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 $\mathbb RP^2\times\mathbb RP^2$, $\mathbb RP^4$ and $\mathbb RP(\gamma\oplus\gamma\oplus\gamma\oplus\underline{\mathbb R})$, obtained from three basic actions by precomposition with automorphisms of $G_3$.