nLab Volodin space

Idea

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.

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

$$d_i(\underline{x}) = (x_0,\ldots, \hat{x_i}, \ldots x_p),$$

so $d_i$ omits $x_i$, whilst $s_j$ repeats $x_j$.

Lemma

The simplicial set, $E(X)$, is contractible.

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

$$g\cdot(g_0,\ldots , g_p) = (gg_0,\ldots, gg_p).$$

(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:

• $d_0(1,g_1,g_1g_2) = (g_1,g_1g_2) \equiv (1,g_2)$, so $d_0[g_1,g_2] = [g_2]$;
• $d_1(1,g_1,g_1g_2) = (1,g_1g_2)$, so $d_1[g_1,g_2] = [g_1g_2]$;
• $d_2(1,g_1,g_1g_2) = (1,g_1)$,

so $d_2[g_1,g_2] = [g_1]$.

(That looks familiar!)

For the degeneracies,

• $s_0(1,g_1,g_1g_2) = (1,1,g_1,g_1g_2)$, so $s_0 [g_1,g_2] =[1,g_1,g_2]$;
• $s_1(1,g_1,g_1g_2) = (1,g_1,g_1,g_1g_2)$, so $s_1 [g_1,g_2] = [g_1,1,g_2] ;$

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.

Lemma

There is a natural simplicial isomorphism,

$$G\backslash E(G)\xrightarrow{\cong}Ner(G[1])= BG.$$

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.

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, $I$.

(cf. Suslin-Wodzicki, (ref. below) p. 65.) We denote by $V(G,\mathcal{H})$, or $V(\mathfrak{H})$, the simplicial subset of $E(G)$ formed by simplices, $(g_0,\ldots,g_p)$, that satisfy the condition that there is some $i\in I$ such that, for all $0\leq j,k\leq p$, $g_j g_k^{-1}\in H_i$.

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

References

• A. A. Suslin and M. Wodzicki, Excision in algebraic K-theory, The Annals of Mathematics, 136, (1992), 51 – 122.

• I. Volodin, 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)