2015/08/17 by Joseph Malkoun, Malkoun, Joseph
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math-ph #math.MG #math.MP #msc:14M15 #msc:14N20 #msc:22E70 #msc:51F99
paper · pdf · doi:10.48550/arxiv.1508.04076
16 pages
arxiv created 2021/05/19 · arxiv updated 2021/05/20
If G is a compact Lie group, T a maximal torus in G (with Lie algebras \mathfrakg and \mathfrakt respectively) and W the corresponding Weyl group, then the Berry-Robbins problem for G, as formulated by Sir Michael Atiyah and Roger Bielawski, asks whether there exists a continuous SU(2) × W equivariant map from the space of regular Cartan triples (an open subset of \mathfrakt ⊗ ℝ3) to G/T, where SU(2) acts via a regular Lie group homomorphism SU(2) → G. This was settled positively by Atiyah and Bielawski, and their maps are even smooth, but they are not explicit. For G=U(n), there exists another construction due to Sir Michael Atiyah and developed further with Paul Sutcliffe, which is explicit, but relies on a linear independence conjecture. The author had previously found a similar type of construction for G=Sp(m), also relying on a linear independence conjecture. In this paper, similar constructions are done for SO(2m+1) and SO(2m), thus exhausting the list of classical groups.