Cohomology of torus manifold bundles

TL;DR

This paper provides algebraic descriptions of the cohomology and K-theory rings of torus manifold bundles, extending previous results to more general toric varieties and their associated bundles.

Contribution

It offers new presentations of cohomology and K-theory rings for torus manifold bundles, generalizing earlier work to include all smooth complete toric varieties.

Findings

Presented cohomology ring of E(X) over B

Described topological K-ring of E(X) over B

Extended results to general smooth complete toric varieties

Abstract

Let $X$ be a torus manifold with locally standard action of a compact torus $T$ of half the dimension and orbit space a homology polytope. Smooth complete complex toric varieties and quasi-toric manifolds are examples of torus manifolds. Consider a principal bundle with total space $E$ and base $B$ with fibre and structure group $T$. Let $E(X)$ denote the total space of the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as an algebra over the singular cohomology ring of $B$ and a presentation of the topological $K$-ring of $E(X)$ as an algebra over the topological $K$-ring of $B$. These are relative versions of the results of M. Masuda and T. Panov [13] on the cohomology ring of a torus manifold and P. Sankaran [14] on the topological $K$-ring of a torus manifold. Further, they extend the results due to P. Sankaran and V. Uma [15] on the cohomology ring and topological $K$-ring of toric bundles with fibre a smooth projective toric variety, to a toric bundle with fibre any smooth complete toric variety.