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\begin{document}

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\section*{Volodin space}

[[!redirects Volodin spaces]]

\hypertarget{volodin_space}{}\section*{{Volodin space}}\label{volodin_space}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{volodin_spaces}{Volodin spaces}\dotfill \pageref*{volodin_spaces} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Volodin spaces} are the [[Vietoris complex]] analogues of the nerve of a family of subgroups discussed in the entry, [[higher generation by subgroups]]. They provide a way of building a geometric object that provides a means of comparing the information on the `big group' that is `stored' by subgroups within the family.

They were essentially introduced by [[Volodin]] as part of his approach to higher [[algebraic K-theory]]. We will discuss them via another approach that is explicit in work by [[Suslin]], on the equivalence of the Volodin K-theory with that of [[Quillen]].

\hypertarget{preliminaries}{}\subsection*{{Preliminaries}}\label{preliminaries}

Let $X$ be a non-empty set, and denote by $E(X)$, the [[simplicial set]] having $E(X)_p = X^{p+1}$, so a $p$-simplex is a $p+1$ tuple, $\underline{x}= (x_0,\ldots, x_p)$, each $x_i \in X$, and in which

\begin{displaymath}
d_i(\underline{x}) = (x_0,\ldots, \hat{x_i}, \ldots x_p),$$and
$$s_j(\underline{x}) = (x_0,\ldots, x_j, x_j, \ldots x_p),
\end{displaymath}
so $d_i$ omits $x_i$, whilst $s_j$ repeats $x_j$.

\begin{ulemma}
The simplicial set, $E(X)$, is contractible.

\end{ulemma}
The proof is fairly easy to construct and is `well known'.

The case we are really interested in is when we replace the general set, $X$, by the underlying set of a group, $G$. (As is often done, we will not introduce a special notation for the underlying set of $G$, just writing $G$ for it.) In this case, we have the simplicial set $E(G)$ and the group, $G$, acts freely on $E(G)$ by

\begin{displaymath}
g\cdot(g_0,\ldots , g_p) = (gg_0,\ldots, gg_p).
\end{displaymath}
(Here we have used a left action of $G$, and leave you to check that the evident right action could equally well be used.)

The quotient simplicial set of orbits, will be denoted $G\backslash E(G)$. It is often useful to write $[g_1,\ldots,g_p]$ for the orbit of the $p$-simplex $(1,g_1,g_1g_2,\ldots, g_1g_2\ldots g_p)\in E(G)_p$.

It is `instructive' to calculate the faces and degeneracy maps in this notation. We will only look at $[g_1,g_2]$ in detail. This element has representative $(1,g_1,g_1g_2)$. We thus have:

\begin{itemize}%
\item $d_0(1,g_1,g_1g_2) = (g_1,g_1g_2) \equiv (1,g_2)$, so $d_0[g_1,g_2] = [g_2]$;
\item $d_1(1,g_1,g_1g_2) = (1,g_1g_2)$, so $d_1[g_1,g_2] = [g_1g_2]$;
\item $d_2(1,g_1,g_1g_2) = (1,g_1)$,

\end{itemize}
so $d_2[g_1,g_2] = [g_1]$.

(That looks familiar!)

For the degeneracies,

\begin{itemize}%
\item $s_0(1,g_1,g_1g_2) = (1,1,g_1,g_1g_2)$, so $s_0 [g_1,g_2] =[1,g_1,g_2]$;
\item $s_1(1,g_1,g_1g_2) = (1,g_1,g_1,g_1g_2)$, so $s_1 [g_1,g_2] =  [g_1,1,g_2] ;$

\end{itemize}
and similarly $s_2 [g_1,g_2] =  [g_1,g_2,1]$.

The general formulae are now easy to guess and to prove - so they will be left to you, and then the following should be obvious.

\begin{ulemma}
There is a natural simplicial isomorphism,

\begin{displaymath}
G\backslash E(G)\xrightarrow{\cong}Ner(G[1])= BG.
\end{displaymath}
\end{ulemma}
We thus have that $G\backslash E(G)$ is a [[classifying space]] for $G$.

This construction of $E(G)$ is exactly that of the nerve of the [[action groupoid]] of the action of $G$ on itself by left multiplication.

\hypertarget{volodin_spaces}{}\subsection*{{Volodin spaces}}\label{volodin_spaces}

We put ourselves in the context of a group, $G$, and a family, $\mathcal{H}$, of subgroups of $G$ as in the context of [[higher generation by subgroups]]. We suppose that $\mathcal{H}= \{H_i\mid i\in I\}$ for some indexing set, $I$.

\begin{udefn}
The simplicial set, $V(G,\mathcal{H})$, will be called the \emph{Volodin space} of $(G,\mathcal{H})$.

\end{udefn}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item A. A. [[Suslin]] and M. Wodzicki, \emph{Excision in algebraic K-theory}, The Annals of Mathematics, 136, (1992), 51 -- 122.


\item [[I. Volodin]], \emph{Algebraic K-theory as extraordinary homology theory on the category of associative rings with unity}, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971) No. 4, 859-887)



\end{itemize}


\end{document}