On GKM fiber bundles and realizability with full flag fibers
Oliver Goertsches, Panagiotis Konstantis, Nikolas Wardenski, Leopold Zoller
TL;DR
The paper analyzes when equivariant fiber bundles of GKM manifolds induce fibrations or fiber bundles of GKM graphs and investigates realizability when the fiber is the GKM graph of a full flag manifold $G/T$. Focusing on base graphs that are 2-regular and fibers isomorphic to $G/T$, it reduces the combinatorial data to a twist automorphism $\Phi=\Phi_1\circ\Phi_2$ with Type 1 (Weyl-group) and Type 2 (outer automorphism) components; realizability occurs precisely when the Type 2 component is trivial. It provides a precise obstruction: if $\Phi_2\neq Id$, the induced cohomology map is not surjective, obstructing geometric realizability, and it offers concrete realizable and non-realizable examples. The work also develops a detailed automorphism theory for the GKM graph of $G/T$ and furnishes explicit constructions (admissible tuples) to produce new realizable GKM fiber bundles, enriching the bridge between algebraic and geometric aspects of GKM theory.
Abstract
We investigate under which conditions an equivariant fiber bundle whose base, total space and fiber are GKM manifolds induces a fibration or fiber bundle of the corresponding GKM graphs. In particular, we give several counterexamples. Concerning the converse direction, i.e., the realization problem for fiber bundles of GKM graphs, we restrict to the setting of fiberwise signed GKM fiber bundles over $n$-gons whose fiber is the GKM graph of a full flag manifold. While it was known that any such bundle is realizable for a $\mathbb{CP}^1$-fiber, we observe that new phenomena occur in higher dimensions where realizability depends on the twist automorphism of the GKM fiber bundle. We classify possible twist isomorphsims and show that realizability can be decided in terms of our classification.