Seminar “On the problem of maximal extension for GKM-graphs”

27 November at 18:00 MSK

Topic: On the problem of maximal extension for GKM-graphs.

In 1998, Goresky, Kottwitz, MacPherson introduced a class of $(S^1)^k$-actions on  smooth manifolds  with degenerate odd-dimensional cohomologies. They calculated the corresponding ring of equivariant cohomologies in terms of so-called GKM-graphs. Roughly speaking, a GKM-graph of type $(n,k)$ is an $n$-valent graph with labels from the lattice $\Z^k$ on its edges. In 2001 Guillemin and Zara introduced GKM-graphs as independent combinatorial objects and studied the corresponding ring. GKM-bundles were introduced by Guillemin, Sabatini, Zara in 2012. This subclass of GKM-graphs consists of combinatorial analogues for torus-equivariant bundles.

In 2019, S. Kuroki defined a free Abelian group for each GKM-graph, which is called the group of axial functions. He showed that this group is generated by labels of edges of each maximal extension of a given GKM-graph. In particular, for given GKM-manifold the rank of the corresponding group of axial functions gives an upper bound on the dimensions of tori, which appear in GKM-extensions of a given GKM-action. An explicit calculation of the group of axial functions is an open problem. It was conjectured recently that the group of axial functions has rank $n$ for each $4$-linearly independent $n$-valent GKM-graph.

In the talk (based on joint work with S. Kuroki) for each $n$-valent GKM-bundle (which satisfies some reasonable conditions) we present a criterion for rank $n$ of corresponding group of axial functions. In addition, we present a new example of an $(n+1)$-linearly independent GKM-graph of type $(n+1+r,n+1)$ with the group of axial functions of rank $n+1$ for each integer $n>1,r>0$, which disproves the hypothesis stated above.

No knowledge of the topic is needed, all definitions will be given during the talk.

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