不动点集为Dold流形P(2,15)的对合

Involutions with fixed point set Dold manifold P(2,15)

为了研究不动点集为Dold流形的对合的等变协边分类,针对一个特定的Dold流形F=P(2,15),确定了以F为不动点集的所有带对合的流形(M,T)的等变协边分类。首先,给出了P(2,15)上切丛和法丛的Stiefel-Whitney示性类。其次,根据Kosniowski-Stong定理,构造合适的对称多项式函数,出现矛盾,证明假设错误,对合不存在;或者证明对任意对称多项式函数都满足Kosniowski-Stong定理,说明对合的存在性。最后,得到以P(2,15)为不动点集的对合(M,T)协边。结果表明,存在以F=P(2,15)不动点集的对合,且能够确定对合的等变协边分类。研究结果推广了不动点集为F=P(2,n)(n=1,3,5)的对合的研究结论,丰富了不动点集为Dold流形的对合的等变协边分类问题,也为研究不动点集其他特殊流形的对合提供了借鉴和参考。

In order to study the equivariant cobordism classification of involutions with fixed point set Dold manifolds,for a special Dold manifold F=P(2,15),the equivariant cobordism classification of all involutions(M,T)with fixed point set was determined.Firstly,the Stiefel-Whitney classes of tangent bundle and normal bundle over P(2,15)were given.Secondly,according to Kosniowski-Stong Theorem,either the contradiction was obtained by constructing a suitable symmetric polynomial function to prove that the hypothesis was wrong and the involution did not exist,or that arbitrary symmetric polynomial functions satisfy Kosniowski-Stong Theorem was proved,which illustrate the existence of involution.Finally,every involution(M,T)with fixed point set P(2,15)bounds was obtained.The results show that there exists an involution with fixed point set P(2,15),and the equivariant cobordism classification of all involutions can be determined.The research results popularize the conclusion of studying involutions fixing F=P(2,n)(n=1,3,5),enrich the equivariant cobordism classification of involutions with fixed point set Dold manifolds,and provide some reference for researching involutions of fixed point set with other special manifold.