A Refined scissors congruence group and the third homology of \textrmSL₂

2023/07/17 by Behrooz Mirzaii, Mirzaii, Behrooz, Elvis Torres Pérez +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.2307.08872

openalex publication_date 2023/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There is a natural connection between the third homology of \textrmSL₂(A) and the refined Bloch group RB(A) of a commutative ring A. In this article we investigate this connection and as the main result we show that if A is a universal \textrmGE₂-domain such that -1 ∈ A^{× 2}, then we have the exact sequence H₃(\textrmSM₂(A),ℤ) → H₃(\textrmSL₂(A),ℤ) → RB(A) → 0, where \textrmSM₂(A) is the group of monomial matrices in \textrmSL₂(A). Moreover we show that RP₁(A), the refined scissors congruence group of A, naturally is isomorph with the relative homology group H₃(\textrmSL₂(A), \textrmSM₂(A),ℤ).

Related

• The homology of SL₂ of discrete valuation rings
• Bloch Groups of Rings
• The third homology of SL₂ of fields with discrete valuation
• The third homology of SL₂ of real quadratically closed fields
• A refined Bloch-Wigner exact sequence in characteristic 2