Equivariant geometric bordisms and universal complexes

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