Quasitoric-流形的最高维上同调的生成元的组合描述及其应用  

The Combinatorial Description of the Highest Dimensional Cohomology Generators of Quasitoric-manifolds and Its Applications

作　　者：叶蔚聪 刘昌莲 刘登品 Wei Cong YE;Chang Lian LIU;Deng Pin LIU(School of Mathematics and Statistics,Guangxi Normal University,Guilin 541004,P.R.China)

机构地区：[1]广西师范大学数学与统计学院,桂林541004

出　　处：《数学学报（中文版）》2024年第5期889-894,共6页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学资金项目(11401118)。

摘　　要：对任意quasitoric-流形,π:M^(2n)→P^(n),其上同调环表示为H^(*)(M^(2n),Z)=Z[F_(1),F_(2),...,F_(m)]/(T_(pn)+J_(pn)),其中F(P)={F_(1),F_(2),…,Fm}是P^(n)中所有余一维面的集合.任取P^(n)的顶点v=F_(i1)∩F_(i2)∩…∩Fin,我们证明了<[F_(i1)F_(i2)…F_(in)],[M^(2n)]>=±1,即[F_(i1)F_(i2)…Fin]是H^(2n)(M^(2n),Z)的生成元.我们进一步利用这一结论讨论quasitoric-流形的刚性问题,并证明如下结论:若f^(*):H^(*)(M_(1)^(2n),Z)→H^(*)(M_(2)^(2n),Z)是一个环同构,则存在一一映射f:Fix(M_(1)^(2n))→Fix(M_(2)^(2n)),这里Fix(M^(2n))是T^(n)-作用在M^(2n)上的不动点.For any quasitoric-manifold,π:M^(2n)→P^(n),its cohomology ring is expressed as H*(M^(2n),Z)=Z[F_(1),F_(2),...,F_(m)]/(T_(pn)+J_(pn)),where F(P)={F_(1),F_(2),...,F_m}is the set of all co-one-dimensional surfaces in P^(n).Taking any vertex v=F_(i1)∩F_(i2)∩…∩F_(in)of P^(n),we prove that<[F_(i1)F_(i2)…F_(in)],[M^(2n)])=±1,that is,[F_(i1)F_(i2)…F_(in)]is the generator of H^(2n)(M^(2n),Z).Further we use this conclusion to discuss the rigidity of quasitoric-manifolds,and prove the following conclusions:If f^(*):H^(*)(M_(1)^(2n),Z)→H~*(M_(2)^(2n),Z)is a ring isomorphism,then there exists a one-to-one mapping f:Fix(M_(1)^(2n))→Fix(M_(2)^(2n)),where Fix(M^(2n))is the fixed point of T~n-acting on M^(2n).

关 键 词：Quasitoric-流形 示性函数 上同调环 

分 类 号：O186.1[理学—数学]