SL(n,Z) cannot act on small spheres

2006/04/11 by Bruno Zimmermann, Zimmermann, Bruno P.
Mathematics · #57S25 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0604239

openalex publication_date 2006/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The group SL(n,Z) admits a smooth faithful action on the (n-1)-sphere S^(n-1), induced from its linear action on euclidean space Rⁿ. We show that, if m < n-1 and n > 2, any smooth action of SL(n,Z) on a mod 2 homology m-sphere, and in particular on the m-sphere S^m, is trivial.

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