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

Abstract

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.

Keywords

• algebraic topology; involution; fixed point set; characteristic class; cobordism class