2021/05/05 by Jason DeVito, DeVito, Jason, David González-Álvaro +1
Mathematics · Medicine · #53C21 19L64 #57R22 #57T15 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.2105.02149
openalex publication_date 2021/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In previous work, the second author and others have found conditions on a homogeneous space G/H which imply that, up to stabilization, all vector bundles over G/H admit Riemannian metrics of non-negative sectional curvature. One important ingredient of their approach is Segal's result that the set of vector bundles of the form G×H V for a representation V of H contains inverses within the class. We show that this approach cannot work for biquotients G/ / H, where we consider vector bundles of the form G×H V. We call such vector bundles biquotient bundles. Specifically, we show that in each dimension n≥ 4 except n=5, there is a simply connected biquotient of dimension n with a biquotient bundle which does not contain an inverse within the class of biquotient bundles. In addition, we show that for n≥ 6 except n=7, there are infinitely many homotopy types of biquotients with the property that no non-trivial biquotient bundle has an inverse. Lastly, we show that every biquotient bundle over every simply connected biquotient Mn = G/ / H with G simply connected and with n∈ \2,3,5\ has an inverse in the class of biquotient bundles.