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 $T^k$-manifolds and closed smooth $(\mathbb{Z}_2)^k$-manifolds. For any unitary $T^k$-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 $(\mathbb{Z}_2)^k$-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.