2012/07/22 by Fukukawa, Yukiko
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Primary 14M15 #Secondary 55N91
paper · doi:10.48550/arxiv.1207.5229
Suppose a compact torus T acts on a closed smooth manifold M. Under certain conditions, Guillemin and Zara associate to (M, T) a labeled graph \mGM where the labels lie in H2(BT). They also define the subring HT^*(\mGM) of \bigoplusv∈ V(\mGM)H^*(BT), where V(\mGM) is the set of vertices of \mGM and we call HT^*(\mGM) the "graph cohomology" ring of \mGM. It is known that the equivariant cohomology ring of M can be described by using combinatorial data of the labeled graph. The main result of this paper is to determine the ring structure of equivariant cohomology ring of a flag manifold of type G2 directly, using combinatorial techniques on the graph \mGM. This gives a new computation of the equivariant cohomology ring of a flag manifold of type G2.