Equivariant bordism classification of five-dimensional $(\mathbb{Z}_2)^3$-manifolds with isolated fixed points

TL;DR

This work classifies the 5-dimensional equivariant unoriented bordism group $Z_5((Z_2)^3)$ for effective $(Z_2)^3$-actions with isolated fixed points. It recasts the problem through the Conner–Floyd representation algebra and the Stong injectivity theorem, encoding fixed-point data via labeled graphs and combinatorial criteria à la Li–Lu–Shen. The authors prove that $\dim_{{\mathbb F}_2} Z_5((Z_2)^3)=77$ and provide an explicit basis given by projectivizations of real vector bundles, yielding a complete description of these equivariant bordism classes. They then construct geometric representatives for all classes using Milnor hypersurfaces and small covers, demonstrating that every class contains a representative that is the projectivization of a real vector bundle over a disjoint union of real projective spaces, thereby achieving a full classification in this case.

Abstract

Denote by $\mathcal{Z}_5((\mathbb{Z}_2)^3)$ the group, which is also a vector space over $\mathbb{Z}_2$, generated by equivariant unoriented bordism classes of all five-dimensional closed smooth manifolds with effective smooth $(\mathbb{Z}_2)^3$-actions fixing isolated points. We show that $\dim_{\mathbb{Z}_2} \mathcal{Z}_5((\mathbb{Z}_2)^3) = 77$ and determine a basis of $\mathcal{Z}_5((\mathbb{Z}_2)^3)$, each of which is explicitly chosen as the projectivization of a real vector bundle. Thus this gives a complete classification up to equivariant unoriented bordism of all five-dimensional closed smooth manifolds with effective smooth $(\mathbb{Z}_2)^3$-actions with isolated fixed points.

Theorems & Definitions (38)

• Theorem 1.1: Theorem \ref{['thm_dimension']}
• Theorem 1.2: Theorem \ref{['thm_representative']}
• Proposition 2.1: ConnerFloyd
• Remark 2.2
• Corollary 2.3
• Theorem 2.4: Stong
• Remark 3.1
• Example 3.2
• Proposition 3.3: LiLuShen_2025arxivLu2008
• Theorem 3.4: LiLuShen_2025arxiv