Representation characterization of equivariant geometric bordism

Abstract

Let $G_k=({\mathbb{Z}}_2{)}^k$. We establish a representation characterization determining when finitely many faithful $G_k$-representations can serve as the fixed-point data of a smooth closed $G_k$-manifold. By partitioning representations via multiplicities of nonzero weights $\rho$ and isomorphic restrictions to $\ker \rho$, we formulate an evenness condition; realizability is equivalent to this condition, yielding a finite local restatement of the tom Dieck--Kosniowski--Stong integrality criterion. As an application, we prove that ${\dim }_{{\mathbb{Z}}_2}{\mathcal{Z}}_4(G_3)=32$ and exhibit 32 basis elements represented by $G_3$-actions on $\mathbb{R}{P}^2\times \mathbb{R}{P}^2$, $\mathbb{R}{P}^4$ and $\mathbb{R}P(\gamma \oplus \gamma \oplus \gamma \oplus \underline{\mathbb{R}})$, obtained from three basic actions by precomposition with automorphisms of $G_3$.

31 pages