Calabi-Yau hypersurfaces and SU-bordism

TL;DR

This paper constructs specific Calabi-Yau hypersurfaces that generate the SU-bordism ring, providing explicit examples and advancing understanding of their role in algebraic topology.

Contribution

It introduces a new family of Calabi-Yau manifolds whose SU-bordism classes generate the entire SU-bordism ring, with explicit low-dimensional representatives.

Findings

Calabi-Yau hypersurfaces generate the SU-bordism ring.

Explicit representatives for low-dimensional generators provided.

Connects toric geometry with algebraic topology.

Abstract

Batyrev constructed a family of Calabi-Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi-Yau manifolds whose SU-bordism classes generate the special unitary bordism ring $\varOmega^{SU}\otimes\mathbb{Z}[\frac{1}{2}]\cong\mathbb{Z}[\frac{1}{2}][y_{i}\colon i\ge 2]$. We also describe explicit Calabi-Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions.