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This paper aims to determine the ring structure of the torus equivariant cohomology of odd-dimensional complex quadrics by computing the graph equivariant cohomology of their corresponding GKM graphs. We show that its graph equivariant cohomology is generated by three types of subgraphs in the GKM graph, which are subject to four different types of relations. Furthermore, we consider the relationship between the two graph equivariant cohomology rings induced by odd- and even-dimensional complex quadrics., Comment: 27 pages, 10 figures

The aim of this paper is to describe the torus equivariant $K$-ring of even-dimensional complex quadrics by studying the graph equivariant $K$-theory of their corresponding GKM graphs. This involves providing a presentation for its graph equivariant $K$- ring in terms of generators and relations. This parallels the description of the equivariant cohomology ring of even-dimensional complex quadrics due to Kuroki., Comment: 17 pages, comments are welcome

The aim of this paper is to describe the equivariant and ordinary Grothendieck ring and the equivariant and ordinary topological $K$-ring of flag Bott manifolds of general Lie type. This will generalize the results on the equivariant and ordinary cohomology of flag Bott manifolds of general Lie type due to Kaji Kuroki Lee and Suh., Comment: 26 pages

The aim of this paper is to describe the topological $K$-ring, in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K-ring and hence the Grothendieck ring of algebraic vector bundles over flag Bott Samelson varieties., Comment: 25 pages, To appear in Forum Mathematicum