Third homology of SL_2 and the indecomposable K_3

TL;DR

This paper investigates the relationship between the third homology of SL_2 over a field and the indecomposable part of K_3, establishing conditions under which a canonical map is an isomorphism.

Contribution

It improves previous results by providing necessary and sufficient conditions for the canonical map to be bijective, linking it to injectivity of certain homology maps.

Findings

The map  is bijective if and only if specific homology maps are injective.

Established equivalence between bijectivity of  and injectivity of maps H_3(GL_2) to H_3(GL_3) and H_3(SL_2) to H_3(GL_2).

Extended understanding of the relationship between algebraic K-theory and homology of special linear groups.

Abstract

It is known that, for an infinite field F, the indecomposable part of K_3(F) and the third homology of SL_2(F) are closely related. In fact, there is a canonical map \alpha: H_3(SL_2(F),Z)_F* --> K_3(F)^ind. Suslin has raised the question that, is \alpha an isomorphism? Recently Hutchinson and Tao have shown that this map is surjective. They also gave some arguments about its injectivity. In this article, we improve their arguments and show that \alpha is bijective if and only if the natural maps H_3(GL_2(F), Z)--> H_3(GL_3(F), Z) and H_3(SL_2(F), Z)_F* --> H_3(GL_2(F), Z) are injective.