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%18.9.13 / 19.2.17
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\bf	FLAGS AND GROTHENDIECK CARTOGRAPHIC GROUP\footnote{translated from Russian by George Shabat. The translator is indebted to Noam Zeilberger and  Daniel Grayson for the idea of translation and for the critical comments.

This text was submitted as a report for a mathematical department of the semi-official body "Center of Scientific and Technical Activity, Research and Social Initiatives'' that existed as a subdivision of the Academy of Sciences of Russian Federation in 1990-91.  Voevodsky was a member of this department  for several months in 1990.}\\
V.A. Voevodsky
\end{center}
\bf1. Cartographic group.\\
\\
1.0. \it The Grothendieck cartographic group \rm is defined by the presentation
$$
\mathcal{C}_n:=\langle\sigma_0,\dots,\sigma_n
\mid \sigma_i^2=1, (\sigma_i\sigma_j)^2=1\text{ if }|i-j|\ge2\rangle
$$

The \it simplicial cartographic group \rm $\mathcal{C}_{n,3}$ is obtained from $\mathcal{C}_{n}$ by imposing the additional relations $(\sigma_i\sigma_{i+1})^3=1$ for $0\le i\le n-2$. Similarly the \it cubic cartographic group \rm $\mathcal{C}_{n,4}$ can be defined by imposing the relations
 $(\sigma_0\sigma_{i})^4=1$, $(\sigma_i\sigma_{i+1})^3=1$  for $0\le i\le n-2$.

We call the \it oriented cartographic group \rm the subgroup $\mathcal{C}_{n}^+$ the subgroup of index 2 in $\mathcal{C}_{n}$, generated by the pairwise products of the generators $\sigma_i$. In the same way the subgroups $\mathcal{C}_{n,3}^+$ and $\mathcal{C}_{n,4}^+$ are defined.\\
\\
\bf1.1. \rm The subgroup of $\mathcal{C}_{n}$ generated by $\sigma_0,\dots, \sigma_{n-1}$ is isomorphic to the group of automorphisms of the $(n+1)$-dimensional simplex $\Delta^n$, i.e., to the symmetric group $\mathsf{S}_{n+1}$. Under this isomorphism the generator $\sigma_i$ corresponds to the transposition $(i+1,i+2)$.

The analogous group $\mathcal{C}_n^+$ is isomorphic to the alternating group $\mathsf{A}_{n+1}$.

The subgroup in $\mathcal{C}_{n,4}$ (correspondingly in $\mathcal{C}_{n,4}^+$), generated by $\sigma_0,\dots,\sigma_{n-1}$ (correspondingly by $\sigma_i\sigma_j$ for $i,j\le n-1$) is isomorphic to
the group of automorphisms of the $(n+1)$-dimensional cube $\mathrm{I}^{n+1}$ (correspondingly to the group of its sense-preserving automorphisms).\\
\\
\bf1.2. \rm Note the isomorphism $\mathcal{C}_{2,3}^+\cong\mathsf{PSL}_2(\mathbb{Z})$. It has the form
$$
\sigma_0\sigma_2\to\left (
 \begin{array}{cc}
  0&-1\\
  1&0\\
 \end{array}
\right )
$$
%%%%%%%%
$$
\sigma_1\sigma_2\to\left (
 \begin{array}{cc}
  1&1\\
  0&1\\
 \end{array}
\right )
$$
%%%%%%%%
$$
\sigma_1\sigma_0\to\left (
 \begin{array}{cc}
  1&-1\\
  1&0\\
 \end{array}
\right )
$$
\bf1.3. \rm By an \it orientation \rm of a $\mathcal{C}_n$-set $X$ we mean a morphism of 
$\mathcal{C}_n$-sets $\nu:X\to\mathcal{C}_n/\mathcal{C}_n^+$. A set $X$ is called \it orientable \rm if it admits an orientation. This property is 
obviously equivalent to the stabilizer of
any element of $X$ being contained in
$\mathcal{C}_n^+$.

For every oriented $\mathcal{C}_n$-set $(X,\nu)$ denote by $X^+$ the preimage of the distinguished point of $\mathcal{C}_n/\mathcal{C}_n^+$ with respect to $\nu$. The association $(X,\nu)\to X^+$ can be extended 
 to an equivalence between the category  of oriented $C_n$-sets and the category of $C_n^+$-sets. 
   Denote the inverse equivalence by the rule
$$
Y\to Y\times\{1,-1\},
$$
where the action of $\mathcal{C}_n$ on $Y\times\{1,-1\}$ has the form
$$
\sigma_i(y,1)=(\sigma_0\sigma_i(y),-1),
$$
%%%%%%%
$$
\sigma_i(y,-1)=(\sigma_i\sigma_0(y),1).
$$
\\
\bf 2. The simplicial set associated to a $\mathcal{C}_n$-set.\rm\\
\\
\bf2.0. \rm For each non-degenerate $k$-dimensional  simplex $\sigma$ in $\Delta^n$ (i.e., for each $(k+1)$-element subset of $\{0,\dots,n\}$) denote by $\mathcal{C}_n(\sigma)$ the subgroup in 
$\mathcal{C}_n$ generated by all the $\sigma_i$ with $i\in\sigma$. Obviously for each $i$ with 
$0\le i\le\dim\sigma$ the inclusion $\mathcal{C}_n(\sigma)\supset\mathcal{C}_n(\partial_i\sigma)$ holds.

For a $\mathcal{C}_n$-set $X$ define the semi-simplicial set $\tilde{N}(X)$ by the rule
$$
\tilde{N}(X)_k=\coprod_{\dim\sigma=k}X/\mathcal{C}_n(\sigma).
$$
The face operators are defined in the obvious way. Denote by $N(X)$ the simplicial set associated with
$\tilde{N}(X)$.\\
\\
\bf2.1. \rm Obviously $N(*)\cong\Delta^n$, 
$N(\mathcal{C}_n/\mathcal{C}_n^+)\cong\Delta^n\coprod_{\partial\Delta_n}\Delta^n$.\\
\\
\bf2.2. \rm The association $X\to N(X)$ can be extended to a functor from  the category
 $\mathcal{C}_n$-Sets to the category $\Delta^{op}$-Sets of simplicial sets.\\
\\
\bf2.3. \rm If $X$ is oriented, then the orientation defines a morphism 
$N(X)\to N(\mathcal{C}_n/\mathcal{C}_n^+)$. The geometric realization
$| N(\mathcal{C}_n/\mathcal{C}_n^+)|$ of the simplicial set $ N(\mathcal{C}_n/\mathcal{C}_n^+)$ is
 homeomorphic to the $n$-dimensional sphere $S^n$, and the continuous map $|X|\to S^n$ is a covering, non-ramified over the complement of the $(n-2)$-dimensional skeleton of the $n$-dimensional simplex, embedded into the $S^n$  in the standard way. Denote this complement by $U_n$. Together with the construction from  \bf1.3 \rm it defines a functor from the category of $\mathcal{C}_n^+$-sets to the category of non-ramified coverings over $U_n$, which is equivalent to the category of 
$\pi_1(U_n)$-sets.\\
\\
\bf2.4. \rm The group $\pi_1(U_n)$ can be presented by  generators $x_{ij}$ with $i,j\in\{0,\dots,n\}$ and relations
$$
x_{ij}x_{ji}=1,
$$
%%%
$$
x_{ij}x_{jk}x_{ki}=1.
$$
The generator $x_{ij}$ corresponds geometrically to the loop around the $(n-2)$-dimensional face
$\{0,\dots,\hat{i},\dots,\hat{j},\dots,n\}$ of the simplex $\Delta^n$. In particular, as an abstract group the group  $\pi_1(U_n)$ is a free group with $n$ generators.\\
\\
\bf2.5. \rm The functor associating to a $\mathcal{C}_n^+$-set $X^+$ the covering $|N(X)|\to S^n$ is isomorphic to the functor induced by the projection  $\pi_1(U_n)\to\mathcal{C}_n^+$ sending $x_{ij}$ to $\sigma_i\sigma_j$. In particular, it is fully faithful.\\
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\bf3. The cartographic group and the flag sets of simplicial pseudomanifolds
\end{center}
\bf3.0. \rm A simplicial set is called a \it simplicial pseudomanifold \rm of dimension $n$ if the following conditions hold:\\
\\
a. Every non-degenerate simplex of $X$ is a face of some non-degenerate $n$-dimensional simplex;\\
b. For each non-degenerate $(n-1)$-simplex $\sigma_{n-1}$ there are exactly two pairs 
$(i,\sigma_{n})$ and $(j,\sigma_{n}')$ with $\sigma_{n},\sigma_{n}'\in X_n$,$0\le i,j\le n$ such that $\sigma_{n},\sigma_{n}'$ are non-degenerate and 
$\sigma_{n-1}=\partial_i\sigma_{n}=\partial_j\sigma_{n}'$.\\
c. If $\sigma, \sigma'$ are non-degenerate simplexes of $X$, then there exists a sequence 
$\sigma=\sigma_0,\sigma_1,\dots, \sigma_k=\sigma'$ of non-degenerate $n$-simplexes, such that
$\sigma_i$ and $\sigma_{i+1}$  have a common non-degenerate $(n-1)$-face.\\
\\
\bf3.1. \rm If the geometric realization of a simplicial set is a connected manifold of dimension $n$, then it is a simplicial pseudomanifold. We call such pseudomanifolds \it nonsingular\rm. The converse is false already in dimension 2. In the general case the geometric realization of a pseudomanifold is a ''conic space'', i.e. a neighborhood of each point is homeomorphic to a cone over the conical space of the preceding dimension.\\
\\
\bf3.2. \rm For any transitive oriented $\mathcal{C}_n$-set $X$ the simplicial set $N(X)$ is an $n$-dimensional simplicial pseudomanifold.\\
\\
\bf3.3. \rm Let $X$ be an $n$-dimensional simplicial pseudomanifold. Introduce the \it flag set of $X$\rm: it is the set $F(X)$ of all sequences $\big((\sigma_n,i_n),\dots,(\sigma_1,i_1),\sigma_0\big)$ such that $\sigma_k$ is a non-degenerate $k$-dimensional simplex in $X$ and
$\partial_{i_k}\sigma_k=\sigma_{k-1}$. For each flag  $f=\big((\sigma_n,i_n),\dots,(\sigma_1,i_1),\sigma_0\big)\in F(X)$ and every $j$ with $0\le j\le n$ there exists a unique flag
 $\big((\sigma_n',i_n'),\dots,(\sigma_1',i_1'),\sigma_0'\big)$ such that $(\sigma_k,i_k)=(\sigma_k',i_k')$ and $(\sigma_j,i_j)\ne(\sigma_j',i_j')$. We let $\sigma_j(f)$ denote it. It defines an action of $\mathcal{C}_{n,3}$ (and hence of $\mathcal{C}_n$) on the set $F(X)$.\\
\\
\bf3.4. \rm The flag set of the $n$-dimensional pseudomanifold is oriented as a $\mathcal{C}_n$-set if and only if it is oriented as a pseudomanifold, i.e. if $\mathrm{H}^n(X,\mathbb{Z})\cong\mathbb{Z}$.\\
\\
\bf3.5. \rm If $X$ is a nonsingular orientable pseudomanifold, then $N(F(X))$ is isomorphic as a simplicial set to the barycentric subdivision of $X$. Generally it is not true that $|N(F(X))|$ is homeomorphic to $|X|$, however, if $X$ is orientable there exists a map  $N(F(X))\to|X|$ realizing the ''normalization''  of $|X|$ in the usual sense. Thus, for example, the only possible singularities in the two-dimensional orientable case are the multiple points, and the  map above ''resolves''  them in the usual sense. In particular, for any oriented $X$ of dimension $\le2$ the pseudomanifold $N(F(X))$ is nonsingular. \\
\\
\bf3.6. \rm Let $M$ be a triangulated manifold. Then  in a way similar to 3.3 one can define the flag set $F(M)$, which will be a $\mathcal{C}_n$-set. The orientation of $M$ defines an orientation of $F(M)$, and hence the covering $M\cong|N(F(M))|\to S^n$ is non-ramified away from the $(n-2)$-skeleton of $\Delta^n$. Under this covering\ the preimage of the face of $\Delta^n$ corresponding to the subset $\{0,\dots,k\}$ coincides with the $k$-dimensional skeleton of the initial triangulation of $M$ for $k\le n-1$.\\
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%\tiny Translated from russian by George Shabat, September 2018
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