$d_is_j= \left\{ \begin{array}{r@{\quad:
\quad}l}s_{j-1}d_i & i<j \\ \text{id} & i=j,i=j+1
\\ s_jd_{i-1} & i>j+1.  \end{array} \right.$$

\[2.3\] A simplicial complex $K_.$ gives a simplicial set where $K_p=\{(a_{i_0},...,a_{i_p})\text{   }|\text{
}\text{some nondecreasing sequences for a given linear ordering of}\text{   }
K_0\}$ is the set of $p$-simplices.

\[2.4\] Given an open cover $\mathcal{U}={U_i}$ of $Z$ we have the nerve $N\mathcal{U}=\{N\mathcal{U}(p)\}$ of the covering, where $$N\mathcal{U}(p)=\bigsqcup_{i_0,...,i_p}U_{i_0}\cap{...}\cap{U_{i_p}},$$ and $(i_0,...,i_p)$ is nondecreasing for a given linear order of the index 