Wei Cong YE, Chang Lian LIU, Deng Pin LIU
For any quasitoric-manifold, $\pi :M^{2n}\to P^{n}$, its cohomology ring is expressed as $H^{\ast}(M^{2n},\mathbb{ Z}) =\mathbb{Z}[F_{1},F_{2},\ldots,F_{m}]/(\mathcal{I}_{P^{n}}+\mathcal{J}_ {P^{n}})$, where $\mathcal{F}(P)=\{F_{1}, F_{2}, \ldots,$ $F_{m}\}$ is the set of all co-one-dimensional surfaces in $P^{n}$. Taking any vertex $\upsilon= F_{i1}\cap F_{i2}\cap\cdots\cap F_{in}$ of $ P^{n}$, we prove that $\langle [F_{i1}F_{i2}\cdots F_{in}],[M^{2n}]\rangle=\pm1$, that is, $[F_{i1}F_{i2}\cdots F_{in}]$ is the generator of $H^{2n}(M^{2n},\mathbb{Z})$. Further we use this conclusion to discuss the rigidity of quasitoric-manifolds, and prove the following conclusions: If $f^{*}:H^{\ast}(M_{1}^{2n},\mathbb{Z})\to H^{\ast}(M_{2}^{2n},\mathbb{Z})$ is a ring isomorphism, then there exists a one-to-one mapping $\tilde{f}:{\rm Fix}(M_{1}^{2n})\to {\rm Fix}(M_{2}^{2n})$, where ${\rm Fix}(M^{2n})$ is the fixed point of $T^{n}$-acting on $M^{2n}$.
