不动点集是RP_1(2m+1)∪RP_2(2m+1)∪RP(2)的对合

INVOLUTIONS WITH THE FIXED POINT SET F=RP_1(2m+1)∪RP_2(2m+1)∪RP(2)

本文旨在证明具有非平凡的光滑对合 T的 p维闭流形 Np ,如果对合的不动点集为 F =RP1 ( 2 m +1 )∪RP2 ( 2 m +1 )∪ RP( 2 ) ,其中 2 m2 =1 ,那么该对合必为下面的情况之一 :( 1 )等价于以 RP( 2 )为不动点集的对合 ( Mr,T) .当 r>2 ,且 r≠ 4 ,那么 ( Mr,T)协边于零 ,当 r=4时 ,( Mr,T)协边于 ( RP( 2 )× RP( 2 ) ,twist) ;( 2 )等价于以 RP( 2 m +1 )∪ RP( 2 )为不动点集的对合 ( Mr,T) .当 r>2 m +2时 ,( Mr,T)协边于 ( RP( 2 m +4 ) ,τ2 ) ,其中τ2 [x0 ,x1 ,… ,x2 n+ 4 ]=[-x0 ,-x1 ,-x2 ,x3,… ,x2 n+ 4 ]

In this paper, it is discussed and proved that for any differentiable involution on an p dismensional manifold ( N p,T ) whose fixed point set is F=RP 1(2m+1)∪RP 2(2m+1)∪RP(2) , the equivariant bordism classes of ( N p,T ) must be one of the following two: (1)the equivariant bordism claases of involutions whose fixed point set is RP(2) ; (2)the equivariant bordism classes of involutions whose fixed point set is RP(2m+1)∪RP(2).$$$$