On the homology description of equivariant unoriented bordism groups

TL;DR

This paper constructs a chain complex to analyze equivariant unoriented bordism groups of manifolds with fixed points, deriving a dimension formula that aligns with recent results for specific cases.

Contribution

It introduces a new chain complex based on a double complex from the universal complex, providing a homological description of equivariant unoriented bordism groups.

Findings

Homology of the chain complex is nontrivial only in degree n-2.

Derived a dimension formula for the bordism group as a -vector space.

Confirmed the formula matches recent results for n=3.

Abstract

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 $n=3$.