The third homology of SL₂(ℚ)

2019/06/27 by Kevin Hutchinson-Lhuissier, Hutchinson, Kevin
Mathematics · #11R70 #20J05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.1906.11650

openalex publication_date 2019/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We calculate the third homology of SL₂(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 K₃ of 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.

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