The third homology of the special linear group of a field

We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory.

💡 Research Summary

The paper investigates the third integral homology group H₃ of the special linear groups SLₙ(F) over an infinite field F, establishing a precise homology stability range. Classical homology stability results are well‑understood for H₁ and H₂, but H₃ has remained elusive because of its intricate connections with algebraic K‑theory. The authors first set up the necessary algebraic and homological machinery: Steinberg modules, standard complexes, and a refined version of the Bar‑Cohomology spectral sequence adapted to linear groups. By analyzing the relative homology of the inclusion iₙ: SLₙ(F) → SLₙ₊₁(F), they obtain vanishing results that hold already for n ≥ 3. Consequently they prove the main stability theorem:

 For every infinite field F and every n ≥ 3,   H₃(SLₙ(F), ℤ) ≅ H₃(SL₃(F), ℤ).

The proof proceeds by showing that any potential “unstable” element in H₃ must arise from Milnor K‑theory, specifically from K₃⁽ᴹ⁾(F). A careful examination of the spectral sequence shows that such elements disappear once the rank reaches three, yielding the claimed isomorphism.

The case n = 2 is treated separately. The map induced by the inclusion i₂: SL₂(F) → SL₃(F) is not surjective; its cokernel is identified with the square of Milnor’s third K‑group. More precisely,

 Coker