Reduced characteristic number criteria for equivariant bordism of $T^k$- and $(\mathbb{Z}_2)^k$-manifolds with isolated fixed points

Abstract: Classical equivariant bordism theories require computing the full collection of equivariant characteristic numbers to detect whether an equivariant manifold bounds equivariantly or not. This paper establishes simplified equivariant bordism characterizations for two families of equivariant manifolds with isolated fixed points: unitary $T<sup>k$-manifolds and closed smooth $(\mathbb{Z}_2)<sup>k$-manifolds. For any unitary $T<sup>k$-manifold $M$ with isolated fixed points, we establish an equivariant unitary bordism criterion built entirely from a single polynomial of equivariant Chern classes. We further introduce the minimal distinguishing degree and obtain two key inequalities that capture the interplay between $\dim M$ and the Euler characteristic $χ(M)$ through this minimal distinguishing degree. These inequalities settle the existence problem of a linear lower bound for $χ(M)$ within the framework of Kosniowski's conjecture and partially verify the conjecture under natural admissible assumptions. We also provide an alternative proof settling the toric generalization of Kosniowski's conjecture when $\dim M=2k$. By contrast, for a closed smooth $(\mathbb{Z}_2)<sup>k$-manifold with isolated fixed points, we derive a more concise equivariant bordism criterion relying solely on the powers of the top equivariant Stiefel-Whitney class. Our new criteria substantially reduce computational demands.

• The paper introduces reduced criteria using a minimal distinguishing polynomial for T^k-manifolds and powers of top Stiefel–Whitney classes for (Z₂)^k-manifolds.
• It links the minimal distinguishing degree to the Euler characteristic and manifold dimension, providing a quantitative lower bound on the number of fixed points.
• The approach streamlines computational checks in equivariant bordism and advances progress on longstanding conjectures such as Kosniowski’s.

Reduced Characteristic Number Criteria for Equivariant Bordism of $T^k$- and $(\mathbb{Z}_2)^k$-Manifolds with Isolated Fixed Points

Overview

The paper "Reduced characteristic number criteria for equivariant bordism of $T^k$- and $(\mathbb{Z}_2)^k$-manifolds with isolated fixed points" (2607.01889) develops new, simplified criteria for detecting equivariant bordism classes of two prominent families of $G$-manifolds with isolated fixed points: unitary $T^k$-manifolds and closed smooth $(\mathbb{Z}_2)^k$-manifolds. Reducing the standard requirement for the computation of all equivariant characteristic numbers, the authors provide criteria based on minimal data, significantly decreasing the computational complexity for determining equivariant boundaries and yielding new results in the context of longstanding conjectures on the minimal number of fixed points.

Equivariant Bordism and Characteristic Numbers

Equivariant bordism theory, since the foundational work of Conner, Floyd, and tom Dieck, has elucidated the relationship between geometric actions of compact Lie groups on manifolds and the algebraic invariants encoded in characteristic numbers. The objects of study here are:

• Unitary $T^k$-manifolds: smooth, closed, oriented manifolds with an effective $(S^1)^k$-action preserving a stable complex structure and having finitely many isolated fixed points.
• Closed smooth $(\mathbb{Z}_2)^k$-manifolds: unoriented manifolds with effective $(\mathbb{Z}_2)^k$0-actions and isolated fixed points.

In the classical setting, equivariant bordism classes are completely determined by all equivariant characteristic numbers: equivariant cohomology Chern numbers for the complex case and equivariant Stiefel–Whitney numbers for the real case. Localization formulas—ABBV for torus actions and tom Dieck–Kosniowski–Stong for $(\mathbb{Z}_2)^k$1—express these global invariants in terms of data at the fixed points, centralizing the role of combinatorial data from tangent representations around isolated fixed points.

Main Results: Reduction to Minimal Invariants

Complex Case: $(\mathbb{Z}_2)^k$2-Manifolds

Theorem 1 provides a decisive answer to the minimal characterization problem for unitary $(\mathbb{Z}_2)^k$3-manifolds: the equivariant bordism class is determined by the vanishing of a prescribed collection of pairings $(\mathbb{Z}_2)^k$4 for $(\mathbb{Z}_2)^k$5 less than the number of fixed points, where $(\mathbb{Z}_2)^k$6 is an explicit “distinguishing polynomial” in equivariant Chern classes. Unlike previous approaches requiring all Chern numbers, the authors show the existence of such a polynomial—constructed from the elementary symmetric functions of the weights of the tangent representations at fixed points—that distinguishes the classes associated with different fixed-point isotropy types.

A key novelty is the introduction of the minimal distinguishing degree $(\mathbb{Z}_2)^k$7, which encodes the lowest possible degree among all such distinguishing polynomials. This degree links the dimension and Euler characteristic of the manifold through derived inequalities. Notably, the authors settle the existence of a positive linear lower bound for the number of fixed points (Euler characteristic) in relation to dimension, thus making substantive analytical progress on Kosniowski’s conjecture and its toric generalization.

Additionally, sharpness and computability are explored through explicit computations and counterexamples. For example, the failure of the top equivariant Chern class alone to serve as an invariant unless in special circumstances clarifies conceptual distinctions between simply using highest degree classes and the more subtle combinatorial information required to distinguish fixed-point data.

Real Case: $(\mathbb{Z}_2)^k$8-Manifolds

Theorem 2 dramatically simplifies the equivariant bordism criterion for $(\mathbb{Z}_2)^k$9-manifolds with isolated fixed points: it suffices to examine powers of the top equivariant Stiefel–Whitney class alone. This is a significant contrast to the complex case—the structure of $T^k$0-representations, specifically the absence of orientation constraints, results in paired distribution of equivariant Euler classes that permits a concise summary for the bounding criterion.

The effectiveness and geometric reasoning underlying this reduction are elucidated, with the paired structure of equivariant Euler classes identified as fundamental. This aligns with and sharpens classical results of Stong and tom Dieck for the setting of isolated fixed points.

Numerical and Structural Consequences

The authors derive two core inequalities connecting the minimal distinguishing degree, the Euler characteristic, and the dimension:

1. If $T^k$1, then $T^k$2 bounds equivariantly.
2. If $T^k$3 is non-equivariantly bounding but not equivariantly, a sharper bound holds: $T^k$4.

These results yield the following consequences:

• The existence problem for a linear lower bound on the number of fixed points in Kosniowski’s conjecture and its generalizations is resolved within the framework of this minimal degree: $T^k$5.
• In the case $T^k$6, the authors show that $T^k$7 and deduce that $T^k$8 for manifolds not bounding equivariantly, confirming the toric generalization of Kosniowski’s conjecture in the full rank case.
• The technique is extended to lower-rank cases (e.g., $T^k$9), revealing that the minimal distinguishing degree can be explicitly calculated (e.g., degree 12 suffices generically, degree 6 for specific homogeneous spaces), leading to quantitatively refined versions of the conjecture.

Theoretical and Practical Implications

On the theoretical side, this reduction to minimal characteristic numbers not only streamlines the computation of equivariant bordism invariants but also enhances structural understanding of how tangent representation data at fixed points dictates global bordism classes. It highlights sharp distinctions between real and complex equivariant settings, arising from the orientation constraints inherent to $(\mathbb{Z}_2)^k$0 versus $(\mathbb{Z}_2)^k$1 actions.

Practically, the simplified characterization enables direct computational checks in equivariant cobordism problems, with immediate consequences for the construction and classification of toric and semi-toric manifolds, especially those entering toric topology, symplectic geometry, and transformation group theory.

Furthermore, these results set the stage for future avenues:

• Investigation of the minimal distinguishing degree in broader classes of $(\mathbb{Z}_2)^k$2-actions.
• Extension of the methodology to settings with non-isolated fixed point components.
• Deeper exploration of the implications for the structure of equivariant cobordism rings, rigidity, and fixed-point formula generalizations.

Conclusion

The paper accomplishes a significant reduction in the complexity required to determine equivariant bordism classes for $(\mathbb{Z}_2)^k$3- and $(\mathbb{Z}_2)^k$4-manifolds with isolated fixed points. By identifying minimal criteria—via powers of a single distinguishing polynomial in the complex case and powers of the top equivariant Stiefel–Whitney class in the real case—the authors consolidate the link between local representation data at fixed points and global equivariant bordism invariants. This leads to new quantitative and structural results for the classification of such manifolds, advances the resolution of Kosniowski's conjecture and its generalizations, and opens new directions for both the computation and theoretical structure of equivariant bordism.