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Extensions of realizable Hamiltonian and complexity one GKM$_4$ graphs

Oliver Goertsches, Grigory Solomadin

Abstract

We prove that the GKM graphs of GKM$_4$ manifolds that are either Hamiltonian or of complexity one extend to torus graphs. The arguments are based on a reformulation of the extension problem in terms of a natural representation of the fundamental group of the GKM graph, using a coordinate-free version of the axial function group of Kuroki, as well as on covers of GKM graphs and acyclicity results for orbit spaces of GKM manifolds.

Extensions of realizable Hamiltonian and complexity one GKM$_4$ graphs

Abstract

We prove that the GKM graphs of GKM manifolds that are either Hamiltonian or of complexity one extend to torus graphs. The arguments are based on a reformulation of the extension problem in terms of a natural representation of the fundamental group of the GKM graph, using a coordinate-free version of the axial function group of Kuroki, as well as on covers of GKM graphs and acyclicity results for orbit spaces of GKM manifolds.

Paper Structure

This paper contains 11 sections, 21 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Graphs, connections, invariant functions
  3. Faces and orbit spaces of GKM manifolds
  4. Definitions and basic facts
  5. Proof of Theorem \ref{['thm:mainthmham']}
  6. Extension for unsigned GKM graphs
  7. Group of label functions
  8. Group of axial functions
  9. Extension criterion for unsigned GKM graphs
  10. Proof of Theorem \ref{['thm:2']}
  11. Extensions of realizable complexity one GKM$_4$ graphs

Key Result

Theorem 2

Let $\Gamma$ be any unsigned (signed, respectively) $\mathop{\mathrm{GKM}}\nolimits_{3}$ graph such that the conjugated $2$-faces in $\Gamma$ generate the fundamental group $\pi_{1}(\Gamma)$. Suppose that the monodromy along any $2$-face of $\Gamma$ is trivial on the transversal edges of this $2$-fa

Figures (4)

  • Figure 1: The same GKM graph with two different acyclic edge orientations providing different fourth Betti numbers (i.e. numbers of vertices that are local maxima with respect to a given orientation)
  • Figure 2: The induction step replaces the edge path $u,v,w$ with the (dashed) edge path $u,z,w$ in a cycle $g$ with a maximal vertex $v$ of multiplicity $2$. The vertices $u,v,w,z$ belong to the same $2$-face $F$. The resulting cycle has the maximal vertex $v$ with multiplicity $1$
  • Figure 3: GKM graph of the flag manifold $Fl_3={\mathrm{SU}}(3)/T^2$. It has a $2$-face $F$ with $b_{4}(F)=2$, depicted by dotted lines
  • Figure 4: The signed GKM graph (with the connection acting by nontrivial permutation along every edge) of the canonical $T^{2}$-action on $G_{2}/SU_{3}=S^6$ does not extend to a torus graph ku-19; the respective unsigned GKM graph (with the different connection sending every edge to itself) extends to an unsigned torus graph, corresponding to the standard action of $T^3$ on $S^6\subset \mathbb C^3\oplus \mathbb R$

Theorems & Definitions (72)

  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Definition 2.2: gu-za-01
  • Definition 2.3: gu-za-01, go-ko-zo-22
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 62 more