Homological Description of Equivariant Geometric Bordism

Abstract

Using the evenness criterion in \cite{LCLS} and duality principle in \cite{CLT}, we construct a finite chain complex ${\mathfrak{B}}_{\bullet }(({\mathbb{Z}}_2{)}^k)$ built from the universal complex $X(({\mathbb{Z}}_2{)}^k)$ and its links. We then show that the equivariant unoriented bordism group ${\mathcal{Z}}_{k+1}(({\mathbb{Z}}_2{)}^k)$ of all $(k+1)$-dimensional smooth closed connected manifolds with effective $({\mathbb{Z}}_2{)}^k$-actions fixing isolated points, is naturally isomorphic to $H_{k-2}({\mathfrak{B}}_{\bullet }(({\mathbb{Z}}_2{)}^k);{\mathbb{Z}}_2)$. The vertical spectral sequence associated with the natural double-complex structure on ${\mathfrak{B}}_{\bullet }(({\mathbb{Z}}_2{)}^k)$ collapses at $E^3$, yielding an explicit formula for ${\dim }_{{\mathbb{Z}}_2}{\mathcal{Z}}_{k+1}(({\mathbb{Z}}_2{)}^k)$ for every $k$.

26 pages, v2 with minor typos corrected