# The third homology of $\mathrm{SL}_2(\mathbb{Q})$

## TL;DR

This paper computes the third homology group of SL_2 over the rationals with half-integral coefficients, revealing a structure linked to prime-specific involutions and the indecomposable K_3 of Q.

## Contribution

It provides an explicit calculation of the third homology of SL_2(Q) and describes the eigenspace decomposition related to prime operators, connecting algebraic K-theory and homology.

## Key findings

- The third homology group is described as a direct sum over primes.
- Each prime p induces an involution with eigenspaces of order related to p+1.
- The kernel of the homology map relates to the indecomposable K_3 of Q.

## Abstract

We calculate the third homology of $\mathrm{SL}_2(\mathbb{Q})$ with half-integral coefficients. Corresponding to each prime $p$ there is an operator on this group with square the identity. The kernel of the (split surjective) homomorphism to the indecomposable $K_3$ of $\mathbb{Q}$ is a the direct sum over all primes of the $(-1)$-eigenspaces of these operators. The $(-1)$-eigenspace of the operator corresponding to the prime $p$ is cyclic of order the odd part of $p+1$.