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

Yuanxin Guan, Zhi Lü

TL;DR

The paper resolves two questions of Mukherjee and Sankaran in the fully effective setting by proving that the fully effective equivariant bordism ring \\mathcal{Z}_*((\\Z_2)^k) is not finitely generated for all k ≥ 3, and by constructing an infinite family of indecomposable elements with unbounded degree. The approach hinges on the Conner–Floyd representation algebra and the injective tangent-representation map \\phi_*, enabling explicit indecomposable generators to be realized as polynomials in the tangent data of fixed points. A key technical step is a decomposition of the non-effective ring \\check{\\mathcal{Z}}_* into direct sums over subgroups, which clarifies the relationship between the effective and non-effective theories. The work also shows GK-dimension constraints, ruling out isomorphism with polynomial algebras in finitely or infinitely many variables, thereby highlighting the rich, constrained algebraic structure of equivariant bordism in higher-rank 2-groups.

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.

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

TL;DR

The paper resolves two questions of Mukherjee and Sankaran in the fully effective setting by proving that the fully effective equivariant bordism ring \\mathcal{Z}_*((\\Z_2)^k) is not finitely generated for all k ≥ 3, and by constructing an infinite family of indecomposable elements with unbounded degree. The approach hinges on the Conner–Floyd representation algebra and the injective tangent-representation map \\phi_*, enabling explicit indecomposable generators to be realized as polynomials in the tangent data of fixed points. A key technical step is a decomposition of the non-effective ring \\check{\\mathcal{Z}}_* into direct sums over subgroups, which clarifies the relationship between the effective and non-effective theories. The work also shows GK-dimension constraints, ruling out isomorphism with polynomial algebras in finitely or infinitely many variables, thereby highlighting the rich, constrained algebraic structure of equivariant bordism in higher-rank 2-groups.

Abstract

In 1998, Mukherjee and Sankaran posed two problems concerning the algebraic structure of the equivariant bordism ring of smooth closed -manifolds with only isolated fixed points. One is the property of being finitely generated as a -algebra, and the other is the existence of indecomposable elements. This paper definitively resolves both problems for the fully effective case. Specifically, let denote the equivariant bordism ring of smooth closed manifolds equipped with fully effective smooth -actions having only isolated fixed points. We prove that is not finitely generated as a -algebra for all . Moreover, the proof explicitly constructs an infinite family of indecomposable elements with unbounded degrees, thereby settling the second problem simultaneously.
Paper Structure (10 sections, 16 theorems, 44 equations)

This paper contains 10 sections, 16 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Tangent representations at fixed points
  3. Relation between and
  4. Proof of the main theorem
  5. Certain -actions on
  6. The case and
  7. The case and
  8. The general case
  9. Proof of the main theorem
  10. Final remark: Gelfand--Kirillov dimension and equivariant bordism ring

Key Result

Proposition 2.1

Suppose that $(\mathbb{Z}_2)^k$ acts on an $n$-dimensional manifold $M$ and $p$ is an isolated fixed point. Then the tangent representation $\tau_pM$ is isomorphic to where $\sum_{\rho\in \operatorname{Hom}((\mathbb{Z}_2)^k, \mathbb{Z}_2)\setminus \{0\}} d(\rho) = n$, and $d(\rho)$ is the dimension of the connected component of the fixed point set $M^{\ker\rho}$ containing $p$.

Theorems & Definitions (27)

  • Proposition 2.1: ConnerFloyd
  • Theorem 2.2: Stong
  • Example 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4
  • Theorem 3.5
  • ...and 17 more