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

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.

22 pages