K-theory of torus manifolds

TL;DR

This paper describes the topological K-theory of certain torus manifolds, generalizing previous results and providing explicit generators and relations for their K-ring, especially for those with homology polytope orbit spaces.

Contribution

It offers a new description of the topological K-ring for a class of torus manifolds, extending earlier work to include manifolds with homology polytope orbit spaces.

Findings

Explicit generators and relations for the K-ring of these torus manifolds.

Generalization of previous results to a broader class including quasi-toric manifolds.

Connection between the combinatorial properties of the orbit space and the K-theory.

Abstract

The {\it torus manifolds} have been defined and studied by M. Masuda and T. Panov (arXiv:math.AT/0306100) who in particular describe its cohomology ring structure. In this note we shall describe the topological $K$-ring of a class of torus manifolds (those for which the orbit space under the action of the compact torus is a {\it homology polytope} whose {\it nerve} is a {shellable} simplicial complex) in terms of generators and relations. Since these torus manifolds include the class of quasi-toric manifolds this is a generalisation of earlier results due to the author and P. Sankaran (arXiv: math.AG/0504107).