2012/04/30 by Aravind Asok, Jean Fasel · 1 citation
Mathematics · #Affine transformation #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Computation #Dimension (graph theory) #Homotopy and Cohomology in Algebraic Topology #Rank (graph theory) #Sheaf #Unimodular matrix #Vector bundle #math.AC #math.AG #math.AT #math.KT #msc:13C10 #msc:14F42 #msc:19A13 #msc:19D45 #msc:55S35
paper · pdf · doi:10.1112/jtopol/jtt046
35 pages; final version (before page proofs) to appear J. Top. Significantly reorganized and incorporates some material from http://arxiv.org/abs/1204.0770 (which will also soon be replaced)
arxiv created 2013/11/21 · openalex publication_date 2014/02/03 · openalex created_date 2016/06/24 · arxiv updated 2017/05/17 · openalex updated_date 2026/08/06
We determine the first non-stable A 1 -homotopy sheaf of SL n . Using techniques of obstruction theory involving the A 1 -Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank at least d - 1 on split smooth affine quadrics of dimension 2 d - 1 . These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations of A 1 -homotopy sheaves with real and complex realization.