Abstract:

Let (M,T) be a smooth closed manifold with a smooth involution T whose fixed point set is F={x(|T(x)=x,) x∈M}, then Fis the disjoint union of smooth closed submanifold of M. It has been proved that whenF=P(2^m,2^m)∪P(2^m,2^m+1)(m≥3), (M,T) doesn’t exist except in the following two cases:  (1) w(λ₁)=(1+a+b)₂^m+2, w(λ₂)=(1+c+d)₂^m+1; (2) w(λ₁)=(1+a)(1+a+b), w(λ₂)=1+c+d, 其中: λ→F=λ₁→P(2^m,2^m)∪λ2→P(2^m,2^m+1) is the normal bundle to F  in M, and λ→F is not bordant to λa∈H₁(P(2^m,2^m);Z₂), b∈H₂(P(2^m,2^m);Z₂), c∈H₁(P(2^m,2^m+1);Z₂), d∈H₂(P(2^m,2^m+1);Z₂) are the generators.

Key words: involution, fixed point set, characteristic class, cobordism class