[Paper Review] K-theory of torus manifolds

V. Uma|ArXiv.org|Jul 31, 2006

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper provides a presentation of the topological K-ring of a class of torus manifolds—specifically, those whose orbit space is a homology polytope with shellable nerve—via generators and relations. It generalizes earlier results on quasi-toric manifolds by showing that the K-ring is isomorphic to a quotient of a polynomial ring in variables corresponding to characteristic submanifolds, with relations arising from empty intersections and equivariant Chern class conditions.

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).

Motivation & Objective

• To extend previous results on the K-theory of quasi-toric manifolds to a broader class of torus manifolds.
• To describe the topological K-ring of torus manifolds whose orbit space is a homology polytope with shellable nerve.
• To provide an explicit presentation of the K-ring using generators corresponding to complex line bundles over characteristic submanifolds and relations from intersection and equivariant topology.
• To establish a ring isomorphism between a quotient of a polynomial ring and the K-ring of such manifolds.

Proposed method

• Use of complex line bundles ${\cal L}_j$ associated to characteristic submanifolds $M_j$, whose first Chern classes correspond to the fundamental classes of $M_j$.
• Construction of a polynomial ring $\mathbb{Z}[v_{Q_1}, \dots, v_{Q_m}]$ with variables indexed by facets of the orbit space $Q$, where $v_{Q_j}$ maps to $[\mathcal{L}_j] - 1$ in $K^0(M)$.
• Definition of an ideal $I$ generated by two types of relations: (i) products $v_{Q_{j_1}} \cdots v_{Q_{j_k}}$ when the corresponding facets have empty intersection, and (ii) equivariant Chern class relations involving $\langle t, a_j \rangle$ for $t \in H^2(BT)$.
• Leveraging a cellular decomposition of $M$ with even-dimensional cells induced by a shelling of the nerve $K$, ensuring $K^0(M)$ is free abelian of finite rank.
• Application of the $\gamma^k$ operation in K-theory to show that certain products of $[\mathcal{L}_j] - 1$ vanish when the corresponding submanifolds have empty intersection.
• Proof of isomorphism via showing surjectivity of the map $\psi: \mathbb{Z}[v_{Q_1}, \dots, v_{Q_m}]/I \to K^0(M)$, and injectivity via rank comparison using the shellable nerve and cell decomposition.

Experimental results

Research questions

• RQ1 How can the topological K-ring of torus manifolds be described in terms of generators and relations when the orbit space is a homology polytope with shellable nerve?
• RQ2 To what extent do the K-theory relations generalize those known for quasi-toric manifolds?
• RQ3 What role does the shellability of the nerve of the orbit space play in constructing a cellular decomposition that simplifies K-theory computation?
• RQ4 How do equivariant Chern class conditions and intersection data of characteristic submanifolds constrain the K-ring structure?

Key findings

• The topological K-ring $K^0(M)$ of a torus manifold $M$ with orbit space $Q$ a homology polytope and shellable nerve is isomorphic to the quotient ring $\mathbb{Z}[v_{Q_1}, \dots, v_{Q_m}]/I$, where $I$ is generated by intersection and equivariant Chern class relations.
• The map $\psi: \mathbb{Z}[v_{Q_1}, \dots, v_{Q_m}]/I \to K^0(M)$ sending $v_{Q_j}$ to $[\mathcal{L}_j] - 1$ is a ring isomorphism, providing an explicit presentation of $K^0(M)$.
• The K-ring is freely generated as a $\mathbb{Z}$-module by $d$ monomials $v_{T_i}$, where $d$ is the number of even-dimensional cells in a perfect cellular decomposition of $M$ induced by the shelling of the nerve.
• The cellular decomposition of $M$ induced by the shelling of the nerve $K$ has cells only in even dimensions, which implies $K^0(M)$ is a free abelian group of finite rank.
• The relations of type (i) arise from the vanishing of sections of direct sums of line bundles over submanifolds with empty intersection, enforced via the $\gamma^k$ operation in K-theory.
• The relations of type (ii) encode the triviality of equivariant line bundles $\mathcal{L}_t = \prod_j \mathcal{L}_j^{\langle t, a_j \rangle}$ for all $t \in H^2(BT)$, reflecting the non-singularity of the characteristic homomorphism $\Lambda$.