[Paper Review] A Bloch-Wigner complex for SL_2

Kevin Hutchinson|arXiv (Cornell University)|Jul 1, 2011

Advanced Algebra and Geometry3 citations

TL;DR

This paper introduces a new Bloch-Wigner complex for $\mathrm{SL}_2(F)$ that computes the integral homology groups $\mathrm{H}_3(\mathrm{SL}_2(F),\mathbb{Z})$ and $\mathrm{H}_2(\mathrm{SL}_2(F),\mathbb{Z})$ directly, refining the classical Bloch-Wigner complex which computes $K$-theory groups. The complex is constructed as a module over $F^\times$, and upon taking $F^\times$-coinvariants, it recovers the classical complex. The key contribution is a precise description of the homology of $\mathrm{SL}_2(F)$ for arbitrary fields, including finite fields, via explicit generators and relations.

ABSTRACT

We introduce a refinement of the Bloch-Wigner complex of a field F. This is a complex of modules over the multiplicative group of the field. Instead of computing K_2 and indecomposable K_3 - as the classical Bloch-Wigner complex does - it calculates the second and third integral homology of SL_2 of the field. On passing to coinvariants for the action of the multiplicative group we recover the classical Bloch-Wigner complex. The case of finite fields is included throughout the article.

Motivation & Objective

• To construct a refined Bloch-Wigner complex that computes the integral homology of $\mathrm{SL}_2(F)$, rather than $K$-theory groups, for any field $F$.
• To extend the classical Bloch-Wigner theorem beyond quadratically closed fields to arbitrary fields, including finite fields.
• To clarify the structure of $\mathrm{H}_3(\mathrm{SL}_2(F),\mathbb{Z})$ and its relation to $K^\mathrm{ind}_3(F)$, particularly through the action of $F^\times$.
• To provide explicit generators and relations for the homology groups in the case of finite fields $\mathbb{F}_q$, including orders of elements like $\{-1\}$ and $C_{\mathbb{F}_q}$.

Proposed method

• Construct a complex of $\mathbb{Z}[F^\times]$-modules whose homology computes $\mathrm{H}_3(\mathrm{SL}_2(F),\mathbb{Z})$ and $\mathrm{H}_2(\mathrm{SL}_2(F),\mathbb{Z})$ directly.
• Define a map $\lambda: \mathcal{P}(F) \to \mathrm{S}^2_{\mathbb{Z}}(F^\times)$, sending $[x] \mapsto (1-x) \otimes x$, and use its kernel to define the Bloch group $\mathcal{B}(F)$.
• Use the action of $F^\times$ on homology, factoring through the Grothendieck-Witt ring $\mathrm{GW}(F)$, to understand the module structure of $\mathrm{H}_2(\mathrm{SL}_2(F),\mathbb{Z})$.
• Analyze the homology of finite subgroups (e.g., quaternion and cyclic groups) in $\mathrm{SL}_2(\mathbb{F}_q)$ to deduce generators and orders in $\mathcal{P}(\mathbb{F}_q)$ and $\mathcal{B}(\mathbb{F}_q)$.
• Apply the universal discrete dilogarithm construction to show that any map $L: \mathbb{F}_q^\times \to A$ satisfying the 5-term functional equation factors through $\mathcal{L}_\theta: \mathcal{P}(\mathbb{F}_q) \to \mathbb{Z}/(q+1)$.
• Use explicit cycle representatives in $\mathrm{H}_3(Q,\mathbb{Z})$ for quaternion subgroups $Q$ to compute images in $\mathcal{B}(\mathbb{F}_q)$, proving $\{-1\}$ has order 2 when $q \equiv 3 \pmod{4}$.

Experimental results

Research questions

• RQ1 How can the Bloch-Wigner complex be refined to compute the integral homology of $\mathrm{SL}_2(F)$ directly, rather than $K$-theory groups?
• RQ2 What is the structure of $\mathrm{H}_3(\mathrm{SL}_2(F),\mathbb{Z})$ as a module over $\mathbb{Z}[F^\times/(F^\times)^2]$ for arbitrary fields $F$?
• RQ3 What are the orders of specific elements like $\{-1\}$ and $C_{\mathbb{F}_q}$ in $\mathcal{B}(\mathbb{F}_q)$ for finite fields $\mathbb{F}_q$?
• RQ4 How does the universal discrete dilogarithm on $\mathbb{F}_q$ relate to the structure of $\mathcal{P}(\mathbb{F}_q)$ and $\mathcal{B}(\mathbb{F}_q)$?
• RQ5 What is the image of the homology of a quaternion subgroup $Q \subset \mathrm{SL}_2(\mathbb{F}_q)$ in $\mathcal{B}(\mathbb{F}_q)$, and how does it determine the order of $\{-1\}$?

Key findings

• The complex introduced computes $\mathrm{H}_3(\mathrm{SL}_2(F),\mathbb{Z})$ and $\mathrm{H}_2(\mathrm{SL}_2(F),\mathbb{Z})$ directly as $\mathbb{Z}[F^\times]$-modules, refining the classical Bloch-Wigner complex.
• For finite fields $\mathbb{F}_q$, the group $\mathcal{P}(\mathbb{F}_q)$ is isomorphic to $\mathbb{Z}/(q+1)$ when $q \equiv 3 \pmod{8}$, with $\{-1\}$ having order 2 in $\mathcal{B}(\mathbb{F}_q)$.
• The element $C_{\mathbb{F}_q} \in \mathcal{B}(\mathbb{F}_q)$ has order $\gcd(6, (q+1)/2)$, with explicit computation showing it maps to $4C_{\mathbb{F}_q}$ under the $\mathrm{H}_3(G,\mathbb{Z}) \to \mathcal{B}(\mathbb{F}_q)$ map for a 3-torsion subgroup $G$.
• The inverse of the isomorphism $\mathcal{R}_\theta: \mathbb{Z}/(q+1) \to \mathcal{P}(\mathbb{F}_q)$ gives a universal discrete dilogarithm, factoring all maps $L: \mathbb{F}_q^\times \to A$ satisfying the 5-term relation.
• The image of $\mathrm{H}_3(Q,\mathbb{Z}) \to \mathcal{B}(\mathbb{F}_q)$ for a quaternion subgroup $Q$ of order 8 is a cyclic group of order 2, proving $\{-1\}$ has order 2 in $\mathcal{B}(\mathbb{F}_q)$ when $q \equiv 3 \pmod{4}$.
• When $q \equiv 7 \pmod{8}$, $\left[-1\right]$ generates the 2-Sylow subgroup of $\mathcal{B}(\mathbb{F}_q)$, and $\left[-1\right]$ has order 4.