On the homology description of equivariant unoriented bordism groups

We construct a chain complex $\mathfrak{B}$ based on a double complex derived from the universal complex $X({\mathbb{Z}}_2^n)$. It is shown that $\mathfrak{B}$ has a nontrivial homology only in degree $n-2$, which is isomorphic to the equivariant unoriented bordism group ${\mathcal{Z}}_{n+1}({\mathbb{Z}}_2^n)$ of all $(n+1)$-dimensional smooth closed ${\mathbb{Z}}_2^n$-manifolds with isolated fixed points. By analyzing the spectral sequence of $\mathfrak{B}$, we derive a dimension formula for ${\mathcal{Z}}_{n+1}({\mathbb{Z}}_2^n)$ as a ${\mathbb{Z}}_2$-vector space, which agrees with a recent result for $n=3$.