Equivariant geometric bordisms and universal complexes

Abstract:

This paper establishes a connection between equivariant bordisms and universal complexes, providing a new approach to the study of equivariant bordisms. We show that $\mathcal {Z}_n(\mathbb {Z}_2^n)$ and $\mathcal {Z}_{2n}^U(T^n)$ are isomorphic to the reduced homology groups $\widetilde {H}_{n-1}(X(\mathbb {Z}_2^n);\mathbb {Z}_2)$ and $\widetilde {H}_{n-1}(X(\mathbb {Z}^n);\mathbb {Z})$, respectively. Here, $\mathcal {Z}_n(\mathbb {Z}_2^n)$ (resp. $\mathcal {Z}_{2n}^U(T^n)$) denotes the group formed by the equivariant unoriented bordism classes of all $n$-dimensional 2-torus manifolds (resp. the equivariant unitary bordism classes of all $2n$-dimensional unitary torus manifolds), and $X(\mathbb {Z}_2^n)$ (resp. $X(\mathbb {Z}^n)$) is the universal complex determined by $\mathbb {Z}_2^n$ (resp. $\mathbb {Z}^n$). Furthermore, by using known theorems on the homotopy type of $X(\mathbb {Z}_2^n)$, we determine the dimension of $\mathcal {Z}_n(\mathbb {Z}_2^n)$ as a $\mathbb {Z}_2$-vector space. As an additional application of this connection, utilizing matroid theory, we show that the geometric generators of the equivariant bordism groups $\mathcal {Z}_n(\mathbb {Z}_2^n)$ and $\mathcal {Z}_{2n}^U(T^n)$ can be chosen from small covers and quasitoric manifolds of Davis–Januszkiewicz theory, respectively.