Combinatorial Algebraic Topology & Applications IV

Cohomology for GKM manifolds and orbit spaces

Not scheduled

40m

Aula Dini (Palazzo del Castelletto)

Contributed Talks

Speaker

Grigory Solomadin (PU Marburg)

Description

In this talk, for any GKM4 manifold M^2n (with (S1)^k torus action) that is either Hamiltonian or satisfies n-k=1 we prove generators-and-relations description of the respective equivariant cohomology ring. The proof follows in two steps. We prove extension of the respective GKM graph to a torus graph (i.e. as in k=n case) using sheaves on graphs and Kuroki's obstruction. An extension of GKM graphs induces epimorphism in cohomology, by analysing Atiyah-Bredon sequence for GKM graphs. If time permits, we prove that: homogeneous GKM3 manifolds are symmetric whose orbit spaces have vanishing fourth homology (with Z or Z/2 coefficients), GKM4 homogeneous manifolds are torus manifolds. Based on arXiv joint preprints 2509.00392, 2602.07734.