2013/10/31 by Ilya Grigoriev · 9 citations
Mathematics · #Algebraic Geometry and Number Theory #Algebraic number #Algebraic topology #Characteristic class #Chern class #Cohomology #Combinatorics #Geometry #Geometry and complex manifolds #Homotopy #Homotopy and Cohomology in Algebraic Topology #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Product (mathematics) #Pure mathematics #Surface (topology) #Topology (electrical circuits) #Vector bundle #math.AT #msc:55R40 #msc:55T10 #msc:57R22
paper · pdf · doi:10.2140/gt.2017.21.2015
published in Geometry & Topology 21(4), 2015-2048 (Mathematical Sciences Publishers) · Accepted version. Numerous minor imporvements. 35 pages, 1 figure
arxiv created 2016/07/18 · openalex publication_date 2017/05/19 · arxiv updated 2017/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study relations among characteristic classes of smooth manifold bundles with highly connected fibers. For bundles with fiber the connected sum of [math] copies of a product of spheres [math] , where [math] is odd, we find numerous algebraic relations among so-called “generalized Miller–Morita–Mumford classes”. For all [math] , we show that these infinitely many classes are algebraically generated by a finite subset. ¶ Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with [math] , according to recent homological stability results. In the case of surface bundles ( [math] ), our approach recovers some previously known results about the structure of the classical “tautological ring”, as introduced by Mumford, using only the tools of algebraic topology.