Non-finitely generated $(\mathbb{Z}_2)^k$-equivariant bordism ring

TL;DR

This paper proves that the equivariant bordism ring of certain smooth manifolds with $(bZ_2)^k$-actions is not finitely generated and constructs infinitely many indecomposable elements, resolving longstanding open problems.

Contribution

It definitively shows the non-finite generation of the equivariant bordism ring and constructs an infinite family of indecomposable elements for all $k \,\geqslant\, 3$ in the fully effective case.

Findings

The equivariant bordism ring is not finitely generated for all $k\geq 3$.

An explicit infinite family of indecomposable elements is constructed.

The results resolve two open problems posed in 1998.

Abstract

In 1998, Mukherjee and Sankaran posed two problems concerning the algebraic structure of the equivariant bordism ring of smooth closed $(\mathbb{Z}_2)^k$-manifolds with only isolated fixed points. One is the property of being finitely generated as a $\mathbb{Z}_2$-algebra, and the other is the existence of indecomposable elements. This paper definitively resolves both problems for the fully effective case. Specifically, let $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ denote the equivariant bordism ring of smooth closed manifolds equipped with fully effective smooth $(\mathbb{Z}_2)^k$-actions having only isolated fixed points. We prove that $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ is not finitely generated as a $\mathbb{Z}_2$-algebra for all $k\geqslant 3$. Moreover, the proof explicitly constructs an infinite family of indecomposable elements with unbounded degrees, thereby settling the second problem simultaneously.