Link Bundles of Compact Toric Varieties of Real Dimension 8

TL;DR

This paper investigates the Betti numbers of links of isolated singularities in 8-dimensional compact toric varieties, constructing intersection spaces and exploring their topological invariants and fiber bundle structures.

Contribution

It determines Betti numbers of links in 8-dimensional toric varieties, constructs intersection spaces, and extends the analysis to link bundles with fiber $S^1$ over toric bases.

Findings

Betti numbers of links contain only one non-combinatorial invariant parameter.

Betti numbers of links and base toric varieties share invariant parameters.

Provides algebraic description of the invariant parameter via the Euler class.

Abstract

The main goal of this work is to determine the Betti numbers of the links of isolated singularities in a compact toric variety of real dimension 8, using the CW-structure of the links. Additionally, we construct the intersection spaces associated with these links. Using the duality of the Betti numbers of intersection spaces, we conclude that, similar to the case of toric varieties of real dimension 6, the Betti numbers of the links contain only one non-combinatorial invariant parameter. In the final section, we extend our discussion to arbitrary compact toric varieties and their associated link bundles. We show that for any given link $\mathcal{L}$, there exists a fiber bundle $\pi: \mathcal{L} \to X$ with fiber $S^{1}$, where the base space $X$ is a compact toric variety. Furthermore, using the Chern-Spanier exact sequences for sphere bundles, we show that for the fiber bundle $\pi:\mathcal{L}\longrightarrow X$, where $\dim_{\mathbb{R}}(X)=6$, the non-combinatorial invariant parameters appearing in the Betti numbers of $\mathcal{L}$ and $X$ are equal. In addition, we provide an algebraic description of the non-combinatorial invariant parameter of $X$ in terms of the cohomological Euler class of the fiber bundle.